3.5 Decimal Numbers Objectives: Name Decimal Numbers Write Decimal Numbers Convert Decimals to Fractions Locate Decimal Numbers on the Number Line Order Decimal Numbers Round Decimal Numbers
Name Decimal Numbers If you have spent money in the United States, you already know quite a bit about decimal numbers. Suppose you buy a sandwich and bottle of water for lunch. The sandwich costs $3.45 and the bottle of water costs $1.25. The tax on your lunch is $.33. What is the total cost of your lunch? $3.45 Sandwich Picture of sandwich and water bottle $1.25 Water + $ .33 Tax $5.03 Total
You pay with a $5 bill and 3 pennies. Do you wait for change? No, $5 and 3 pennies is the same as $5.03 . How many pennies does it take to equal the value of one dollar? 100 pennies = $1. So each 1 1 of one dollar. We write the value of one penny as $.01. .01 = . penny is 100 100 Writing numbers in decimal notation is a way of showing parts of a whole, very much like fraction notation. In decimals, the number of parts of the whole is always a power of 10. Our system of counting numbers is based on the number 10. 1 .1 = 10 1 1 10 = 10 .01 = 100 100 = 10 ⋅ 10 1 .001 = 1000 = 10 ⋅ 10 ⋅ 10 1000 10,000 = 10 ⋅ 10 ⋅ 10 ⋅ 10 1 .0001 = 10,000 The names of the decimal places correspond to their fraction values. .1 is “one tenth” .01 is “one hundredth” Chart of place value .001 is “one thousandth” .0001 is “one ten-thousandth”
Remember that $5.03 lunch? How do you say $5.03? “Five dollars and three cents.” Naming decimal numbers (those that don’t represent money!) is done in a similar way. When we talk about the number 5.03, we say “five and three hundredths.” 3.5 Chapter 3b manuscript July 09.doc
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To name a decimal number: Name the number to the left of the decimal Write ”and” for the decimal Name the “number” part to the right of the decimal as if it were a whole number and then name the decimal place of the last digit
Example 1: Naming Decimal Numbers Name the decimal number: .3 Solution: Name the number to the left of the decimal - nothing to name Name the number to the right of the decimal Name the decimal place of the last digit The 3 is in the tenths place Quick Check
.3 There is no number to the left of the decimal. three three tenths
Name the decimal number:
1) .7
2) .08
Sometimes you’ll see .3 written as 0.3. You still call it ‘three tenths.’ When the number before the decimal is zero, it is not included in the name. Example 2: Naming Decimal Numbers Name the decimal number: .009 Solution: Name the number to the left of the decimal - nothing to name Name the number to the right of the decimal Name the decimal place of the last digit-The 9 is in the thousandths place
Quick Check
.009 There is no number to the left of the decimal. nine nine thousandths
Name the decimal number:
3) .005
4) .018
Example 3: Naming Decimal Numbers Name the decimal number: 2.45 Solution: Name the number to the left of the decimal Write ”and” for the decimal Name the number to the right of the decimal The 5 is in the hundredths place.
Quick Check
2.45 two two and two and forty-five two and forty-five hundredths.
Name the decimal number:
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5) 3.57
6) 19.58
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Example 4: Naming Decimal Numbers Name -15.571 Solution:
Name the number to the left of the decimal Write ”and” for the decimal Name the number to the right of the decimal The 1 is in the thousandths place
Quick Check
-15.571 negative fifteen negative fifteen and negative fifteen and five hundred seventy-one__ negative fifteen and five hundred seventy-one thousandths
Name the decimal number:
7) -13.461
8) -2.053
Write Decimal Numbers When you write a check you write both the numerals and the name of the number; someone at the bank looks to make sure these match. Picture of check
Let’s see how to write the decimal number when you are given the name. To write a decimal number: Look for the word ‘and’—it locates the decimal point. Translate the words before ‘and’ into the whole number to the left of the decimal point. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place Fill in zeroes for place holders as needed
Example 5: Writing Decimal Numbers Write “fourteen and thirty-seven hundredths” as a decimal number Solution:
fourteen and thirty-seven hundredths When given the words for a decimal number, look for the word ‘and’-it locates the decimal point Translate the words before ‘and’ Mark two decimal places for hundredths Translate the words after ‘and’. Write the number 37 in the places putting the 7 in the hundredths place
fourteen and thirty-seven hundredths 14 14 ___ ___ tenths hundredths
14 3
7
tenths hundredths
14.37
Quick Check Write as a decimal number: 9) thirteen and sixty-eight hundredths 10) one and four hundredths
Example 6: Writing Decimal Numbers Write “four thousandths” as a decimal number 3.5 Chapter 3b manuscript July 09.doc
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Solution:
Look for the word “and” Put three decimal places for thousandths
four thousandths There is no “and’ so there is no whole number ___ ___ ___ tenths hundredths thousandths
Put the number 4 in the thousandths place
___
Put zeroes as placeholders in the remaining decimal places.
___
4
0
4
tenths hundredths thousandths
0
tenths hundredths thousandths
.004
Quick Check Write as a decimal number: 11) eight thousandths 12) one hundred seven thousandths
Think about money again. You know that $1 is the same as $1.00. The way you write it depends on the context. In the same way, all integers can be written as decimal numbers with as many zeros as needed. 5 = 5.0 - 2 = -2.0 5 = 5.00 - 2 = -2.00 5 = 5.000 - 2 = -2.000 and so on….
Convert Decimals to Fractions 3 . We convert decimal 100 numbers into fractions by using the fraction that corresponds to the place value of the last (farthest right) digit. Notice that in the decimal number .03 the 3 is in the hundredths place, and 3 100 is the denominator of the equivalent fraction. .03 = . So the decimal number 5.03 is 100 3 Picture of sandwich and water . equivalent to the mixed number 5 bottle-part gone 100
Back to lunch now. $5.03 means 5 dollars and 3 pennies. .03 means
Convert a Decimal to a Fraction or Mixed Number Write the whole number, if there is a number to the left of the decimal. Determine the place value of the final digit Write a fraction o numerator—the ‘numbers’ to the right of the decimal the decimal point o denominator—the place value corresponding to the final digit Example 7: Converting a Decimal to a Fraction Write .09 as a fraction Solution:
.09 There is no number to the left of the decimal Determine the place value of the final digit
. 0
9
tenths hundredths
Write the fraction numerator is 9- the number to the right of the decimal denominator is 100 the place value of 9 100
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Quick Check
Write as a fraction:
14) -.03
13) .07
Example 8: Converting a Decimal to a Fraction Write .374 as a fraction Solution:
.374 There is no number to the left of the decimal Determine the place value of the final digit
. 3
7
4
tenths hundredths thousandths
Write the fractionnumerator is 374- the number to the right of the decimal denominator is 1000 the place value of 374 1000
the final digit 4 Quick Check
Write as a fraction:
15) -.231
.374 =
374 1000
16) .027
Example 9: Converting a Decimal to a Fraction Write 3.7 as a mixed number Solution:
Write the whole number part Determine the place value of the final digit
3.7 3 _____ 3. 7
Write the fraction
3
tenths
7 10
The mixed number 3 Quick Check
Convert to a fraction:
3.7 = 3
7 10
7 is equivalent to the decimal 3.7. 10
17) 5.3
18) 11.69
Locate Decimal Numbers on the Number Line Since decimal numbers are forms of fractions, locating decimal numbers on the number line is similar to locating fractions on the number line. Example 10: Locating a Decimal Number on the Number Line Locate 0.4 on the number line Solution: Activity Worksheet available
4 , a proper fraction, so it is located between 0 and 1 10 We divide the interval between 0 and 1 into 10 equal parts Label these 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 We can write 0 as 0.0 and 1 as 1.0, so that the numbers are consistently in tenths. Mark 0.4
0.4 is equivalent to
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| | | | | | | | | | ·| ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 0.8
Quick Check
Locate on the number line:
19) 0.6
20) 0.9
Example 11: Locating a Decimal Number on the Number Line Locate -0.8 on the number line Solution: -0.8 is equivalent to -
8 , so it is located between 0 and -1 10
Mark off and label the tenths in the interval between 0 and -1 . ·| | | | | | | | | | | ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 .0
Quick Check
Locate on the number line:
21) -0.6
22) -0.7
Order Decimal Numbers Which is larger, .04 or .40? If you think of this as money, you know that $.40 is greater than $.04. So, .40 .04 Remember, you can use the number line to order numbers. a b ‘ a is less than b ’ when a is to the left of b on the number line a b ‘ a is greater than b ’ when a is to the right of b on the number line Where are .04 and .40 located on the number line? .04
| · | | | | | | | | | ·| ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 0.8
We see that .40 is to the right of .04 on the number line. So, again, .40 .04 . How does .31 compare to .308? If we convert them to fractions we can tell which is larger. .31
.308
31 100
308 1000
You need a common denominator to compare them.
3110 10010 310 1000
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310 308 1000 1000
so .31> .308
Notice what we did in converting .31 to a fraction – we started with the fraction
31 and ended 100
310 310 . Converting back to a decimal number gives .310. So 1000 1000 .31 is equivalent to .310. Writing zeroes at the end of decimal numbers does not change their value! 31 310 .31 .310 100 1000
with the equivalent fraction
.31 = .310
Equivalent Decimal Numbers Two decimal numbers are equivalent if they convert to equivalent fractions.
Writing zeroes at the end of a decimal number does not change its value.
So let’s look again at comparing .308 to .31. It is helpful to place one number above the other. To compare .308 and .31, write one number under the other, lining up the decimals. .308 .31 It is easiest to compare the two numbers if they have the same number of decimal places. Since these numbers do not have the same number of decimal places, add zeros to even out the number of decimal places. (We showed above why this is correct mathematically.) .308 .310 Now they are both thousandths. How does 308 compare to 310? Since 308 is less than 310, we see that 308 thousandths is less than 310 thousandths. So.308 is less than .310 and .308 to order .9 __.5 Solution:
.9 __.5 Write the numbers one under the other, lining up the decimals
.9 .5
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Since 9 is larger than 5, 9 tenths is larger than 5 tenths
.9 > .5
Quick Check: Order each of the following pairs of numbers, using < or >. 23) .2 ___ .3 24) .4 ___ .7
Example 13: Ordering Decimal Numbers Use < or > to order .64 __.7
.64 __.7
Solution:
Write the numbers one under the other, lining up the decimals
.64 .7
They do not have the same number of digits. .64 .70
Write one zero at the end of .7 Since 64. 25) .42 ___.5 26) .18 ___ .2 Example 14: Ordering Decimal Numbers Use < or > to order .83 ___ .803 Solution:
.83 ___ .803 Write the numbers one under the other, lining up the decimals
.83 .803 .830
They do not have the same number of digits.
.803
Write one zero at the end of .83 Since 830 >803, 830 thousandths is greater than 803 thousandths
.830 > .803 .83 > .803
Quick Check: Order each of the following pairs of numbers, using < or >. 27) .76 ___ .706 28) .305 __.35
When ordering negative decimal numbers, remember how to order negative integers. Larger numbers are to the right on the number line 2 3
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Smaller numbers are to the left on the number line 9 6 | | | | | | | | | | | ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ -10 0 -9 -8 -7 -6 -5 -4 -3 -2 -1
In the same way .2 .3 and .9 .6 | | | | | | | | | | | ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 .0
Example 15: Ordering Decimal Numbers Use < or > to order -.1__- .8 Solution: -.1__- .8
Write the numbers one under the other, lining up the decimals They have the same number of digits. Since -1 > -8 , 1 tenths is greater than
8 tenths
- .1 - .8
-.1 > -.8
Quick Check: Order each of the following pairs of numbers, using < or >. 30) -.6 ___- .7 29) -.3 ___- .5
Example 16: Ordering Decimal Numbers Use < or > to order -.915 ___- .9501 Solution: -.915 ___- .9501
Write the numbers one under the other, lining up the decimals Write one zero at the end of
- .915 - .9501
.915 - .9150 - .9501
Since
-.9150 < -.9501 -.915 < -.9501
9150< 9501,
Quick Check: Order each of the following pairs of numbers, using < or >. 32) -.5041___- .541 31) -.832 ___- .8023
Round Decimal Numbers Is $2.72 closer to $2.70 or to $2.80? Is it closer to $2 or to $3? Knowing how to round decimal numbers will help you answer these questions. You round numbers to get approximate values when the exact values are not needed.
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Let’s look at the number line to answer the questions above. The first question ’is $2.72 closer to $2.70 or to $2.80?’, asks us to round 2.72 to the nearest tenth. The number 7 is in the tenths place. 2.72 is closer to 2.70 than it is to 2.80. So, rounding 2.72 to the nearest tenth would give 2.7. Let’s consider the second question now. Is $2.72 closer to $2 or to $3? We are asked to round 2.72 to the nearest whole number. 2.72 is closer to 3.00 than to 2.00. Rounding 2.72 to the nearest whole number gives 3. Can you do this without the number lines? We will use a method similar to the one we used to round whole numbers
Round Whole Numbers To round a whole number to a given place value: Locate the given place value with an arrow Underline the digit to the right of the given place value o Is this digit greater than or equal to 5? Add 1 to the digit in the given place value o Is this digit less than 5? Do not add 1 Rewrite the number, deleting all digits to the right of the rounding digit
Example 17: Rounding Decimal Numbers Round 18.359 to the nearest a) hundredth
b) tenth
c) whole number
Solution: Round 18.359 a) to the nearest hundredth hundredths place
Locate the given place value with an arrow
18.35 9 hundredths place
Underline the digit to the right of the given place value Since 9 is greater than or equal to 5, add 1 to the 5
18.35 9 18.35 9
add1 delete
Rewrite the number, deleting all digits to the right of the rounding digit 18.36 18.36 is 18.359 rounded to the nearest hundredth b) to the nearest tenth tenths place
Locate the given place value with an arrow
18.3 5 9 tenths place
Underline the digit to the right of the given place value Since 5 is greater than or equal to 5, add 1 to the 3
18.3 59
18.359
add1
delete
Rewrite the number, deleting all digits to the right of the rounding digit 18.4 18.4 is 18.359 rounded to the nearest tenth
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c) to the nearest whole number ones place
Locate the given place value with an arrow
18.3 59 ones place
Underline the digit to the right of the given place value Since 3 is not greater than or equal to 5, do not add 1 to the 8
18.3 59
18.3 . 59
do not add1 delete
Rewrite the number, deleting all digits to the right of the rounding digit 18 18 is 18.359 rounded to the nearest whole number Quick Check 33) Round 6.582 to the nearest 34) Round 15.2175 to the nearest
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b) tenth b) hundredth
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3.5 Exercises Writing Exercises
How does knowing about US money help you learn about decimal numbers? Explain how you write “three and nine hundredths” as a decimal number. Jim ran a 100-meter race in 12.32 seconds. Tim ran the same race in 12.3 seconds. Who had the fastest time, Jim or Tim? How do you know? Gerry saw postcards marked for sale at .40¢. What is wrong with the marked price?
Practice Makes Perfect Name Decimal Numbers Name each decimal number 1) 0.5 2) 0.8
3) 0.71
4) 0.64
5) 0.002
6) 0.005
7) 0.381
8) 0.479
9) 5.01
10) 14.02
11) -17.9
12) -31.4
Write Decimal Numbers Write as a decimal number
13) Eight and three hundredths
14) Nine and seven hundredths
15) Twenty-nine and eighty-one hundredths
16) Sixty-one and seventy-four hundredths
17) One thousandth
18) Nine thousandths
19) Twenty-nine thousandths
20) Thirty-five thousandths
21) Seven tenths
22) Six tenths
23) Negative eleven and nine ten-thousandths
24) Negative fifty-nine and two ten-thousandths
Convert Decimals to Fractions Convert each decimal number to a fraction
25) 0.13
26) 0.19
27) 0.7
28) 0.1
29) 0.239
30) 0.373
31) 0.011
32) 0.049
33) 1.99
34) 5.83
35) 0.007
36) 0.003
37) 0.25
38) 0.75
39) 0.006
40) 0.008
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41) 6.4
42) 5.2
43) 0.125
44) 0.375
45) 0.55
46) 0.45
47) 1.324
48) 2.482
Locate Decimal Numbers on the Number Line Locate on the number line.
