MAT001 – Chapter 3 - Decimals

Decimal Fractions

Section 3.1

Decimal fractions are fractions with 10, 100, 1000, and so on in the denominator. 8 63 450 , , and 10 100 1000

Using Decimal Notation

Decimal fractions can be written in many ways.

1 10 (in fractional form) 0.1 (in decimal form)

One-tenth (in words)

1

CQ3-01. What decimal is indicated by the word “five and four hundred sixtyseven ten-thousandths”? 5.467 1.

Place Values Hundreds

Tens

Ones

Decimal Point

Tenths

Hundredths

Thousandths

Ten-Thousandths

3

4

9

.

3

8

6

1

100

10

1

“and”

1 10

1 100

2

1 1 1000 10000

The decimal 349.3861 can be written in words as “three hundred forty-nine and three thousand eight hundred sixty-one ten-thousandths.”

2.

5.0467

3.

5,467,000

4.

0.5467

3

Fractions & Equivalent Decimals

Changing from Fractions to Decimals Example: 61 1. Write the fraction 100 as a decimal.

Note the relationship between fractions and their equivalent numbers’ decimal form. 3 10

0.3

one zero

49 1000

61 100

0.049

0.61

2. Write the fraction 5

three zeros one decimal place

4

three decimal places

5

27 1000

three zeros 5

27 as a decimal. 1000

5.027 three decimal places 6

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MAT001 – Chapter 3 - Decimals

Changing from Decimals to Fractions

CQ3-02. Write 0.0155 in fractional form and reduce, if possible.

Example: 1. Write the decimal 0.371 as a fraction.

0.371

1.

371 1000

2. 3.