49) 0.8
50) 0.3
51) -.2
52) -0.9
53) 3.1
54) 2.7
55) -2.5
56) -1.6
Order Decimal Numbers Order each of the following pairs of numbers, using < or >. 57) .9__ .6 58) .7__ .8 59) .37__ .63
60) .86__ .69
61) .6__.59 65)
.5__ .3
62) .27__.3
63) .91__.901
64) .415__.41
66) −.1__ −.4
67)
68)
Round Decimal Numbers Round each number to the nearest tenth. 69) .67 70) .49
.62__
.619
71) 2.68
Round each number to the nearest hundredth. 73) .845 74) .761 75) 5.7962 77) .299 78) .697 79) 4.098 Round each number to the nearest 81) 5.781 82)1.638
a) hundredth b) tenth 83) 63.479
7.31__ 7.3
72) 4.63 76) 3.6284 80) 7.096 c) whole number 84) 84.281
Every Day Math 85) Danny got a raise and he now makes $58,965 a year. Round this number to the nearest thousand and then ten thousand. 86) Selena’s new car cost $23,795. Round this number to the nearest thousand and then ten thousand.
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3.5 Answers Quick Check 1) seven tenths
2) eight hundredths
5) three and fifty-seven hundredths
3) five thousandths
4) eighteen thousandths
6) nineteen and fifty-eight hundredths
7) negative thirteen and four hundred sixty-one thousandths 8) negative two and fifty-three thousandths 9) 13.68 13)
10) 1.04
7 100
17) 5
3 10
14) -
11) .008
3 100
15) -
12) .107
231 1000
16)
27 1000
69 18) 11 100
19, 20) 19
20
| | | | | | | ·| | ·| | ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 21,22) 22
21
| | | | | | | | | ·| ·| ¬¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ .0 -1 -. 9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1
Practice Makes Perfect 1) five tenths
2) eight tenths
3) seventy-one hundredths
4) sixty-four hundredths
5) two thousandths
6) five thousandths
7) three hundred eighty-one thousandths
8) four hundred seventy-nine thousandths
9) five and one hundredth
10) fourteen and two hundredths
11) negative seventeen and nine tenths
12) negative thirty-one and four tenths
13) 8.03
14) 9.07
15) 29.81
16) 61.74
17) 0.001
18) 0.009
19) 0.029
20) 0.035
21) 0.7
22) 0.6
23) -11.0009
24) -59.0002
25)
13 100
26)
19 100
27)
31)
11 1000
32)
49 1000
99 33) 1 100
37)
1 4
38)
3 4
39)
3 500
40)
1 125
41) 6
43)
1 8
44)
3 8
45)
11 20
46)
9 20
47) 1
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7 10
28)
1 10
34) 5
287
83 100
29)
239 1000
30)
373 1000
35)
7 1000
36)
3 1000
2 5
42) 5
1 5
81 250
48) 2
241 500
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49-56)
55
56
52
51
50
49
54
53
· ¾¾ · ·¾| ¾ ¾¾| ¾ ¾ ¾¾ · · | ¾ ¾¾| ¾ ¾ ·¾¾| ¾ ¾ | · ¾ ¾· ¾ | ¾¾ |¾ ¬¾ 0 1 2 3 -4 -3 -2 -1
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69) .7
70) .5
71) 2.7
72) 4.6
73) .85
74) .76
75) 5.80
76) 3.63
77) .30
78) .70
79) 4.10
80) 7.10
81a) 5.78
81b) 5.8
81c) 6
82a) 1.64
82b) 1.6
82c) 2
83a) 63.48
83b) 63.5
83c) 63
84a) 84.28
84b) 84.3
84c) 84
85) $59,000; $60,000
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3.6 Decimal Numbers Objectives: Add and Subtract Decimal Numbers Multiply Decimal Numbers Divide Decimal Numbers Convert Fractions to Decimals Simplify Expressions using the Order of Operations Find the Mean of a Set of Numbers Find the Median of a Set of Numbers Use Decimal Numbers in Applications Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
Add and Subtract Decimal Numbers Let’s look at the lunch order from the start of section 3.4 one more time, this time noticing how the numbers were added together. $3.45 Sandwich Picture of sandwich and $1.25 Water water bottle mostly gone $ .33 Tax $5.03 Total All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them, too. Then we just added each column, as if we were adding whole numbers. By lining up decimal numbers this way, we can add or subtract the corresponding place values.
Add or Subtract Decimal Numbers Write the numbers so the decimal points line up vertically Use zeros as place holders, as needed Add or subtract the numbers as if they were whole numbers---temporarily ignoring any decimal point The decimal in the answer lines up with the decimal points in the given numbers
Example 1: Adding and Subtracting Decimal Numbers Add 3.7 12.4 Solution: 3.7 12.4
3.7 + 12.4
Write the numbers so the decimals line up vertically
16.1
Quick Check
Add:
1) 5.7 11.9
2) 18.32 14.79
Example 2: Adding and Subtracting Decimal Numbers Add 23.5 41.38
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Solution:
23.5 41.38 23.5
Remember: 510
41.38
Write the numbers so the decimals line up vertically
1010 50
23.50
100
41.38
Put a 0 as a placeholder after the 5 in 23.5
So,
We can do this because .5 = .50 5 50 (in fractions, ) 10 100 Now that the place values line up, add the numbers as if they were whole numbers. Then place the decimal in the sum. Quick Check
Add:
3) 4.8 11.69
5
50 10 100 .5 .50
23.50 41.38 64.88
4) 5.123 18.47
Example 3: Adding and Subtracting Decimal Numbers Subtract 20 14.65 Solution: 20 14.65
Write the numbers so the decimals line up vertically Remember 20 is a whole number, so place the decimal 20.
after the 0
14.65 20.00
Put in zeroes as placeholders
14.65 20.00
14.65
Subtract:
5.35
Quick Check Subtract: 5) 10 9.58
6) 50 37.42
Example 4: Adding and Subtracting Decimal Numbers Subtract 2.51 7.4 Solution: If we subtract 7.4 from 2.51 , the answer will be negative, since 7.4 2.51 . To subtract easily, we place the 7.4 on top and subtract the 2.51 2.51 7.4
Write the numbers so the decimals line up vertically
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7.40
Put in zeros as placeholders
2.51 7.40
Subtract
2.51
Remember the answer is negative.
4.89 2.51 7.4 4.89
Quick Check Subtract: 7) 4.77 6.3
8) 8.12 11.7
Multiply Decimal Numbers The rules for multiplying decimal numbers will make sense if we first review multiplying fractions. Remember? To multiply fractions, you multiply the numerators and then multiply the denominators. So let’s see what we’d expect for the product of decimals by converting them to fractions. (0.3)(0.7) (0.2)(0.46) Convert to fractions Multiply Convert to decimals
3 7 10 10 21 100 0. 21
2 46 10 100 92 1000 . 092
2places
3places
Do you see a pattern? In the first example, we multiplied two numbers that each had one decimal place and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places. How many decimal places would you expect for the product of (0.01)(0.004)? If you said “five”, you understand the pattern! Add up all the decimal places in the factors – two plus three – to get five decimal places in the answer. (0.01)(0.004 ) 0.00004 2places 3places
5places
4 1 4 100 1000 100,000
Multiply Decimal Numbers Line up numbers on the right Multiply the numbers as if they were whole numbers-temporarily ignore the decimal. The number of decimal places in the product is the sum of the decimal places in the factors
Example 5: Multiplying Decimal Numbers Multiply 3.9 4.075
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Solution: (3. 9)(4.075)
3places
1place
4.075
Line up numbers on the right
3.9 4.075 3.9
Multiply as if there were no decimals
36675 12225 158925
Count the decimal places in the factors (3 and 1) 4.075 3.9
Place the decimal point 4 places from the right
36675 12225
.
15 8925
4places
Quick Check
Multiply: 9) 4.5 6.107
10) 10.79 8.12
The rules for multiplying positive and negative numbers apply to decimal numbers, too. When multiplying two numbers, if their signs are the same the product is positive if their signs are different the product is negative So when you multiply signed decimal numbers, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign. Example 6: Multiplying Decimal Numbers Multiply 8.2 5.19 Solution: 8. 2 5.19 1place 2 places The product is negative.
The signs are different. Line up the numbers on the right.
5.19 8.2
Multiply as if they were whole numbers
1038 4152 42558
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The number of decimal places in the product is the sum of the decimal places in the factors. Place the decimal point 3 places from the right. Remember, the product is negative. Quick Check
1 place 2 places
42.558 42.558
Multiply: 11) 4.63 2.9
12) 7.78 4.9
Divide Decimal Numbers To understand decimal division, let’s look at this multiplication problem: (0.2)(4) 0.8 So, how many 0.2’s are there in 0.8? There are 4 of them. This means that 0.8 divided by 0.2 is 4.
.
.
.
.
2 2 2 2
| | | | | | 0
0.2
0.4
0.6
1
0.8
0.8 0.2 4 You’d get the same answer if you divide 8 2 , both whole numbers! Why is this so? 0.8 0.2 0.8 10
0.2 10 8 2
To divide decimal numbers we use the equivalent fractions property to make the denominator a whole number. The effect is to move the decimals in the numerator and denominator the same number of places to the right.
Divide Decimal Numbers Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal in the dividend the same number of places-adding zeroes as needed Divide Place the decimal in the quotient above the decimal in the dividend
Example 7: Dividing Decimal Numbers Divide 2.89 3.4 Solution:
2.89 3.4
The signs are the same
The quotient is positive
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.85 3.4 2.890 2 72 170 170
Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal in the dividend the same number of places-adding zeroes as needed Divide
Write the quotient with the appropriate sign. Quick Check
The quotient is .85
Divide: 13) 1.989 5.1
14) 2.04 5.1
Example 8: Dividing Decimal Numbers Divide 25.68 1.07 25.68 1.07
Solution:
The signs are different
.
Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal in the dividend the same number of places-adding zeroes as needed Divide
The quotient is negative. 24 1.07 25.68 21 4 428 428
Write the quotient with the appropriate sign. Quick Check Divide:
24
15) 48.3 1.15
16) 85.5 2.25
Example 9: Dividing Decimal Numbers Divide 25.65 .06 Solution:
25.65 .06
The signs are the same
The quotient is positive
Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal in the dividend the same number of places-adding zeroes as needed Divide
427.5 .06 25.650 24 16 12 45 42 30 30
427.5
Write the quotient with the appropriate sign.
Quick Check
Divide:
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The rules for dividing positive and negative numbers also apply to decimal numbers, of course. When dividing signed decimal numbers, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
Convert Fractions to Decimals Now that we can convert decimals to fractions, we also need to be able to do the reverse— convert fractions to decimals. Remember that the fraction bar is actually an operation symbol 4 telling us to divide. So means 4 5 , so take the number 4 and divide it by 5. When you do 5 .8 Notice we needed to write 4 as 4.0 in 5 4.0 this division, you get the answer 0.8. order to do the division! 40 Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator of the fraction
Example 10: Converting a Fraction to a Decimal Number 3 Write as a decimal number 4 Solution: 3 4
Divide 3 by 4
.75 4 3.00 28 20 20
.75
3 .75 4
Quick Check Write as a decimal number:
19)
2 5
20)
3 5
Example 11: Converting a Fraction to a Decimal Number 11 as a decimal number Write 20 Solution:
11 20 Divide 11 by 20
.55
.55 20 11.00 100 100 100
11 .55 20
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Quick Check Write as a decimal number:
21)
13 20
22)
23 50
Example 12: Converting a Fraction to a Decimal Number 7 Write as a decimal number 8 Solution:
Divide 7 by 8
7 8 .875 8 7.0 00 64 60 56 40 40
.875
7 .875 8
Quick Check Write as a decimal number:
23)
5 8
24)
5 16
Example 13: Converting a Fraction to a Decimal Number 7 Write as a decimal number 2 Solution: 7 2
Divide 7 by 2
3.5 2 7.0 6 10 10
3.5
7 3.5 2 7 is an improper fraction; its value is greater than 1. And see that the equivalent 2 decimal number is also greater than 1.
Notice that
Quick Check Write as a decimal number:
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9 4
26)
11 2
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So far, in all the examples converting fractions to decimals the division ended with no remainder. 4 This is not always the case. Let’s see what happens when we try to convert the fraction to a 3 decimal number. We need to divide 4 by 3. 1.333......... 3 4.000 3 10 9 10 9 10 9 1
No matter how many more zeroes we add, we’ll always have a remainder of 1. Those threes in the quotient will go on forever! 1.333….. is called a repeating decimal. Repeating Decimal Number A repeating decimal number is a decimal number in which the last digit or group of digits repeats endlessly.
How do you know how many ‘repeats’ to write? We use a shorthand notation – we place a line over the digits that repeat. The repeating decimal number 1.333…. is written 1.3 . The line above the 3 tells you that the 3 repeats endlessly. 1.333..... 1.3 Here are some more examples of the notation used for repeating decimal numbers: 4.1666... 4.16 only the 6 repeats 4.161616.... 4.16 16 is the repeating block .272727..... .27 9.090909..... 9.09 Example 14: Converting a Fraction to a Decimal Number 11 Write as a decimal number 9 11 9
Solution:
Divide 11 by 9 1.22 9 11.00 9 20 18 20 18 2
This division will continue to have remainder 2 forever!
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Quick Check
Write as a decimal number:
27)
16 3
28)
25 6
Example 15: Converting a Fraction to a Decimal Number 3 Write as a decimal number 22 Solution: 3 22
Divide 3 by 22 .13636 22 3.00000 22 80 66 140 132 80 66 14 3 .136 22
The remainders will keep repeating the same pattern.
Quick Check
Write as a decimal number: 29)
27 11
30)
51 22
Simplify Expressions Using the Order of Operations All the order of operations we have used so far also apply to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?
Example 16: Simplifying Expressions Using the Order of Operations Simplify: 7 21.7 18.3 Solution:
7 21.7 18.3
Subtract
7 3.4
Multiply
23.8
Quick Check Simplify: 31) 8 37.5 14.6
32) . 25 56.74 25.69
Example 17: Simplifying Expressions Using the Order of Operations Simplify: 6 .6 .2 4 .12
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Solution:
6 .6 .2 4 .12 6 .6 .2 4 .01
Simplify exponents first
10 .8 .01 10.8 .01 10.79
Divide and multiply Add Subtract
Quick Check Simplify:
33) 9 .9 .4 3 .22
34) 12 .4 1.5 2 .1 2
Example 18: Simplifying Expressions Using the Order of Operations 3 Simplify: 6.8 4 Solution: First we must change one number so both numbers are in the same form. We can change the fraction to a decimal, or change the decimal to a fraction. It does not matter, but generally we change the one that seems easier. 3 6.8 4 3 Change to a decimal .75 6.8 4 Add 7.55 Quick Check Simplify:
35)
3 4.9 8
36) 5.7
13 20
Find the Mean of a Set of Numbers One application of decimal numbers that arises often is finding the mean of a set of numbers. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size. The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values.
Mean The mean of a set of n numbers is the arithmetic average of the numbers. sum of all the numbers mean = n
Let’s see how to use this formula to find the mean of the numbers 8, 12, 15, 9, and 6. There are 5 numbers in the set, so n 5 . sum of all the numbers mean = n
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8+12+15+9+6 5 50 mean 5 mean 10 The mean is 10.
mean
When you calculate the mean of a set of numbers, you should always check to make sure your answer makes sense. The mean, or average, represents a ‘typical’ number in that set. So it should not be less than the smallest number nor greater than the biggest number. The smallest numbers above is 6 and the largest is 15; 10 is neither less than 6 nor greater than 15.
Find the Mean of a Set of Numbers To find the mean of a set of numbers: sum of all the numbers
Write the formula for the mean:
Count how many numbers are in the set. Call this n and write it in the denominator. Write the sum of the numbers in the numerator. Simplify the fraction. Check to see that the mean is ‘typical’ - neither less than the smallest number nor greater than the largest number in the set.
mean =
n
Example 19: Finding the Mean of a Set of Numbers Kathryn went to tea with her mother-in-law, Felecia, and three of Felecia’s friends. The ages of the women are 53, 89, 91, 93, and 97. Find the mean age of the women. Solution:
Write the formula for the mean
mean =
sum of all the numbers n
Count how many numbers are in the set. Call this n and write it in the denominator
mean =
Write the sum of all the numbersin the numerator
mean =
sum of all the numbers 5 53 89 91 93 97 5
423 mean 5 mean = 84.6 The mean is 84.6
Simplify the fraction
Check:
Is 84.6 ‘typical’? Yes, it is neither less than 53 nor greater than 97.
Quick Check 37) The ages of the four students in Ben’s carpool are 25, 18, 21, and 22. Find the mean age of the students. 38) Yen counted the number of emails she received from Monday through Friday. The numbers were 9, 15, 12, 10, and 8. Find the mean number of emails.
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Did you notice that in the last example, while all the numbers were whole numbers, the mean was 84.6, a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money - it will make sense to report the mean in dollars and cents. Example 20: Finding the Mean of a Set of Numbers For the past 4 months, Daisy’s cell phone bills were $42.75, $50.12, $41.54, $48.15. Find the mean cost of Daisy’s cell phone bills. Solution:
Write the formula for the mean
mean =
sum of all the numbers n
Count how many numbers are in the set. Call this n and write it in the denominator
mean =
Write the sum of all the numbersin the numerator
mean =
sum of all the numbers 4 42.75 50.12 41.54 48.15 4
Simplify the fraction mean =
182.56 4
mean = 45.64 The mean cost of her cell phone bill was $45.64. Check:
Does $45.64 seem ‘typical’ of this set of numbers? Yes, it is neither less than $41.54 nor greater than $50.12.
Quick Check 39) Last week Ray recorded how much he spent for lunch each workday. He spent $6.50, $7.25, $4.90, $5.30, and $12.00. Find the mean. 40) Lisa had the receipts from the past 4 times she filled her car with gas. The receipts showed $34.87, $42.31, $38.04, $43.26. Find the mean.
Example 21: Finding the Mean of a Set of Numbers Each time Pilar filled her car with gas, she recorded the miles per gallon of that tankful. For the past 8 fill-ups the miles per gallon were 28.8, 32.9, 32.5, 27.9, 30.4, 32.5, 31.6, and 32.7. Find the mean. Solution:
Write the formula for the mean
mean =
sum of all the numbers n
Count how many numbers are in the set. Call this n and write it in the denominator
mean =
sum of all the numbers 8
Write the sum of all the numbers in the numerator
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mean =
28.8 32.9 32.5 27.9 30.4 32.5 31.6 32.7
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Simplify the fraction
mean =
249.3 8
mean = 31.1625 The original numbers had 1 decimal place, so round the mean to 2 decimal places. mean = 31.16 The mean is 31.16 miles per gallon. Check: Is 31.16 ‘typical’? Yes, it is neither less than 27.9 nor greater than 32.9.
Quick Check 41) Elisa measured the temperature of 6 patients on her floor of the hospital. The temperatures were 99.2, 101.9, 98.6, 99.5, 100.8, and 99.8. Find the mean. 42) At the end of every workday, Rocky reported how many miles he had driven the company van. Last week the number of miles was 48.58, 42.91, 60.49, 45.22, and 40.35. Find the mean.
Find the Median of a Set of Numbers When Anh, Beyonce, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights are: Anh Beyonce Dora Eve Francine 59” 60” 65” 68” 70” Dora is in the middle of the group. Her height, 65”, is the median of the girls’ heights. Half of the heights are less than or equal to Dora’s height, and half are greater than or equal. The median is the middle value.
59 60 65 68 70 2 below
2 above
Median: The median of a set of numbers listed in numerical order is the middle value. Half the numbers are less than the median Half the numbers are greater than the median
If Carmen, the pianist, joins the girls on stage, she fits, height-wise, between Beyonce and Dora. Anh 59”
Beyonce 60”
Carmen 62”
Dora 65”
Eve 68”
Francine 70”
Now what is the median height? The group of six girls can be divided into two equal parts. 59 60 62
65 68 70
Mathematicians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median height of the girls on stage is the mean of Carmen’s height 62 65 to get 63.5” . and Dora’s height. The mean of 62” and 65” can be found by simplifying 2
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median 59 60 62
63.5
3 below
65 68 70 3 above
The median height of the 6 girls is 63.5”. Notice that when the number of girls was 5, the median was the third height, but when the number of girls was 6, the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.
Find the median of a set of numbers To find the median of a set of numbers: List the numbers in order from smallest to largest. Count how many numbers are in the set. Call this n . n 1 If n is an odd number, evaluate the fraction . The median is the number located in 2 this position. n If n is an even number, evaluate the fraction . The median is the mean of the number 2 in this position and the next number. Example 22: Finding the Median of a Set of Numbers Find the median of the numbers 12, 13, 19, 9, 11, 15, 18 Solution: List the numbers in order from smallest to largest Count how many numbers are in the set. Call this n . n 1 Since 7 is an odd number, evaluate the fraction 2 7 1 2 n 1 2 4
9, 11, 12, 13, 15, 18, 19 n7
median
th
The median is the number located in the 4 position
9, 11, 12, 13 ,15, 18, 19 3 below
3 above
The median is 13. Quick Check Find the median of the numbers 43) 43, 38, 51, 40, 46
44) 15, 35, 20, 45, 50, 25, 30
Example 23: Finding the Median of a Set of Numbers Find the median of the numbers 83, 79, 85, 86, 92, 100, 76, 90, 88, 64 Solution: List the numbers in order from smallest to largest Count how many numbers are in the set. Call this n .
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64, 76, 79, 83, 85, 86, 88, 90, 92, 100 n 10
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n 2 10 2 5 The median is the mean of the 5th and 6th numbers.
Since 10 is an even number, evaluate the fraction
64, 76, 79, 83, 85, 86 , 88, 90, 92, 100 5 numbers
Find the mean of 85 and 86.
5 numbers
85 86 2 mean 85.5
mean
The median is 85.5. Quick Check Find the median of the numbers: 45) 8, 7, 5, 10, 9, 12 46) 21, 25, 19, 17, 22, 18, 20, 24
Use Decimal Numbers in Applications There are many applications of decimal numbers in real life – most of them involving money! If we use our strategy for applications, it will give us a plan for finding the answer!
Strategy for Applications Identify what you are asked to find Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Write a complete sentence that answers the question
Example 24: Using Decimal Numbers in Applications Jessie put 8 gallons of gas in her car. One gallon of gas costs $3.52. How much did Jessie owe for all the gas? Solution: What are you asked to find?
Write a phrase Translate into algebra Simplify Write a sentence
Picture of car being
How much did Jessie owe for all the gas? gassed up! Jessie put 8 gallons of gas in her car. 8 times the cost of one gallon of gas. 8($3.52) $28.16
Jessie owes $28.16 for her gas purchase.
Quick Check 47) Hector put 13 gallons of gas in his car. One gallon of gas costs $3.17. How much did Hector owe for all the gas? 48) Christopher bought 5 pizzas for the team. Each pizza cost $9.75 How much did all the pizzas cost? 3.6 Chapter 3b manuscript July 09.doc
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Example 25: Using Decimal Numbers in Applications Four friends went out for dinner. They shared a large pizza and a pitcher of soda. The total cost of their dinner was $31.76. If they divide the bill equally among themselves, how much should each friend pay? Solution: What are you asked to find? Write a phrase Translate Simplify Write a sentence
How much should each friend pay? $31.76 is divided equally among the four friends. $31.76 4 $7.94 Each friend should pay $7.94 for his share of the dinner.
Quick Check 49) Six friends went out for dinner. The total cost of their dinner was $92.82. If they divide the bill equally among themselves, how much should each friend pay? 50) Chad worked 40 hours last week and his paycheck was $ 570. How much does he earn per hour?
Example 26: Using Decimal Numbers in Applications Marla buys 6 bananas that cost $0.22 each and 4 oranges that cost $0.49 each. How much is the total cost of the fruit? Solution: What are you asked to find? How much is the total cost of the fruit? 6 times the cost of each banana plus 4 times the cost of each orange. 6($0.22) 4($0.49) Translate Simplify $1.32 $1.96 $3.28 Add Write a sentence Marla’s total cost for the fruit is $3.28
Quick Check 51) Suzanne buys 3 cans of beans that cost $0.75 each and 6 cans of corn that cost $0.62 each. How much is the total cost of the groceries? 52) Lydia purchased movie tickets for the family. She bought two adult tickets for $9.50 and four children tickets for $ 6.00. How much did the tickets cost Lydia?
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Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers You learned about counting numbers, whole numbers, and fractions, and located them on the number line. Now you have negative numbers on your number line, too. The whole numbers and their opposites are called the integers. Numbers like 5, 2, 0, 143, and 99 are integers.
Integers: ... 3, 2, 1, 0, 1, 2, 3,...
Take out your number line and put a black square around each integer. Notice that some numbers now have several symbols around them. The number 2, for example, is a counting number, a whole number, and an integer. 5 is just an integer. Show the number line as it is from the activities here. 4 What about fractions? Look at on your number line. It has no symbols around it; it is not a 5 counting number, whole number, or an integer. The set of fractions – positive and negative – is called the set of rational numbers. Rational numbers are numbers that can be written as a ratio of two integers. In precise mathematical terms, we say:
Rational number
A rational number is a number of the form
All fractions, such as
p , where p and q are integers and q o . q
4 7 13 20 are rational numbers. Each numerator and denominator is , , , 5 8 4 3
an integer. What about the integers? Can you write an integer as a fraction? 3 8 0 3 8 =0 1 1 1 So, all integers are also rational numbers! Any number that can be expressed as a fraction of integers is rational. Are there any numbers that are not rational? Yes, but you may not have encountered them very much. The number , which is very important in the study of circles, is not rational. Neither is 2 , (read “the square root of 2”) which is the number that when multiplied by itself gives 2. Numbers that are not rational are called irrational.
Irrational number
An irrational number is one that is not rational. Another way to decide if a number is rational or irrational is to look at its decimal representation. Every integer is a rational number and we could write them as decimals by simply adding a decimal point and a zero.
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....., - 2,
- 1, 0, 1, 2, 3, ...... ....., - 2.0, - 1.0, 0.0, 1.0, 2.0, 3.0, .....
These decimals stop.
Look at the decimal form of the fractions listed above. (How do you find the decimal form?) 4 , 5
7 , 8
13 , 4
.875
3.25
.8
20 3
6.6
These decimals stop or repeat.
It turns out that the decimal form of each rational number either stops or repeats. The decimal representation of our example of an irrational number, π, never repeats and never stops. p = 3.141592654....... This is true of all irrational numbers. Decimal Representation of Rational and Irrational Numbers If the decimal representation of a number, the number is: repeats or stops, rational does not repeat and does not stop irrational
When we put together these two sets of numbers, the rational numbers and the irrational numbers, they make the set of real numbers. We can illustrate how the numbers fit together with a diagram like this: Real numbers Rational numbers
Irrational numbers
Integers Whole numbers Counting numbers
Example 27: Identifying Integers, Rational Numbers, Irrational Numbers, and Real Numbers 14 2 1 Given the numbers: 7, , 8, , 0, 6 , 5 5 7 9 list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers Solution: a) whole numbers: b) integers:
7, 8, 0
c) rational numbers:
7,
d) irrational numbers:
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We’ll look more closely at square roots later!
8, 0 14 2 1 , 8, , 0, 6 5 7 9
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e) real numbers:
7,
14 2 1 , 8, , 0, 6 , 5 5 7 9
Quick Check For the given numbers list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers e) real numbers 5 1 1 8 4 2 54) 7, 6, , 3, , 0, 1 , 53) 12, , 5, , 0, 2 , 3 4 3 10 3 5 7
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real” and, if so, what could they be? For centuries, the only numbers people knew about were the real numbers but then some mathematicians started questioning the basic rules for operating with real numbers and invented the set of “imaginary numbers”. This was like changing the basic rules of a board game to see what happens to the game! You won’t encounter imaginary numbers in this course, but you will later on in your studies of algebra.
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Strategies for Success
Name____________________
Test Stress Reduction You can reduce your test stress by taking control of your success with some strategies that are easy to incorporate into your test prep routine. Take control by being prepared mathematically. Prepare yourself mathematically for the test so you will have confidence in your ability to succeed. If you feel prepared and confident, you will believe you can do well. These positive thoughts will carry over to your actions on the test. Lack of preparation causes students to be nervous and ‘blank out’ and get discouraged and overwhelmed. Follow the Test Preparation Skills strategies for preparing for the test. In order to be prepared mathematically I will:
Take control by taking care of your body.
Maintain your exercise routine. Exercise helps reduce stress. Get a good night’s sleep. Your brain becomes refreshed as you sleep. Cramming all night will not result in your best performance. Eat properly and maintain good nutrition. Give your body the food it needs to work hard during the test. Dress for your success. Dress in a way that makes you feel most confident and comfortable. Some students like to dress up a bit for tests and others prefer to wear their favorite jeans-choose what works for you! In order to take care of my body I will:
Take control by planning ahead.
Plan your transportation so that you arrive early and relaxed. Make sure that you have all the required materials in your backpack. o Pencils/erasers/highlighter o Calculator o Scantron /Blue Book or other materials required by your teacher o Any assignment that you need to turn in. Pack personal items that add to your comfort such as tissues, water, a jacket or sweatshirt. In order to plan ahead I will:
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3.6 Exercises Writing Exercises
Find the quotient of .12 .04 and explain in words all the steps you took.
What is the difference between the mean and the median of a set of numbers?
Practice Makes Perfect Add and Subtract Decimal Numbers Add or Subtract
1) 16.92 7.56
2) 18.37 9.36
3) 256.37 85.49
4) 248.25 91.29
5) 21.76 30.99
6) 15.35 20.88
7) 37.5 12.23
8) 38.6 13.67
9) 4.2 ( 9.3)
10) 8.6 ( 8.6)
11) 100 64.2
12) 100 65.83
13) 72.5 100
14) 86.2 100
15) 91.75 10.462
16) 94.69 12.678
17) 55.01 3.7
18) 59.08 4.6
19) 2.51 7.4
20) 3.84 6.1
Multiply Decimal Numbers Multiply
21) (0.2)(0.4)
22) 0.6 0.7
23) ( 4.3)(2.71)
24) 8.5 1.69
25) (0.06)(21.75)
26) 0.09 24.78
27) ( 5.18)( 65.23)
28) 9.16 68.34
Divide Decimal Numbers Divide
29) 0.6 0.2
30) 0.8 0.4
31) 1.44 0.3
32) 1.25 0.5
33) 1.75 0.05
34) 1.15 0.05
35) 5.0 2.5
36) 6.5 3.25
Convert Fractions to Decimals Convert each fraction to a decimal number
37)
2 5
38)
4 5
39)
41)
17 20
42)
13 20
43)
11 4
44)
17 4
47)
951 40
48)
849 40
45)
310 25
46)
284 25
3 8
40)
5 8
Simplify Expressions using the Order of Operations Simplify
49) 6 12.4 9.2
50) 3 15.7 8.6
51) 24 .5 .3
53) 18 .75 .15
54) 27 .55 .35
55) 15 .5 2.2 3 .4
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56) 14 .7 3.1 5 .2 59) 2.4
5 8
2
60) 3.9
57)
1 6.5 2
58)
1 10.75 4
9 20
Find the Mean of a Set of Numbers 61) The ages of the five women in a book club are 34, 45, 29, 61, and 41. Find the mean age of the women.
62) Miranda took her five children to the library. The number of books each checked out was 3, 8, 2, 2, and 5. Find the mean number of books checked out. 63) The number of hours Jessica worked for each of the last four weeks was 18, 23, 15, and 22. Find the mean number of hours. 64) The number of minutes it took Jim to ride his bike to school for each of the past six days was 21, 18, 16, 19, 24, and 19. Find the mean number of minutes. 65) Pete called six pizza shops to ask for the cost of a medium cheese pizza. The costs were $12.45, $12.99, $10.50, $11.25, $9.99, and $12.72. Find the mean cost of a pizza. 66) Four girls leaving a mall were asked how much money they had just spent. The amounts were $0, $14.95, $35.25, and $25.16. Find the mean amount of money spent. 67) Juan bought 5 shirts to wear to his new job. The cost of the shirts was $32.95, $38.50, $29.99, $17.48, and $24.25. Find the mean cost. 68) Norris bought 6 books for his classes this semester. The cost of the books was $74.28, $120.95, $52.40, $10.59, $35.89, and $59.24. Find the mean cost. 69) The amount of rainfall in Santa Ana for the first 6 months of 2009 was 0.75, 3.27, 0.04, 0.17, 0.00, and 0.10 inches. Find the mean amount of rainfall. 70) The amount of snow at Mammoth for 6 months in the winters of 2008 and 2009 was 60.3, 79.7, 50.9, 28.0, 47.4, and 46.1 inches. Find the mean amount of snow. 71) The top eight baseball players in spring 2009 had batting averages of .373, .360, .321, .321, .320, .312, .311, and .311. Find their mean. 72) The top nine college basketball players in the 2008-09 season had the following total points: 916, 847, 830, 812, 798, 764, 751, 739, and 729. Find their mean. Find the Median of a Set of Numbers Find the median of the numbers 73) 24, 19, 18, 29, 21 75) 65, 56, 35, 34, 44, 39, 55, 52, 45 77) 4, 8, 1, 5, 14, 3, 1, 12 79) 99.2, 101.9, 98.6, 99.5, 100.8, 99.8
74) 48, 51, 46, 42, 50 76) 121, 115, 135, 109, 136, 147, 127, 119, 110 78) 3, 9, 2, 6, 20, 3, 3, 10 80) 28.8, 32.9, 32.5, 27.9, 30.4, 32.5, 31.6, 32.7
81) Juan bought 5 shirts to wear to his new job. The cost of the shirts was $32.95, $38.50, $29.99, $17.48, and $24.25. Find the median cost. 82) Last week Ray recorded how much he spent for lunch each workday. He spent $6.50, $7.25, $4.90, $5.30, and $12.00. Find the median. 83) Michaela is in charge of 6 two-year olds at a daycare center. Their ages, in months, are 25, 24, 28, 32, 29, and 31. Find the median age. 84) Norris bought 6 books for his classes this semester. The cost of the books was $74.28, $120.95, $52.40, $10.59, $35.89, and $59.24. Find the median cost.
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Use Decimal Numbers in Applications Solve
85) Adam bought a t-shirt for $18.49 and a book for $8.92. The sales tax was $1.65. How much did Adam spend? 86) Roberto’s restaurant bill was $20.45 for the entrée and $3.15 for the drink. He left a $4.40 tip. How much did Roberto spend? 87) Brenda got $40 from the ATM. She spent $15.11 on a pair of earrings. How much money did she have left? 88) Marissa found $20 in her pocket. She spent $4.82 on a smoothee. How much of the $20 did she have left? 89) Alan got his first paycheck from his new job. He worked 30 hours and earned $382.50. How much does he earn per hour? 90) Maria got her first paycheck from her new job. She worked 25 hours and earned $362.50. How much does she earn per hour? 91) Mayra earns $9.25 per hour. Last week she worked 32 hours. How much did she earn? 92) Peter earns $8.75 per hour. Last week he worked 19 hours. How much did he earn? 93) Emily bought a box of cereal that cost $4.29. She had a coupon for $0.55 off, and the store doubled the coupon. How much did she pay for the box of cereal? 94) Diana bought a can of coffee that cost $7.99. She had a coupon for $0.75 off, and the store doubled the coupon. How much did she pay for the can of coffee? 95) The grocery store had a special on macaroni and cheese. The price was $3.87 for 3 boxes. How much did each box cost? 96) The pet store had a special on cat food. The price was $4.32 for 12 cans. How much did each can cost? 97) The parking meter at the beach costs $0.25 for every 20 minutes. How much does it cost to park for 2 hours? 98) The parking lot at the airport charges $0.75 for every 15 minutes. How much does it cost to park for 1 hour? 99) Leo took part in a diet program. He weighed 190 pounds at the start of the program. During the first week, he lost 4.3 pounds. During the second week, he had lost 2.8 pounds. The third week, he gained 0.7 pounds. The fourth week, he lost 1.9 pounds. What did Leo weigh at the end of the fourth week? 100) On April 1, the snowpack at the ski resort was 4 meters deep, but the next few days were very warm. By April 5, the snow depth was 1.6 meters less. On April 8 it snowed and added 2.1 meters of snow. What was the total depth of the snow? Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers For the given numbers, list the: a) whole numbers, b) integers, c) rational numbers, d) irrational
numbers, e) real numbers 101) 8,
12 3 5 , 9, , 0, 4 , 7 5 7 9
8 3 2 103) 7, 1, , , 0, , 3 7 9
3.6 Chapter 3b manuscript July 09.doc
1 5, 3 ,6 4
102) 9,
11 2 4 , 7, , 0, 3 , 3 6 5 9
5 2 3 104) 6, 3, , , 0, , 2 7 5
312
1 3, 2 ,5 5
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Every Day Math 105) Annie has two jobs. She gets paid $14.04 per hour for tutoring at City College and $8.75 per hour at a coffee shop. Last week she tutored for 8 hours and worked at the coffee shop for 15 hours. a) How much did she earn? b) If she had worked all 23 hours as a tutor instead of working both jobs, how much more would she have earned? 106) Jake has two jobs. He gets paid $7.95 per hour at the college cafeteria and $20.25 at the art gallery. Last week he worked 12 hours at the cafeteria and 5 hours at the art gallery. a) How much did he earn? b) If he had worked all 17 hours at the art gallery instead of working both jobs, how much more would he have earned? 107) Jeannette and her friends love to order mud pie at their favorite restaurant. They always share just one piece of pie among themselves. With tax and tip, the total cost is $6.00. How much does each girl pay if the total number sharing the mud pie is a) 2? b) 3? c) 4? d) 5? e) 6? 108) Alex and his friends go out for pizza and video games once a week. They share the cost of an $15.60 pizza equally among themselves. How much does each person pay if the total number sharing the pizza is a) 2? b) 3? c) 4? d) 5? e) 6?
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3.6 Answers Quick Check 1) 17.6
2) 33.11
3) 16.49
4) 23.593
5) 0.42
6) 12.58
7)
8)
9) 27.4815
10) 87.6148
11)
12) 38.122
16)
38
17) 587.3
18) 34.25
22) 0.46
23) .625
24) .3125
1.53
3.58
13.427
13) .39
14) .4
15)
19) 0.4
20) 0.6
21) 0.65
25) 2.25
26) 5.5
27) 5. 3
28) 4.16
29) 2.45
30) 2.318
31) 183.2
32) 7.7625
33) 11.16
34) 32.99
35) 5.275
36) 6.35
37) 21.5
38) 10.8
39) $7.19
40) $39.62
41) 99.97
42) 47.51
43) 43
44) 30
45) 8.5
46) 20.5
47)$41.21
48)$48.75
49)$15.47
50)$14.25
51)$5.97
52)$43
53) a) 0, 5
53b) 12, 0, 5
53c) 12,
5 1 1 , 5, , 0, 2 4 3 10
53d)
3
53e) 12,
54c) 6,
8 4 2 , 3, , 0, 1 3 5 7
54) a) 0, 6
54b) 3, 0, 6
42
__
__
__
__
5 1 1 , 5, , 0, 2 , 3 4 3 10
54d)
7
8 4 2 7, 6, , 3, , 0, 1 , 3 5 7
54e)
Practice Makes Perfect 1) 24.48
2) 27.73
3) 170.88
7) 49.73
8) 52.27
9)
13.5
4) 156.96
5)
10)
11) 35.8
12) 34.17
17.2
9.23
6)
5.53
13)
27.5
14)
13.8
15) 102.212
16) 107.368
17) 51.31
18) 54.48
19)
4.89
20)
2.26
21) 0.08
22) 0.42
23)
24)
11.653
25) 1.305
26) 2.2302
27) 337.8914
28) 625.9944
29) 3
30) 2
31)
32)
33) 35
34) 23
35) 2
36) 2
4.8
2.5
14.365
37) 0.4
38) 0.8
39) 0.375
40) 0.625
41) 0.85
42) 0.65
43) 2.75
44) 4.25
45) 12.4
46) 11.36
47) 23.775
48) 21.225
49) 19.2
50) 21.3
51) 12.09
52) 7.81
53) 20
54) 30
55) 36.44
56) 4.54
57) 7
58) 11
59) 3.025
60) 4.35
61) 42
62) 4
63) 19.5
64) 19.5
65) $11.65
66) $18.84
67) $28.63
68) $58.89
69) .722
70) 52.1
71) .3276
72) 798.4
73) 21
74) 48
75) 45
76) 121
77) 4.5
78) 4.5
79) 99.65
80) 32.05
81) $29.99
82) $6.50
83) 28.5
84) $55.82
85) $29.06
86) $28.00
87) $24.89
88) $15.18
89) $12.75
90) $14.50
91) $296.00
92) $166.25
93) $3.19
94) $6.49
95) $1.29
96) $0.36
97) $1.50
98)$3.00
99) 181.7 pounds
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100) 4.5 meters 8/25/09 ©
101b) 8, 9, 0, 101c) 8,
101a) 9, 0, 101e) 8, 102d)
12 5
,9,
3
5 ,0, 4 , 7 7 9
102e) 9,
3
11 6
12 5
102a) 7, 0
,7,
8
3
2
1
3
7
9
4
8
3
2
1
3
7
9
4
5
2
3
1
2
7
5
5
2 5
4 ,0, 3 , 3 9
103c) 7, 1, , ,0, ,3 ,6
,9,
3 7
,0, 4
101d)
9
7
102b) 9, 7, 0 102c) 9,
104a) 0, 5
11 6
,7,
2 5
,0, 3
4 9
103b) 7, 1, 0, 6
103a) 0, 6 103d)
103e) 7, 1, , ,0, , 5,3 ,6
5
5
104b) 6, 3, 0, 5 5
2
3
1
2
7
5
5
104c) 6, 3, , ,0, ,2 ,5
104d)
105a) $243.57 105b) $79.35
106a) $196.65 106b) $147.60 107a) $3.00
107b) $2.00
107c) $1.50
107d) $1.20
107e) $1.00
108c) $3.90
108d) $3.12
1084e) $2.60
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104e) 6, 3, , ,0, , 3,2 ,5
108a) $7.80
315
108b) $5.20
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3.7 Square Roots Objectives: Simplify Expressions with Square Roots
Simplify Expressions with Square Roots Remember, when a number n is multiplied by itself, we write it n 2 and read it ‘n squared’. For example, 82 read ‘8 squared’ 64 64 is called the square of 8. Similarly, 121 is the square of 11, because 112 is 121. Square of a number If n 2 = m , then m is the square of n .
Complete this table to show the squares of the counting numbers 1 through 15. Number Square
Activity Worksheet available.
n n2
1
2
3
4
5
6
7
8 64
9
10
11 121
12
13
14
15
Put about 50 color counters on your desk. Take some of the counters and arrange them to make a square. For example,
is a square made from four counters. It has two counters on each side. Make as many squares as you can with your counters. Draw a picture of each square that you create. Record your results in the table below:
Picture of square
Total number of counters in the square Number of counters on each side
4 2
The numbers you have in the second row of the table (the total number of counters) are called perfect squares. Each of them is the square of the number below it. Do you see the similarity between this table and the one you filled in with the squares of the counting numbers 1 through 15? Could you make a square with 6 counters?
Why or why not?
Could you make a square with 100 counters?
Why or why not?
How many counters would be on each side of a square made of 100 counters? What number squared gives 100?
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Learn to recognize the perfect square numbers!
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100 is 102 A number whose square is m is called a square root of m . 102 = 100
so
2
(-10) = 100
10 is a square root of 100.
so -10 is a square root of 100.
Square Root of a Number If n 2 = m , then n is a square root of m .
Square Root Notation: radical sign
m¬
m is read ‘the square root of m ’
radicand 2
If m = n , then
m = n , for n ³ 0 .
Using square root notation, we write: 100 square root of 100 10 Complete the following table showing square roots. 1
4
9
16
25
36
49
64
81
Example 1: Simplifying Expressions with Square Roots Simplify: a) 25 b) 121 c) 0 Solution: a) Since 52 = 25
c) Since 02 = 0
100 10
d)
121
5
121 11
0 0
d) Since 72 = 49
49 7
Quick Check Simplify: b) 169 1) a) 36 b) 196 2) a) 16
c) c)
1 4
d) d)
169
49
b) Since 112 = 121
25
144
81 64
You have seen that
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196
225
102 = 100
so
2
(-10) = 100
10 is a square root of 100.
so -10 is a square root of 100.
To avoid confusion mathematicians have agreed that the symbol indicates the positive square root. To show the negative square root, a negative sign must be placed in front of the radical. 25 - 25 5 -5 Example 2: Simplifying Expressions with Square Roots Simplify: a) - 9 b) - 144
Solution: a) The negative is in front of the radical sign.
- 9 -3
b) The negative is in front of the radical sign.
- 144 -12
Quick Check
Can you simplify
Simplify: 3) a) - 4
b) - 225
4) a) - 64
b) - 625
-25 ? Is there a number whose square is -25 ?
(
2
) = -25?
There is no real number whose square is -25 . Any positive number squared is positive. Any negative number squared is positive. So there is no real number equal to -25 . You’ll come back to this idea in a future math class! Example 3: Simplifying Expressions with Square Roots Simplify: a) -169 b) - 64 Solution: a)
-169 not a real number
b) The negative is in front of the radical.
Quick Check
Simplify:
- 64 -8
5) a)
-196
b) - 81
6) a) - 49
b) - -64 When using the order of operations to simplify an expression that has square roots, we treat square roots as exponents and the radical sign as a grouping symbol. Example 4: Simplifying Expressions with Square Roots Simplify: a) 25 + 144 b) 25 + 144 25 + 144
Solution: a) 3.7 Chapter 3b manuscript July 09.doc
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5 + 12 17
Use the order of operations Simplify
25 + 144
b) Simplify under the radical sign Simplify
169 13
Notice the different answers in part a and b . Quick Check Simplify: b) 9 + 16 7) a) 9 + 16
8) a)
64 + 225
b)
64 + 225
The square root expressions that we have looked at so far have not had any variables. What 9x 2 . Think of an
happens when we want a square root of a variable expression? Consider 2
expression whose square is 9 x . 2
( ? ) = 9x 2 2
(3 x ) = 9 x 2
so,
9x 2 = 3x
When we use a variable in a square root expression, we assume that the variable represents a non-negative number. Example 5: Simplifying Expressions with Square Roots x2
Simplify a)
b)
c) - 81y 2
16 x 2
d)
36 x 2 y 2
d)
100a 2 b 2
d)
225m 2 n 2
Solution: x2
a) 2
Since ( x ) = x 2
x
b)
16x 2
Since
2
(4 x ) = 16 x 2
4x - 81y 2
c) Since
2
(9y ) = 81y 2
-9 y 36 x 2 y 2
d) 2
Since (6 xy ) = 36 x 2 y 2
6 xy
Quick Check
Simplify
9) a)
y2
b)
10) a)
m2
b) - 100 p 2
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64x 2
319
c) - 121y 2 c)
169 y 2
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Link to Literacy Objective: Recognize perfect square numbers
Read the children’s book Sea Squares, by Joy N. Hulme . 1. What is the story about? 2. Imagine there was another page that showed 12 sailboats with 12 sailors in each boat. How many sailors would there be? 3. How does the story help children learn about square numbers? 4. Make up 2 examples, similar to those in Sea Squares, but with things found around your house, that show 32 and 82. 5. Read Sea Squares to a child in 3rd or 4th grade. Ask the child to verify the numbers on each page, and explain her reasoning to you – for example, how does she know there are 16 feet on the 4 seals? Did she count all 16 or did she multiply 4x4? Which method is faster? Write a brief paragraph about the child’s comments.
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3.7 Exercises Writing Exercises
Why is there no real number equal to
2
What is the difference between 9 and
-64 ?
9?
Practice Makes Perfect Simplify Expressions with Square Roots Simplify 1) 36 2) 4
5)
6)
9
9) - 4
16
10) - 100
3)
64
4)
169
7)
100
8)
144
11) - 1
12) - 121
13)
-121
14)
-36
15)
-9
16)
-49
17)
9 + 16
18)
25 + 144
19)
9 + 16
20)
25 + 144
21)
y2
22)
b2
23)
49x 2
24)
100 y 2
27)
144 x 2 y 2
28)
196a 2 b 2
25) - 64a 2
26) - 25 x 2
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3.7 Answers Quick Check 1a) 6 2 a) 4 3a) -2 5a) not a real number 7a) 7 9a) y 10a) m
1b) 13 2b) 14 3b) -15 5b) -9 7b) 5 9b) 8 x 10b) -10 p
1c) 1 2c) 2 4a) -8 6a) -7 8a) 17 9c) -11y 10c) 13 y
1d) 9 2d) 8 4b) -25 6b) not a real number 8b) 23 9d) 10ab 10d) 15mn
3) 8 7) 10 11) -1 15) not a real number 19) 7 23) 7 x 27) 12 xy
4) 13 8) 12 12) -11 16) not a real number 20) 17 24) 10 y 28) 14ab
Practice Makes Perfect 1) 6 5) 3 9) -2 13) not a real number 17) 5 21) y 25) -8a
2) 2 6) 4 10) -10 14) not a real number 18) 13 22) b 26) -5 x
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3.8 PROPERTIES OF REAL NUMBERS Objectives: Use the Commutative Properties of Addition and Multiplication Use the Associative Properties of Addition and Multiplication Identify the Commutative and Associative Properties of Addition and Multiplication Simplify Expressions Using the Distributive Property Identify the Identity Properties of Addition and Multiplication Use the Inverse Properties of Addition and Multiplication Use the Properties of Zero Identify the Properties of Real Numbers Simplify Expressions Using the Properties of Real Numbers
Now we can take a look at the properties of real numbers. Keep in mind that many of these properties describe things you know intuitively, but it will help to give names to the properties and define them formally. This way you’ll be able refer to them and use them as you solve equations in the next chapter.
Use the Commutative Properties of Addition and Multiplication Think about adding two numbers, say 5 and 3. The order you add them doesn’t affect the result, does it? 5 + 3 = 8 and 3 + 5 = 8 What about multiplying 5 and 3? 5 ⋅ 3 = 15 and 3 ⋅ 5 = 15 . Again, the order doesn’t matter, 5 + 3 = 3 + 5 and 5 ⋅ 3 = 3 ⋅ 5 . These are examples of the commutative property. The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. Notice, you use the same numbers, but the order has changed.
Commutative Property If a, b are real numbers, then of additionof multiplication If a, b are real numbers, then
a+b = b +a a⋅b = b⋅a
In addition and multiplication, changing the order of the numbers gives the same result.
When adding or multiplying, changing the order gives the same result.
Example 1: Using the Commutative Properties of Addition and Multiplication Use the commutative property to rewrite the following: a) 7 + 6 = _____ b) -1 + 3 = _____ c) 4 ⋅ 9 = _____ d) 7 (-8) = _____ Solution: Commutative property of addition: a) 7 + 6 = _____
a+b = b+a Change the order. b) -1 + 3 = _____
-1 + 3 = 3 + (-1)
7+6 = 6+7
Commutative property of multiplication: c) 4 ⋅ 9 = _____
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a⋅b = b⋅a Change the order. d) 7 (-8) = _____
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7 (-8) = -8 ⋅ 7
4⋅9 = 9⋅ 4
Quick Check Use the commutative property to complete the following: 1) a) 5 + 8 = _____ b) -4 + 7 = _____ c) 6 ⋅ 12 = _____ d) 9 (-3) = _____
2) a) 1 + 7 = _____
b) -2 + 14 = _____
c) 3 ⋅ 5 = _____
d) 12 (-4) = _____
Notice we have not said that subtraction is commutative. Is it? Can you think of an example to show that subtraction is or is not commutative? 7-3 ¹ 3-7 Subtraction is not commutative. For example, 4 ¹ -4 What about division? Is it commutative? Give an example that shows that division is or is not commutative. 12 ¸ 4 ¹ 4 ¸ 12 1 3¹ 3
Division is not commutative.
Use the Associative Properties of Addition and Multiplication If you were asked to simplify this expression, how would you do it and what would your answer be? 7+8+2 Some people would think 7 + 8 = 15 and then 15 + 2 = 17 . Others might start with 8 + 2 = 10 and then10 + 7 = 17 . Either way gives the same result.
(7 + 8) + 2 = 7 + (8 + 2) 15 + 2 = 7 + 10 17 = 17
You probably know this, but the terminology may be new to you. This example illustrates the Associative Property. The associative property has to do with grouping. If you change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order--the only difference is the grouping. Using the Associative Property can make your work easier.
Associative Property of addition of multiplication
(a + b) + c = a + (b + c ) If a, b, c are real numbers, then (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c ) If a, b, c are real numbers, then
In addition and multiplication, changing the grouping of the numbers gives the same result.
When adding or multiplying, changing the grouping gives the same result.
Example 2: Using the Associative Properties of Addition and Multiplication Use the associative property to rewrite the following: a) (3 + 0.6) + 0.4 = __________
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Solution: a) Associative Property of Addition
(a + b) + c = a + (b + c )
Change the grouping
(3 + 0.6) + 0.4 = __________ (3 + 0.6) + 0.4 = 3 + (0.6 + 0.4)
Change the grouping
(-7 ⋅ 5) ⋅ 2 = __________ (-7 ⋅ 5) ⋅ 2 = -7 ⋅ (5 ⋅ 2)
b) Change the grouping
Quick Check Use the associative property to rewrite the following: 3) a) (1 + 0.7) + 0.5 = __________ b) (-9 ⋅ 3) ⋅ 6 = __________
4) a) (4 + 0.6) + 0.4 = __________
b) (-2 ⋅ 7) ⋅ 5 = __________
Besides using the Associative Property to make calculations easier, we will often need it to simplify expressions. Example 3: Using the Associative Properties of Addition and Multiplication Simplify: 6 (3 x ) Solution: Without having a value for x , we can’t multiply 3 times x . But we can still simplify this expression. 6 (3 x )
Use the associative property for multiplication
(6 ⋅ 3) x
Multiply
Quick Check 5) 8 (4 x )
18 x Simplify: 6) -9 (7 y )
Identify the Commutative and Associative Properties of Addition and Multiplication Algebra students often confuse the commutative and associative properties. But these same words are used in ordinary English conversation. Picture of commuters – traffic?
Commutative refers to order. Think of the commuters on the freeway-they live in one city and work in another. In the morning they drive from home to work and in the evening they drive from work to home. What changes? The order! Associative refers to grouping. Think about the friends you associate with - your group of friends - it will help you remember that associative refers to grouping. Example 4: Identifying the Commutative and Associative Properties of Addition and Multiplication 3.8 Chapter 3b manuscript July 09.doc
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Identify whether each example is using the commutative or associative property a) (8 + 13) + 7 = 8 + (13 + 7) b) 5 ⋅
2 2 = ⋅5 3 3
d) 15 + x + 25 = x + 15 + 25 æ æ1 ö 1ö e) çç12 ⋅ ÷÷ ⋅ 3 = 12 ⋅ çç ⋅ 3÷÷ ÷ çè èç 3 ÷ø 3ø
c) 4 ⋅ y ⋅ 6 = 4 ⋅ 6 ⋅ y Solution: Often it is easiest to look at the left side, look at the right side and see what is different-order or grouping.
same numbers, same order, grouping is different
(8 + 13) + 7 = 8 + (13 + 7) associative property
same numbers, different order
2 2 = ⋅5 3 3 commutative property
same numbers, different order
4⋅ y ⋅6 = 4⋅6⋅ y commutative property
same numbers, different order
15 + x + 25 = x + 15 + 25 commutative property
same numbers, same order, grouping is different
æ ö æ ö çç12 ⋅ 1 ÷÷ ⋅ 3 = 12 ⋅ çç 1 ⋅ 3÷÷ ÷ çè ç è 3 ÷ø 3ø associative property
a)
5⋅
b)
c)
d)
e)
Quick Check Identify whether each example is using the commutative or associative property 3 3 b) (-1 + 10) + 6 = -1 + (10 + 6) c) 15 ⋅ = ⋅ 15 7) a) 5 ⋅ x ⋅ 7 = 5 ⋅ 7 ⋅ x 5 5 æ 1 ö÷ æ1 ö÷ b) çç8 ⋅ ÷ ⋅ 16 = 8 ⋅ çç ⋅ 16÷ c) 11 + a + 17 = a + 11 + 17 8) a) 9 ⋅ (3 x ) = (9 ⋅ 3) ⋅ x çè 4 ÷ø çè 4 ø÷
Simplify Expressions Using the Distributive Property
Picture of ticket window at movie theater
Now for some more math terms that describe something you may already know! But first an example: Three friends are going to the movies. They each need $9.25 – that’s 9 dollars and 1 quarter – to pay for their tickets. How much money do they need all together? You can think about the dollars separately from the quarters. They need 3 times $9, so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.
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Distributive Property If a, b, c are real numbers, then
a (b + c ) = ab + ac
Back to our friends at the movies, we could write it like this: 3 (9 + .25) = 3(9) + 3(.25) 3(9.25) = 27 + .75 27.75 = 27.75
In algebra, we often use the distributive property to remove parentheses as we simplify expressions. For example, if you are asked to simplify the expression 5( x + 2) , the order of operations says to work in the parentheses first. But you cannot add x and 2 ; they are not like terms. So we use the distributive property to get 5( x + 2) 5 x + 52 5 x + 10 5( x + 2) Example 5: Simplifying Expressions Using the Distributive Property Simplify: 3 ( x + 4) Solution: 3 ( x + 4) 3⋅ x + 3⋅ 4 3 x + 12
Distribute Multiply Quick Check 9) 4 ( x + 2)
Simplify: 10) 6 ( x + 7)
Example 6: Simplifying Expressions Using the Distributive Property Simplify: 6 (5 x + 1) Solution: 6 (5 x + 1)
Distribute Multiply Quick Check Simplify: 11) 9 (3 x + 8)
6 ⋅ 5x + 6 ⋅1 30 x + 6
12) 5 (5 x + 9)
The Distributive Property can be used to simplify expressions that look slightly different from a (b + c ) . Here are two other forms.
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Distributive Property If a, b, c are real numbers, then
a (b + c ) = ab + ac
Other forms: (b + c )a = ba + ca a (b - c ) = ab - ac
Example 7: Simplifying Expressions Using the Distributive Property Simplify: 2 ( x - 3) Solution: 2 ( x - 3)
2⋅ x - 2⋅3 2x - 6
Distribute Multiply Quick Check Simplify: 13) 8 ( x - 5)
14) 7 ( x - 6)
Example 8: Simplifying Expressions Using the Distributive Property Simplify: 6 (5n - 9) Solution: 6 (5n - 9) 65n - 69 30n - 54
Distribute Simplify Quick Check 15) 5 (6n - 4)
Simplify 16) 9 (12k - 15)
Example 9: Simplifying Expressions Using the Distributive Property 3 Simplify: ( x + 12) 4 Solution:
Distribute
3 ( x + 12) 4 3 3 ⋅ x + ⋅ 12 4 4 3 x +9 4
Simplify Quick Check Simplify: 2 17) ( x + 10) 5
18)
3 ( x + 21) 7
Example 10: Simplifying Expressions Using the Distributive Property Simplify: m(n + 4)
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Solution: m(n + 4) m n + m 4 mn + 4m
Distribute Simplify
Quick Check Simplify: 19) r (s + 2)
Notice that we wrote m 4 as 4m . We can do this by the commutative property of multiplication. Generally we put the numbers first in an expression.
20) y (z + 8)
Example 11: Simplifying Expressions Using the Distributive Property Simplify: ( x + 8) p Solution:
( x + 8) p xp + 8 p
Distribute Quick Check Simplify: 21) ( x + 2) p
22)
( y + 4) p
Example 12: Simplifying Expressions Using the Distributive Property Simplify: -2(4 y + 1) Solution: -2(4 y + 1) -2 4 y + (-2)1
Distribute Simplify
-8 y - 2
Quick Check Simplify 23) -3(6 y + 5)
24) -6(8 y + 11)
Example 13: Simplifying Expressions Using the Distributive Property Simplify: -11(4 - 3a ) Solution: -11(4 - 3a ) Distribute -11 4 - (-11)3a -44 + 33a Simplify Quick Check 25) -5(2 - 3a )
Simplify 26) -7(8 - 15 y )
Example 14: Simplifying Expressions Using the Distributive Property Simplify: -( y + 5) Solution: -( y + 5) Multiplying by -1 results in the opposite -1( y + 5) Distribute -1 y + (-1)5 3.8 Chapter 3b manuscript July 09.doc
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Simplify
-y + (-5)
Simplify
-y - 5
Quick Check 27) -( x - 4)
Simplify 28) -( z - 11)
Example 15: Simplifying Expressions Using the Distributive Property Simplify: 9 (8 x - 3) - (-2) Solution: 9 (8 x - 3) - (-2)
Distribute
9 ⋅ 8 x - 9 ⋅ 3 - (-2) 72 x - 27 - (-2)
Multiply
72 x - 25
Subtract Quick Check Simplify: 29) 4 (6 x - 1) - (-8)
30) 7 (3 x - 9) - (-1)
Example 16: Simplifying Expressions Using the Distributive Property Simplify: 8 - 2 ( x + 3) Solution: 8 - 2 ( x + 3)
Distribute Combine like terms Quick Check Simplify: 31) 9 - 3 ( x + 2)
8 - 2x - 6 -2 x + 2
32) 7 x - 5 ( x + 4)
Example 17: Simplifying Expressions Using the Distributive Property Simplify: 4 ( x - 8) - ( x + 3) Solution: 4 ( x - 8) - ( x + 3)
Distribute Combine like terms Quick Check Simplify: 33) 6 ( x - 9) - ( x + 12)
4 x - 32 - x - 3 3 x - 35
34) 8 ( x - 1) - ( x + 5)
Identify the Identity Properties of Addition and Multiplication What follows are some important properties that they will be useful when you learn how to solve equations in the next chapter.
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Identity Property For any real number a : of addition 0 is the additive identity of multiplication For any real number a : 1 is the multiplicative identity
a+0 = a
0+a = a
a ⋅1 = a
1a = a
When you add 0 to any number, you get that same number as the result. Adding 0 doesn’t change the value. We call 0 the additive identity. For example, 0 + (-8) = -8 13 + 0 = 13 -14 + 0 = -14 When you multiply any number by 1, you get that same number as the result. Multiplying by 1 doesn’t change the value. We call 1 the multiplicative identity. For example, 3 3 -27 ⋅ 1 = -27 1 = 43 ⋅ 1 = 43 5 5 Example 18: Identifying the Identity Properties of Addition and Multiplication Identify whether each example is using the additive or multiplicative identity. a) 7 + 0 = 7 b) -16 (1) = -16 c) 5 ⋅ 1 + (-8) = 5 + (-8) d) (5 + 0) + 14 = 5 + 14 Solution: a) adding zero - zero is the additive identity
7+0 = 7 additive identity
b)
-16 (1) = -16
multiplying by 1 – 1 is the multiplicative identity
multiplicative identity
c)
5 ⋅ 1 + (-8) = 5 + (-8)
multiplying by 1 – 1 is the multiplicative identity
multiplicative identity
d)
(5 + 0) + 14 = 5 + 14
adding zero - zero is the additive identity
additive identity
Quick Check Identify whether each example is using the additive or multiplicative identity. 35) a) 23 + 0 = 23 b) -37 (1) = -37
36) a) 9 ⋅ 1 + (-3) = 9 + (-3)
b) (7 + 0) + 11 = 7 + 11
Use the Inverse Properties of Addition and Multiplication What number added to 5 gives the additive identity? 5 + _____ = 0 We know 5 + (-5) = 0 What number added to -6 gives the additive identity? -6 + _____ = 0 We know -6 + 6 = 0 Notice that in each case, the missing number was the opposite of the number! 3.8 Chapter 3b manuscript July 09.doc
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We call -a the additive inverse of a . The opposite of a number is the additive inverse of the number. A number and its opposite add to zero -- the additive identity.
2 gives the multiplicative identity? 3 2 2 3 ⋅ _____ = 1 We know ⋅ = 1 3 2 3
What number multiplied by
What number multiplied by 2 gives the multiplicative identity? 1 2 ⋅ _____ = 1 We know 2 ⋅ = 1 2 Notice that in each case, the missing number was the reciprocal of the number! 1 the multiplicative inverse of a . The reciprocal of a number is the multiplicative a inverse of the number. A number and its reciprocal multiply to one-- the multiplicative identity.
We call
This leads to the inverse properties.
Inverse Property of addition
a + (-a) = 0
For any real number a ,
-a is the additive inverse of a A number and its opposite add to zero
of multiplication
For any real number a , a ¹ 0
a⋅
1 is the multiplicative inverse of a . a A number and its reciprocal multiply to one
A number and its opposite add to zero!
1 =1 a A number and its reciprocal multiply to one!
Example 19: Using the Inverse Properties of Addition and Multiplication 5 Find the additive inverse of a) 13 b) c) .6 8 Solution: a) The additive inverse is the opposite Find the opposite of 13
additive inverse of 13 -13 -13 is the additive inverse of 13 13 + (-13) = 0
b)
additive inverse of
5 8
The additive inverse is the opposite 3.8 Chapter 3b manuscript July 09.doc
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Find the opposite of
5 8
5 8 5 5 - is the additive inverse of 8 8 -
5 æç 5 ö÷ + ç- ÷ = 0 8 çè 8 ø÷
c) The additive inverse is the opposite Find the opposite of .6
additive inverse of .6 -.6 -.6 is the additive inverse of .6 .6 + (-.6) = 0
Quick Check Find the additive inverse of: 7 37) a) 18 b) c) 1.2 9
38) a) 47
b)
7 13
c) 8.4
Example 20: Using the Inverse Properties of Addition and Multiplication 4 Find the additive inverse of a) –8 b) c) -7.5 3 Solution: a) additive inverse of –8. Find the opposite of -8 -(-8)
8 8 is the additive inverse of –8 8 + (-8) = 0
additive inverse of -
b) Find the opposite of -
4 3
4 3
æ 4ö -çç- ÷÷÷ çè 3 ø
4 3 4 4 is the additive inverse of 3 3 4 æç 4 ö÷ + ç- ÷ = 0 3 çè 3 ÷ø
c) Find the opposite of -7.5
additive inverse of –7.5. -(-7.5) 7.5 7.5 is the additive inverse of –7.5
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7.5 + (-7.5) = 0
Quick Check Find the additive inverse of: 9 39) a) -14 b) c) 1.7 4
40) a) -46
b) -
5 2
c)
4.1
Example 21: Using the Inverse Properties of Addition and Multiplication Find the multiplicative inverse of 4. Solution:
multiplicative inverse of 4. The multiplicative inverse is the reciprocal Find the reciprocal of 4
1 is the reciprocal of 4 4 1 is the multiplicative inverse of 4 4 1 4⋅ = 1 4
Quick Check Find the multiplicative inverse of: 41) 9 42) 18
Example 22: Using the Inverse Properties of Addition and Multiplication 1 Find the multiplicative inverse of - . 9 Solution: 1 multiplicative inverse of 9 1 1 -9 is the reciprocal of Find the reciprocal of 9 9
-9 is the multiplicative inverse of -
1 9
1 (-9) = 1 9
Quick Check Find the multiplicative inverse of: 4 1 44) 43) 5 7
Use the Properties of Zero The Identity Property of Addition says that when you add 0 to any number, you get that same number as a result. What happens when you multiply a number by 0? 3 ⋅0 = 0 9⋅0 = 0 -15 ⋅ 0 = 0 10
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Property of Zero For any real number a, a⋅0 = 0 0a = 0 The product of any number and 0 is 0.
The product of any number and 0 is 0.
Example 23: Use the Properties of Zero Simplify
a) 23 ⋅ 0
b) -8 ⋅ 0
c)
5 ⋅0 12
Solution: a) The product of any number and 0 is 0.
23 ⋅ 0 0
b) The product of any number and 0 is 0.
-8 ⋅ 0 0
The product of any number and 0 is 0.
5 ⋅0 12 0
d) The product of any number and 0 is 0.
0(2.94) 0
c)
d) 0(2.94)
Quick Check Simplify
2 3
45) a) -14 ⋅ 0
b) 0 ⋅
46) a) (1.95) ⋅ 0
b) 0(-17)
c) (16.5) ⋅ 0
d) 300 ⋅ 0
c) 79 ⋅ 0
d) 0 ⋅
5 4
What about dividing with 0? What is 0 ¸ 3 ? Think about a real example: if there are no cookies in the cookie jar and 3 people to share them, how many cookies does each person get? There are no cookies to share! Each person gets 0 cookies. 0¸3 = 0 Remember that you can always check division with the related multiplication fact. 12 ¸ 6 = 2 is correct because 2 ⋅ 6 = 12 . So we know 0 ¸ 3 = 0 is correct because 0 ⋅ 3 = 0 . Zero divided by any real number except itself is zero.
We’ll explain later why we need to say “except itself”
Property of Zero For any real number a, except 0 0 =0. a Zero divided by any real number except itself is zero. 3.8 Chapter 3b manuscript July 09.doc
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Example 24: Use the Properties of Zero Simplify 0 0 c) a) 0 ¸ 5 b) -2 33 Solution: a) Zero divided by any real number except itself is zero.
d) 0 ¸ 0¸5 0 0 33 0
b) Zero divided by any real number except itself is zero. c) Zero divided by any real number except itself is zero.
0 -2 0 0¸
d) Zero divided by any real number except itself is zero. Quick Check Simplify 0 47) a) 0 ¸ 11 b) -6 8 0 b) 48) a) 0 ¸ 3 17
c) 0 ¸
7 8
3 10
c) 0 ¸ (-10)
7 8
0
d)
0 42
d) 0 ¸ 12.75
Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact. Is there a number that multiplied by 0 gives 4? 4 ¸ 0 = ? means ? 0 = 4 . Since, any number multiplied by 0 gives 0, the result will never be 4. There is no answer to 4 ¸ 0 ! Mathematicians say that division by zero is undefined. Property of Zero For any real number a,
a is undefined 0 Division by zero is undefined
Division by zero is undefined.
Example 25: Use the Properties of Zero Simplify 25 - 32 c) a) 7 ¸ 0 b) 0 0 Solution: a) Division by zero is undefined 3.8 Chapter 3b manuscript July 09.doc
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4 ¸0 9
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25 0 undefined
b) Division by zero is undefined
- 32 0 undefined
c) Division by zero is undefined
4 ¸0 9 undefined
d) Division by zero is undefined Quick Check Simplify 18 49) a) 16 ¸ 0 b) 0
50) a)
-5 0
b) 96 ¸ 0
c)
-2 0
d)
1 ¸0 5
c)
4 ¸0 15
d)
22 0
These three properties of zero are listed all together below:
Properties of Zero For any real number a, a⋅0 = 0
0 a = 0
0 = 0 for a ¹ 0 a a is undefined 0
Maybe you’re wondering why the second property of zero says that 0 divided by any real number 0 except itself is zero? If 0 were allowed in the denominator, that would make . This is covered 0 by the third property.
Identify the Properties of Real Numbers Here is a summary of the properties of real numbers.
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Properties of Real Numbers Commutative Property of addition If a, b are real numbers, then of multiplication If a, b are real numbers, then Associative Property of addition If a, b, c are real numbers, then of multiplication If a, b, c are real numbers, then
a+b = b+a a⋅b = b⋅a
(a + b) + c = a + (b + c ) ( a ⋅ b )⋅ c = a ⋅( b ⋅ c )
Distributive Property
If a, b, c are real numbers, then
a (b + c ) = ab + ac
Identity Property of addition For any real number a : 0 is the additive identity of multiplication For any real number a : 1 is the multiplicative identity Inverse Property of addition For any real number a ,
a+0 = a 0+a = a a ⋅1 = a 1a = a a + (-a) = 0
-a is the additive inverse of a
of multiplication
a⋅
For any real number a , a ¹ 0
1 =1 a
1 is the multiplicative inverse of a . a Properties of Zero a⋅0 = 0 0a = 0 0 =0 a
For any real number a, For any real number a , a ¹ 0
a is undefined 0
For any real number a ,
Example 26: Identifying the Properties of Real Numbers Which of the properties of real numbers is being used? a) 15( y - 5) = 15 y - 75 b) -3 (7 x ) = (-37) x c) 51 + 4 x - 51 = 4 x + 51- 51 Solution: a) a (b + c ) = ab + ac
15( y - 5) = 15 y - 75
the distributive property
b)
-3 (7 x ) = (-37) x
Same order, same order, grouping is different
the associative property of multiplication
c) Same numbers, different order
51 + 4 x - 51 = 4 x + 51- 51 the commutative property of addition
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Quick Check Which of the properties of real numbers is being used? 51) a) 8 + x - 8 = x + 8 - 8 b) 7(m - 2) = 7m - 14 c) 2 (7 y ) = (2 ⋅ 7) y
52) a) 8 (15n) = (8 ⋅ 15) n
b) 21 + c - 21 = c + 21- 21
Example 27: Identifying the Properties of Real Numbers Which of the properties of real numbers is being used? æ10 7 ö a) -43 + 43 = 0 b) çç ⋅ ÷÷÷ x = 1⋅ x çè 7 10 ø
c) -4( x - 9) = -4 x + 36
c) (37u )(0) = 0
Solution:
a) A number and it’s opposite add to zero
-43 + 43 = 0 the inverse property of addition
A number and its reciprocal multiply to one.
æ10 7 ö ççç ⋅ ÷÷÷ x = 1⋅ x è 7 10 ø the inverse property of multiplication
c)
(37u )(0) = 0
The product of any number and 0 is 0
properties of zero
b)
Quick Check Which of the properties of real numbers is being used? æ2 3ö 53) a) çç ⋅ ÷÷÷ x = 1⋅ x b) 49 + (-49) = 0 c) (24 p)(0) = 0 çè 3 2 ø æ 1ö b) (13a)(0) = 0 c) çç5 ⋅ ÷÷÷ k = 1⋅ k 54) a) -17 + 17 = 0 çè 5 ø
Simplify Expressions Using the Properties of Real Numbers Using the properties of real numbers can make your work easier when you simplify expressions. Be on the lookout for special relationships between numbers, like inverses, and for pairs of numbers whose sums or products are multiples of 10.
Example 28: Simplifying Expressions Using the Properties of Real Numbers Simplify: 23 + 18 + (-23) Solution: 23 + 18 + (-23)
Notice the additive inverses
23 + 18 + (-23)
Use the commutative property of addition
23 + (-23) + 18 0 + 18
Use the inverse property of addition
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Use the identity property of addition Quick Check Simplify:
18
55) 19 + 63 + (-19)
56) -47 + 88 + 47
Example 29: Simplifying Expressions Using the Properties of Real Numbers Simplify: (n + 0.37) + 4.63 Solution: The order of operations says to do the work inside the parentheses first. But you cannot add n + 0.37 (n + 0.37) + 4.63 Use the associative property of addition n + (0.37 + 4.63) Add n +5 Quick Check Simplify:
57) ( y + 0.15) + 2.85
58) ( x + 18) + 12
Example 30: Simplifying Expressions Using the Properties of Real Numbers 11 12 Simplify: ⋅ (-28) ⋅ 12 11 Solution: 11 12 ⋅ (-28) ⋅ 12 11 11 12 ⋅ (-28) ⋅ Notice the multiplicative inverses 12 11 11 12 ⋅ ⋅ (-28) Use the commutative property of multiplication 12 11 Use the inverse property of multiplication 1⋅ (-28) -28 Use the identity property of multiplication Quick Check Simplify:
59)
4 7 ⋅ (-23) ⋅ 7 4
60)
8 15 ⋅ (4 y ) ⋅ 15 8
Example 31: Simplifying Expressions Using the Properties of Real Numbers 1 Simplify: ⋅ 5 ⋅ 12 4 Solution: 1 ⋅ 5 ⋅ 12 4 1 ⋅ 12 ⋅ 5 Use the commutative property of multiplication 4 Multiply 3⋅5 Multiply 15 Quick Check Simplify:
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2 ⋅ 17 ⋅ 12 3
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62)
5 ⋅ 24 ⋅ 18 9
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Example 32: Simplifying Expressions Using the Properties of Real Numbers Simplify: 3(2 x - 5) + 6 Solution: 3(2 x - 5) + 6 Distribute 6 x - 15 + 6 Combine like terms 6x - 9 Quick Check Simplify:
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64) 7(4a - 8) - 11
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Strategies for Success Test Taking Skills
Name____________________
Always, Sometimes, Never. Respond to each statement by checking Always, Sometime or Never A S N Before the test ___ ___ ___ I arrive on time or even early so I feel calm and ready. ___ ___ ___ I set out the required materials so I feel prepared. ___ ___ ___ If a problem in the rest of my life may interfere with my test performance, I write it down on a card and zip it in my backpack until after the test. ___ ___ ___ I ignore others in the room- I do not want to pick up their negativity or anxiety. I am prepared and confident. ___ ___ ___ I check my inner voice. I turn any negative thoughts into positive statements. “I am prepared; I’ve done what I can; I am ready to succeed; I can do math!” ___ ___ ___ I use the restroom. Most teachers do not allow exit/re-entry during the test. ___ ___ ___ The test itself ___ ___ ___ I do a “data dump” as soon as I get the test. I no longer need to think about remembering these facts/formulas. ___ ___ ___ I scan the test, reading all problems before I begin to work any. ___ ___ ___ I read directions carefully – I circle, underline or highlight key words and directions. ___ ___ ___ I note easy problems and do them first to build my confidence and ensure those points. ___ ___ ___ If I can’t do a problem immediately, I write down anything I can think of such as formulas, pictures, etc., then I move on and return to it later. The solution may come to me as I work on the other problems. ___ ___ ___ I show all my work. I write all steps, reasoning, and supporting evidence. This is really helpful when my teacher awards partial credit. ___ ___ ___ I check my work. ___ ___ ___ I check answers. I make sure word problems have reasonable answers. ___ ___ ___ I pace myself. ___ ___ ___ I do not turn in my test early. I use the time to carefully go over my work. ___ ___ ___ I ignore others. I remember that those done early may be turning in a blank test. ___ ___ ___ If I do not know how to do something, I try to relate it to something I do know. ___ ___ ___
___ ___ ___ ___ ___ ___ ___
Stress reduction during the test ___ ___ I check my inner voice. I turn any negative thoughts into positive statements. ___ ___ I imagine and visualize that I am in my favorite pleasant relaxing situation. ___ ___ I take mental breaks. ___ ___ I do stress reducing exercises. ___ ___ I do deep breathing. ___ ___ I do muscle tensing and relaxing. ___ ___
Look at your checklist. Can you think of any technique(s) that you use regularly that is not on the checklist? Add it (them) in the checklist. Look at the ‘Sometimes’ and ‘Never’ categories. List three techniques that you may try during the next test. 1)
2) 3)
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Links to Literacy
Each Orange Had 8 Slices, by Paul Giganti, Jr. Objective: Use the associative property of multiplication
Read the children’s book Each Orange Had 8 Slices, by Paul Giganti, Jr
1. What is the story about? 2. How does it help children learn to multiply numbers? 3. The first page of the book shows 3 flowers with 6 petals and 2 bugs. a. To find the total number of bugs, did you start by finding the number of bugs on each flower, or did you find the total number of petals first? b. Would it make a difference in the final answer? 4. Explain how this book relates to the Associative Property of Multiplication, (a b ) c a (b c ) . 5. Read Each Orange Had 8 Slices to a young child.
Ask the child to draw a picture that could be another page of the book, and then talk with the child about the total number of items in the picture.
What did the child draw? How many items were in the picture? Attach the child’s drawing.
Write a short paragraph about the child’s comments and reactions.
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3.8 Exercises Writing Exercises
In your own words, state the commutative property of addition. In your own words, state the associative property of multiplication. What is the difference between the additive inverse and the multiplicative inverse of a number? How can the use of the properties of real numbers make it easier to simplify expressions?
Practice Makes Perfect Use the Commutative Properties of Addition and Multiplication Use the commutative property to complete the following:
1) 8 + 9 = ___
2) 7 + 6 = ___
3) 8 (-12) = ___
4) 7 (-13) = ___
5) (-19)(-14) = ___
6) (-12)(-18) = ___
7) -11 + 8 = ___
8) -15 + 7 = ___
9) x + 4 = ___
10) y + 1 = ___
11) -2a = ___
12) -3m = ___
Use the Associative Properties of Addition and Multiplication Use the associative property to complete the following:
13) (11 + 9) + 14 = ___ 14) (21 + 14) + 9 = ___ 15) (12 ⋅ 5) ⋅ 7 = ___
16) (14 ⋅ 6) ⋅ 9 = ___
æ 4ö 17) (-7 + 9) + 8 = ___ 18) (-2 + 6) + 7 = ___ 19) çç16 ⋅ ÷÷÷ ⋅ 15 = ___ çè 5ø
æ 2ö 20) çç13 ⋅ ÷÷÷ ⋅ 18 = ___ çè 3ø
21) 3 (4 x ) = ___
22) 4 (7 x ) = ___
23) (12 + x ) + 28 = ___ 24) (17 + y ) + 33 = ___
Identify the Commutative and Associative Properties of Addition and Multiplication Identify whether each example is using the commutative or associative property æ æ1 ö 1ö 25) (5 + 1.2) + 0.8 = 5 + (1.2 + 0.8) 26) çç16 ⋅ ÷÷ ⋅ 10 = 16 ⋅ çç ⋅ 10÷÷ ÷ ÷ çè ç è5 ø 5ø 27) 9 ⋅ x ⋅ 14 = 9 ⋅ 14 ⋅ x 28) 3.75 + 1.62 + 0.25 = 3.75 + 0.25 + 1.62 3 3 29) 33 + n + 26 = n + 33 + 26 30) 6 ⋅ = ⋅ 6 5 5 æ ö æ ö 1 1 32) (9 + 2.4) + 0.6 = 9 + (2.4 + 0.6) 31) çç22 ⋅ ÷÷÷ ⋅ 9 = 22 ⋅ çç ⋅ 9÷÷÷ çè 3 ø èç 3ø Simplify Expressions Using the Distributive Property Simplify using the distributive property
33) 4( x + 8)
34) 3(a + 9)
35) 8 (4 y + 9)
36) 9 (3w + 7)
37) 3(1 + 4n ) 41) 8(2u - 6)
38) 5(2 + 3 x ) 42) 6(3m - 7)
39) 6(c - 13) 43) 7(3 p - 8)
40) 7( y - 13) 44) 5(7u - 4)
1 (n + 8) 2 49) x( y 9)
1 (u + 9) 3 50) c(d 3)
1 (3q + 12) 4 51) r (s 18)
1 (4m + 20) 5 52) u(v 10)
53) ( y + 4)p
54) (a + 7)x
55) -2( y + 13)
56) -3(a + 11)
45)
46)
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57) 7(4 p 1) 61) -9(3a - 7)
58) 9(9a 4) 62) -6(7 x - 8)
59) -3( x - 6) 63) -7(4 - 2m )
60) -4(q - 7) 64) -8(5 - 3c )
65) -(r + 7)
66) -(q + 11)
67) -(3 x - 7)
68) -(5 p - 4)
69) 73) 77) 80) 83) 86) 89) 92)
4(3 y - 7) - (-9) 70) 6(5 p - 2) - (-3) 71) 16 - 3( y + 8) 72) 18 - 4( x + 2) 7( y - 5) - 2( y + 4) 74) 4( x - 2) - 3( x + 2) 75) 14(c - 1) - 8(c - 6) 76) 11(n - 7) - 5(n - 1) 6(7 y + 8) - (30 y - 15) 78) 7(3n + 9) - (4n - 13) 79) -20 + 7(a + 4) -8 + 3( y + 15) 81) 5 + 9(n - 6) 82) 12 + 8(u - 1) 4 - 11(3c - 2) 84) 9 - 6(7n - 5) 85) -12 - (u + 10) 22 - (a + 3) 87) 8 - (r - 7) 88) -4 - (c - 10) 5(2n + 9) + 12(n - 3) 90) 9(5u + 8) + 2(u - 6) 91) (5m - 3) - (m + 7) (4 y - 1) - ( y - 2) 93) (7u - 3) - 2(u - 4) 94) (4a - 9) - 3(a + 5)
Identify the Identity Properties of Addition and Multiplication Identify whether each example is using the additive or multiplicative identity. 3 3 95) 101 + 0 = 101 96) (1) = 97) -9 ⋅ 1 = -9 5 5 99) 7 ⋅ 1-11 = 7 - 11 100) (9n + 0) - 5n = 9n - 5n 98) 0 + 64 = 64 101) (3 x + 0) + 8 x = 3 x + 8 x 102) 4 ⋅ 1+ 53 = 4 + 53 Use the Inverse Properties of Addition and Multiplication Find the additive inverse 103) –20 104) -19 105) –13.2
106) -31.7
107) 15
108)
37.5
110)
13 24
112)
11 17
114) -
111)
22 15 22
109)
113) -
56.9 14 19
Find the multiplicative inverse 7 12 3 119) 10
115)
8 13 5 120) 12
116)
117) 8
118) 9
121) -17
122) –19
Use the Properties of Zero Simplify
123) 48 ⋅ 0 127) 0 ¸ 131)
11 12
3 ⋅0 8
0 6 6 128) 0 2 132) ¸ 0 9 124)
3 0 0 129) 3 1 133) ¸0 10 125)
126) 22 ⋅ 0 130) 0 ¸ 134)
7 15
5 ⋅0 12
Identify the Properties of Real Numbers Which of the properties of real numbers is being used? 135) -3 + 8 = 8 + (-3) 136) -14 ⋅ 1 = -14
137) 1⋅ x = x
138) 7( x + 2) = 7(2 + x )
139) 15 + 0 = 15
140) (-5 + 7) + 9 = -5 + (7 + 9)
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3 3 +0 = 4 4
141) 7 + (8 + 4) = (7 + 8) + 4
142)
5 8 ⋅ =1 8 5 æ 1ö æ 1ö 145) çç- ÷÷(-2) = (-2)çç- ÷÷ èç 2 ø÷ èç 2 ø÷
144) 8 (-12) = (-12) 8
143)
0 =0 -2 149) 3 ( y - 9) = 3 y - 27
5 æ 9ö 146) - çç- ÷÷÷ = 1 9 çè 5 ø
147)
148) 19 x 0 = 0
151) (2 x ) y = 2 ( x ⋅ y )
150) 6 éë5 (-4)ùû = [ 6 ⋅ 5 ](-4) 152) -4 ( y - 1) = -4 y + 4
Simplify Expressions Using the Properties of Real Numbers Simplify 153) -15 + 8 + 15 154) -34 + 19 + 34 155) ( x + 0.49) + 7.51 156) ( y + 0.38) + 9.62 5 18 (47) 18 5 123 159) 0 3 161) (30k ) 5 1 163) ⋅ 8 ⋅ 18 3
157)
165) (275)(1.19 y )(0) 167) 6 (8m - 3) - 4
7 13 (68) 13 7 2 160) (12n) 3 95 162) 0
158)
164) (180 x )(5.73)(0) 4 166) ⋅ 7 ⋅ 25 5 168) 2 (9u - 5) + 1
Everyday Math 169) Trader Joe’s grocery stores sell a bottle of wine they call “Two Buck Chuck” for $1.99. They sell a case of 12 bottles for $23.88. Use the distributive property to multiply 12(2 .01) and see if it is a bargain to buy “Two Buck Chuck” by the case.
170) Adele's shampoo sells for $3.99 per bottle at the grocery store. a) Use the distributive property to calculate how much it would cost to buy 3 bottles of shampoo 3(3.99) = 3(4 -.01) b)The same shampoo is offered as a three-pack at the warehouse store for $10.49. How much would Adele save by buying 3 bottles at the warehouse store?
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3.8 Answers Quick Check 1a) 5 + 8 = 8 + 5
1b) -4 + 7 = 7 + (-4)
1c) 6 (12) = (12) 6
1d) 9 (-3) = (-3) 9
2) a) 1 + 7 = 7 + 1
2b) -2 + 14 = 14 + (-2)
2c) 3 ⋅ 5 = 5 ⋅ 3
2d) 12 (-4) = (-4)12
3a) (1 + 0.7) + 0.5 = 1 + (0.7 + 0.5)
3b) (-9 ⋅ 3) ⋅ 6 = (-9)(3 ⋅ 6)
4a) (4 + 0.6) + 0.4 = 4 + (0.6 + 0.4)
4b) (-2 ⋅ 7) ⋅ 5 = -2 (7 ⋅ 5)
5) 32 x 7a) Commutative Property of Multiplication
6) -63 y 7b) Associative Property of Addition
7c) Commutative Property of Multiplication
8a) Associative Property of Multiplication
8b) Associative Property of Multiplication
8c) Commutative Property of Addition
9) 4 x + 8
10) 6 x + 42
11) 27 x + 72
12) 25 x + 45
13) 8 x - 40
14) 7 x - 42
15) 30n - 20
16) 108k - 135
19) rs + 2r
20) yz + 8 y
17)
2 x+4 5
3 x +9 7
18)
21) xp + 2 p
22) yp + 4 p
23) -18 y - 15
24) -48 y - 66
25) -10 + 15a
26) -56 + 105 y
27) -x + 4
28) -z + 11
29) 24 x + 4
30) 21x - 62
31) -3 x + 3
32) 2 x - 20
33) 5 x - 66
34) 7 x - 13
35a) Additive identity
35b) Multiplicative identity
36a) Multiplicative identity
36b) Additive identity
37a) -18 38b) -
7 13
39c) 1.7 41)
1 9
37b) -
7 9
37c) -1.2
38a) -47
38c)
8.4
39a) 14
39b)
40a) 46 42)
1 18
40b)
5 2
9 4
40c) 4.1
43) -7
44)
5 4
45a) 0
45b) 0
45c) 0
45d) 0
46a) 0
46b) 0
46c) 0
46d) 0
47a) 0
47b) 0
47c) 0
47d) 0
48a) 0
48b) 0
48c) 0
48d) 0
49a) undefined
49b) undefined
49c) undefined
49d) undefined
50a) undefined
50b) undefined
50c) undefined
50d) undefined
51a) Commutative Property of Addition
51b) Distributive Property
51c) Associative Property of Multiplication
52a) Associative Property of Multiplication
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52b) Commutative Property of Addition
52c) Distributive Property
53a) Inverse Property of Multiplication
53b) Inverse Property of Addition
53c) Properties of Zero
54a) Inverse Property of Addition
54b) Properties of Zero
54c) Inverse Property of Multiplication
55) 63
56) 88
57) y + 3
58) x + 30
59) -23
60) 4 y
61) 136
62) 240
63) 16 x - 25
64) 28a - 67
Practice Makes Perfect 1) 8 + 9 = 9 + 8
2) 7 + 6 = 6 + 7
3) 8 (-12) = (-12) 8
4) 7 (-13) = (-13) 7
5) (-19)(-14) = (-14)(-19)
6) (-12)(-18) = (-18)(-12)
7) -11 + 8 = 8 + (-11)
8) -15 + 7 = 7 + (-15)
9) x + 4 = 4 + x
10) y + 1 = 1 + y
11) -2a = a (-2)
12) -3m = m (-3)
13) (11 + 9) + 14 = 11 + (9 + 14)
14) (21 + 14) + 9 = 21 + (14 + 9)
15) (12 ⋅ 5) ⋅ 7 = 12 (5 ⋅ 7)
16) (14 ⋅ 6) ⋅ 9 = 14 (6 ⋅ 9)
17) (-7 + 9) + 8 = -7 + (9 + 8)
18) (-2 + 6) + 7 = -2 + (6 + 7)
æ æ4 ö 4ö 19) çç16 ⋅ ÷÷ ⋅ 15 = 16 çç ⋅ 15÷÷ èç èç 5 ø÷ 5 ø÷
æ æ2 ö 2ö 20) çç13 ⋅ ÷÷ ⋅ 18 = 13 çç ⋅ 18÷÷ ÷ø çè 3 èç 3 ø÷
21) 3 (4 x ) = (3 ⋅ 4) x
22) 4 (7 x ) = (4 ⋅ 7) x
23) (12 + x ) + 28 = 12 + ( x + 28)
24) (17 + y ) + 33 = 17 + ( y + 33)
25) Associative
26) Associative
27) Commutative
28) Commutative
29) Commutative
30) Commutative
31) Associative
32) Associative
33) 4 x + 32
34) 3a + 27
35) 32y + 72
36) 27w + 63
37) 3 + 12n
38) 10 + 15 x
39) 6c - 78
40) 7 y - 91
41)16u - 48
42) 18m - 42
43) 21p - 56
44) 35u - 20
45)
1 n+4 2
46)
1 u +3 3
47)
3 q +3 4
48)
4 m+4 5
49) xy + 9 x
50) cd + 3c
51) rs - 18r
52) uv - 10u
53) yp + 4 p
54) ax + 7 x
55) -2y - 26
56) -3a - 33
57) -28 p - 7
58) -81a - 36
59) -3 x + 18
60) -4q + 28
61) -27a + 63
62) -42 x + 48
63) -28 + 14m
64) -40 + 24c
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65) -r - 7
66) -q - 11
67) -3 x + 7
68) -5 p + 4
69) 12y - 19
70) 30 p - 9
71) -3 y - 8
72) 10 - 4 y
73) 5 y - 43
74) x - 14
75) 6c + 34
76) 6n - 72
77) 12y + 63
78) 17n + 76
79) 7a + 8
80) 3 y + 37
81) 9n - 49
82) 8u + 4
83) 26 - 33c
84) 39 - 42n
85) -22 - u
86) 19 - a
87) 15 - r
88) 6 - c
89) 22n + 9
90) 47u + 60
91) 4m - 10
92) 3 y + 1
93) 5u + 5
94) a - 24
95) Additive
96) Multiplicative
97) Multiplicative
98) Additive
99) Multiplicative
100) Additive
101) Additive
102) Multiplicative
103) 20
104) 19
105) 13.2
106) 31.7
107) -15
108) -22
109) -37.5
110) -56.9
111) -
113)
11 17
114)
14 19
115)
117)
1 8
118)
1 9
119) -
121) -
1 17
122) -
1 19
13 24
12 7
10 3
112) 116)
15 22
13 8
120) -
123) 0
124) 0
12 5
125) undefined
126) 0
127) 0
128) undefined
129) 0
130) 0
1331) 0
132) undefined
133) undefined
134) 0
135) Commutative Property of Addition
136) Identity Property of Multiplication
137) Identity Property of Multiplication
138) Commutative Property of Addition
139) Identity Property of Addition
140) Associative Property of Addition
141) Associative Property of Addition
142) Identity Property of Addition
143) Inverse Property of Multiplication
144) Commutative Property of Multiplication
145) Commutative Property of Multiplication
146) Inverse Property of Multiplication
147) Property of Zero
148) Property of Zero
149) Distributive Property
150) Associative Property of Multiplication
151) Associative Property of Multiplication
152) Distributive Property
153) 8
154) 19
155) x + 8
156) y + 10
157) 47
158) 68
159) undefined
160) 8n
161) 18k
162) undefined
163) 48
164) 0
165) 0
166) 140
167) 48m - 22
168) 18u - 9
170a) $11.97
170b) $1.48
169) The cost is the same 3.8 Chapter 3b manuscript July 09.doc
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Chapter 3 Review Exercises 3.1 Visualizing Positive and Negative Numbers Locate Positive and Negative Numbers on the Number Line Locate and label the following points on the number line. 1 1) 5 2) 2 3) 2 7 5) 6) 3 21 3 Order Positive and Negative Numbers Order each of the following pairs of numbers, using < or >. 1 7) 4 ___ 1 8) 5 ___ 2 9) ___ 1 3 5 7 1 3 12) ____ 11) ___ 12 12 2 5 Find Opposites For each of the following numbers find it’s a) opposite and
13) 6
14) 2
Simplify Absolute Values Evaluate. 17) x when x 14 18) r when r 27
15)
4)
3 4
10) 2 21 ___ 2
b) absolute value.
3 8
16)
19) y when y 33
5 6
20) n when n 4
Fill in < , >, or = for each of the following pairs of numbers.
21) 4 ____ 4
22) 2 ___ 2 23) 6 ___ 6
Simplify. 25) 12 5
26) 9 7
27) 14 8 2
24)
3 3 ___ 4 4
28) 5 4 15 3
Translate Phrases to Expressions with Positive and Negative Numbers Translate each phrase into an algebraic expression. 29) the opposite of 16 30) the opposite of 8 31) negative 3 32) 19 minus negative 12 33) a temperature of 10 below zero 34) an elevation of 85 feet below sea level
3.2 Addition of Positive and Negative Numbers Model Addition of Positive and Negative Numbers Use counters to model the following. 35) 3 7 36) 2 6 37) 5 ( 4)
38) 3 ( 6)
Simplify Expressions with Positive and Negative Numbers Simplify each expression. 39) 14 82 40) 33 ( 67) 41) 75 25
42) 54 ( 28)
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43) 11 ( 15) 3
44) 19 ( 42) 12
45)
2 7 5 5
46)
3 5 10 8
Evaluate Variable Expressions with Positive and Negative Numbers Evaluate each expression. 47) n 4 when n a) 1, b) 20 48) x ( 9) when x a)3, b) 3
49) y
1 1 5 when y a) b) 3 3 6
50) (u v )2 when u 4, v 11
Translate English Phrases to Algebraic Expressions Translate each phrase into an algebraic expression and then simplify. 51) the sum of 8 and 2 52) 4 more than 12 53) 10 more than the sum of 5 and 6 54) the sum of 3 and 5 , increased by 18 Solve Applications Involving Positive and Negative Numbers Solve the following applications. 55) On Monday, the high temperature in Denver was 4 degrees. Tuesday’s high temperature was 20 degrees more. What was the high temperature on Tuesday?
56) Frida owed $75 on her credit card. Then she charged $21 more. What was her new balance?
3.3 Subtraction of Positive and Negative Numbers Model Subtraction of Positive and Negative Numbers Use counters to model the following. 57) 6 1 58) 4 ( 3) 59) 2 ( 5)
60) 1 4
Simplify Expressions with Positive and Negative Numbers Simplify each expression. 61) 24 16 62) 19 ( 9) 63) 31 7
64) 40 ( 11)
65) 52 ( 17) 23
68) 32 72
69)
3 5 3 5 8 8 8 8
66) 35 ( 3 9) 70)
67) (1 7) (3 8)
3 3 2 2
Evaluate Variable Expressions with Positive and Negative Numbers Evaluate each expression. 71) x 7 when x a)5, b) 4 72) 10 y when y a)15, b) 16
73) n
1 1 1 b) when n a) 6 6 6
74) 8 3x 2 when x 4
Translate English Phrases to Algebraic Expressions Translate each phrase into an algebraic expression and then simplify. 75) the difference of 12 and 5 76) subtract 23 from 50 Solve Applications with Negative Numbers Solve the following applications. 77) One morning the temperature in Bangor, Maine was 18 degrees. By afternoon, it had dropped 20 degrees. What was the afternoon temperature?
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78) On January 4, the high temperature in Laredo, Texas was 78 degrees, and the high in Houlton, Maine was 28 degrees. What was the difference in temperature of Laredo and Houlton?
3.4 Multiplication and Division with Positive and Negative Numbers Multiply Positive and Negative Numbers Multiply.
79) 9 4
80) 5( 7)
81) ( 11)( 11)
1 82) ( 32) 4
Find Reciprocals of Negative Numbers Find the reciprocal.
83)
9 10
84)
7 1 85) 86) 7 3 4
Divide Positive and Negative Numbers Divide.
87) 56 ( 8)
88) 120 ( 6)
89)
10 15
Simplify Expressions with Positive and Negative Numbers Simplify each expression. 91) 5( 9) 3( 12) 92) ( 2)5 93) 3 4 95) 42 4(6 9) 96) (8 15)(9 3) 97) 2( 18) 9
99)
4 15 20
100)
90)
0 2
94) ( 3)(4)( 5)( 6) 98) 45 ( 3) 12
3( 9) 7 5 10 18
Evaluate Variable Expressions with Positive and Negative Numbers Evaluate each expression. 101) 7 x 3 when x 9 102) 16 2n when n 8
103) 5a 8b when a 2, b 6
104) x 2 5 x 4 when x 3
Translate English Phrases to Algebraic Expressions Translate each phrase into an algebraic expression and then simplify. 105) the product of 12 and 6 106) the ratio of 14 and 28 107) the quotient of 20 and the sum of p and q 108) the product of 3 and the sum of 7 and s
3.5 Decimal Numbers Name Decimal Numbers Name each decimal number. 109) 0.8 110) 0.375 113) −12.5 114) −4.09
111) 0.007
112) 5.24
Write Decimal Numbers Write as a decimal number.
115) three tenths
116) nine hundredths
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118) ten and thirty-five thousandths 120) negative five hundredths
119) negative twenty and three tenths
Locate Decimal Numbers on the Number Line Locate on the number line.
121) 0.6
122)
0.9
123) 2.2
124)
1.3
Order Decimal Numbers Order each of the following pairs of numbers, using < or >. 125) .6___.8 126) .2___.15 127) .803____.83 128) .56____-.562
Round Decimal Numbers
129) Round 12.529 to the nearest
a) hundredth
b) tenth c) whole number
130) Round 4.8447 to the nearest
a) thousandth
b) tenth
c) whole number
3.6 Operations with Decimal Numbers Add and Subtract Decimal Numbers Add or Subtract.
131) 5.75 8.46
132) 32.89 8.22
133) 24 19.31
134) 10.2 14.631
Multiply Decimal Numbers Multiply.
135) (0.3)(0.7)
136) (6.4)(0.25)
Divide Decimal Numbers Divide.
137) 0.8 0.2
138) 1.65 0.15
Convert Fractions to Decimals Convert each fraction to a decimal number. 3 5 1 143) 3
7 8 6 144) 11
139)
140)
151)
19 20
142)
21 4
Simplify Expressions using the Order of Operations Simplify.
145) (10.3 5.8) 4
146) 52.36(.5) 1.08(90)
147) 30 (.45 .15)
148) (.4) (.7) 2
2
Find the Mean of a Set of Numbers 149) Each workday last week, Yoshie kept track of the number of minutes she had to wait for the bus. The number of minutes were 3, 0, 8, 1, and 8. Find the mean
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150) On his last 6 homework assignments, Edison made the following number of errors: 2, 4, 1, 0, 1, and 1. Find the mean. 151) For the past 4 weeks, Jay’s paycheck was $270, $310.50, $243.75, and $252.15. Find the mean. 152) In the last 3 months, Raoul’s water bill was $31.45, $48.76, and $42.60. Find the mean.
Find the Median of a Set of Numbers Find the median of the numbers. 153) 41, 45, 32, 60, 58 154) 18, 7, 12, 16, 20 155) 25, 23, 24, 26, 29, 19, 18, 32 156) 52, 63, 45, 51, 55, 75, 60, 59 Use Decimal Numbers in Applications Solve.
157) Miranda got $40 from her ATM. She spent $9.32 on lunch and $16.99 on a book. How much money did she have left? 158) A pack of 16 water bottles cost $6.72. How much did each bottle cost?
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers For the given numbers, list the: a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers. 2 159) 5, , 0, 2, 4 3
160) 9, 5,
3 , 2, 0, 3 21 4
3.7 Square Roots Simplify Expressions with Square Roots Simplify. 161) 64 162) 144
163) - 25
164) - 81
165)
-9
166)
-36
167)
64 + 225
168)
64 + 225
169)
100q 2
170)
49 y 2
171)
4a 2 b 2
172)
121c 2 d 2
3.8 Properties of Real Numbers Use the Commutative Properties of Addition and Multiplication Use the commutative property to complete the following:
173) 6 4 ____
174) 14 5 ____
175) 3n ____
176) a 8 ____
Use the Associative Properties of Addition and Multiplication Use the associative property to complete the following:
177) (13 5) 2 _____
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179) (4 9 x ) x _____
180)
1 (22y ) _____ 2
Identify the Commutative and Associative Properties of Addition and Multiplication Identify whether each example is using the commutative or associative property. 2 3 2 3 181) (13 0.6) 1.4 13 (0.6 1.4) 182) 17 17 3 2 3 2 184) 5(9 x ) (59)x 183) 4 y 8 4 8 y Simplify Expressions Using the Distributive Property Simplify using the distributive property.
185) 7( x 9) 189)
1 (15n 6) 3
186) 9(u 4)
187) 3(6m 1)
188) 8( 7a 12)
190) ( y 10) p
191) (a 4) (6a 9)
192) 4( x 3) 8( x 7)
Identify the Identity Properties of Addition and Multiplication Identify whether each example is using the additive or multiplicative identity. 193) 35(1) 35 194) 29 0 29 Use the Inverse Properties of Addition and Multiplication Find the additive inverse. 3 195) 32 196) 19.4 197) 5 Find the multiplicative inverse. 1 9 199) 200) 5 201) 10 2
198)
7 15
202)
4 9
Use the Properties of Zero Simplify.
203) 830
204)
0 9
205)
5 0
206) 0
2 3
Identify the Properties of Real Numbers Identify the property of real numbers that justifies each statement. 207) 2 12 12 ( 2) 208) 11 (3 d ) (11 3) d
209) 1n n 2 2 11x 11 x 11 11 213) 9(7 x ) 63 9 x
211)
210)
3 8 1 8 3
212) 23k 0 0 214) 3 y 19 319 y
Simplify Expressions Using the Properties of Real Numbers Simplify. 5 13 215) 43 39 ( 43) 216) (n 6.75) 0.25 217) 57 13 5 2 3 220) 9(6 x 11) 15 219) 28 3 7
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Chapter 3 Review Exercises Answers 3.1 1-6)
3
1
1
3
7
2 5| 2 2 4 3 | | | | | | | | | 4
7) < 11) >
3
2
1
0
9) > 10) < 13a) 6 13b) 6 3 3 14a) 2 14b) 2 15a) 15b) 8 8 5 5 16b) 17) 14 18) 27 16a) 6 6 19) 33 20) 4 21) < 22) < 23) = 24) < 25) 7 26) 16 27) 4 28) 53 29) 16 30) ( 8) 31) 3 32) 19 ( 12) 33) 10 degrees
1
2
3
4
5
8) < 12) >
34) 85 feet
3.2 35) 10 36) 4 37) 1 39) 96 40) 100 43) 1 44) 49
38) 9 41) 50
42) 26 37 45) 1 46) 40 48 a) 6 48b) 12
47a) 3 47b) 16 1 50) 49 51) 8 2; 6 49a) 0 49b) 2 52) 12 4; 8 53) 10 [ 5 ( 6)]; 1 54) [3 ( 5)] 18; 16 56) 96
55) 16 degrees
3.3 57) 5 58) 1 61) 8 62) 28 65) 58 69) 2 70) 3
59) 7 60) 5 63) 38 64) 29 66) 47 67) 1 68) 40 71 a) 2 71b) 11 1 72a) 5 72b) 26 73a) 0 73b) 3 76) 50 23; 73 74) 4075) 12 5; 17 78) 106 degrees
77) 2 degrees
3.4 79) 36 10 83) 9
80) 35 81) 121 82) 8 3 1 84) 85) 4 86) 7 7
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87) 7 88) 20 89)
2 3
90) 0 93) 81 94) 360 97) 4 98) 27
91) 9 92) 32 95) 54 96) 42 31 99) 3 100) 4 103) 58 107)
101) 66
104) 2
20 pq
102) 32
105) 12 6; 72
106)
14 1 ; 2 28
108) 3( 7 s )
3.5 109) eight tenths 110) three hundred seventy-five thousandths 111) seven thousandths 112) five and twenty-four hundredths 113) negative twelve and five tenths 114) negative four and nine hundredths 115) 0.3 116) 0.09 117) 0.27 118) 10.035 120) –0.05 119) 20.3 121-124) 0.9 0.6 2.2 | 1.3 | | | | | | | | 1.5
1.0
0.5
0
125) < 126) > 127) < 128) > 129a) 12.53 129b) 12.5 129c) 13 130b) 4.8 130c) 5
0.5
1.0
1.5
2.0
2.5
130a) 4.845
3.6 131) 14.21 135) 0.21 139) 0.6 __
132) 24.67 136) 1.6 140) 0.875
133) 4.69 134) 24.831 137) 4 138) 11 142) 5.25 141) 0.95
___
144) 0.54 145) 18 146) 123.38 143) 0. 3 147) 50 148) 0.65 149) 4 150) 1.5 151) $269.10 152) $40.94 153) 45 154) 16 155) 24.5 156) 57 157) $13.69 158) $0.42 2 159b) 5,0, 4 159c) 5, ,0, 4 159d) 2 159a) 5,0 3 2 3 160a) 9, 0 160b) 9, 2, 0 160c) 9, , 2, 0, 3 21 159e) 5, ,0, 2, 4 3 4 3 160e) 9, 5, , 2, 0, 3 21 4
160d)
5
3.7 161) 8 162) 12 163) 5164) 9 165) not a real number 166) not a real number 167) 23 168) 17 170) 7 y 171) 2ab 172) 11cd 169) 10q
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3.8 173) 4 6
174) 5( 14)
175) n 3
176) 8 a
1 180) 22 y 2 181) associative 182) commutative 183) commutative 184) associative 185) 7 x 63 186) 9u 36 187) 18m 3 188) 56a 96 189) 5n 2 190) yp 10 p 191) 5a 13 192) 4 x 68 193) multiplicative identity 194) additive identity 3 7 195) 32 196) 19.4 197) 198) 5 15 2 1 9 200) 201) 10 202) 199) 9 5 4 203) 0 204) 0 205) undefined 206) 0 207) commutative property of addition 208) associative property of addition 209) multiplicative identity 210) multiplicative inverse 211) associative property of multiplication 212) property of zero 213) distributive property 214) commutative property of multiplication 217) 57 218) 34 215) 39 216) n 7 219) 8 220) 54 x 84
177) 13(5 2)
178) 22 (7 3)
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Chapter 3 Practice Test 3 1 1 , - 2 , 1.5, on the number line. 4 3 2 3 find its 2) For the number 5 a) opposite______ b) reciprocal_____
1) Locate
c) absolute value______
Simplify 3) 4 - 9
4) -8 + 6
5) -15 + (-12)
6) -7 - (-3)
7) 10 - (5 - 6)
8) -3 ⋅ 8
9) -6(-9)
10) 70 ¸ (-7)
11) -
13) 15.4 3.02
14)
16) 0.6 0.3
17) (.5)(24.5 - 3.16)
0 8 15) 20 5.71 æ 4ö 3 18) (-29)çç ÷÷÷ çè 3 ø 4 21) 6(5 x - 4) 12)
24)
4 0
7(-5) + 3 ⋅ 9 2-4 22) m(n 2)
19)
25)
36 64
5 æç 3 ö÷ ¸ ç- ÷ 12 çè 4 ÷ø
0.64 0.3
20) -3 + 15 y + 3 23) (12a + 4) - (9a + 6) 26)
144n 2
Evaluate
27) x
when x = -23
29) 35 - a when a = -4
2 2 28) - + y when y = 3 3 30) (-2r )2 when r = 3
a+b when a = -25, b = 15 10 5 as a decimal number. 34) a) Write 33) 3m - 2n when m = 6, n = -8 8 b) Write 1.73 as a fraction. 35) Write in order from smallest to largest: .25, .4, .1, .209 7 5 31) - f when f = 3 11
32)
36) Round 16.749 to the nearest a) tenth b) hundredth c) whole number 1 5 37) From the numbers -4, - 1 , 0, , 2, 7 which are 2 8 a) Integers? b) Rational numbers? c) Irrational numbers? d) Real numbers? 7 38) For the number , find its 2 a) additive inverse ______ b) multiplicative inverse ________ 39) Identify as an example of the commutative property or the associative property: b) x ⋅ 14 = 14 x c) (8 ⋅ 2) ⋅ 5 = 8 ⋅ (2 ⋅ 5) a) ( y + 6) + 3 = y + (6 + 3) 40) Early one morning, the temperature in Ithaca was –4o. By afternoon, it had risen 15o. What was the afternoon temperature?
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41) One day last winter, the temperature in Santa Ana was 84 o and the temperature in Boston was –3o. What was difference in temperature between Santa Ana and Boston? 42) Three friends went out to dinner and agreed to split the bill evenly. The bill was $79.35. How much should each person pay? 43) The price of a box of cereal is $4.29. Vicky has a coupon for $0.35 off, and the store will double the coupon. How much will Vicky pay for the cereal? 44) Tamar had 86, 91, and 99 on her first three math quizzes. What was the mean of her quiz scores? 45) For six days last week the number of customers in Sam’s sandwich shop at noon were 28, 15, 22, 29, 35, and 30. What was the median?
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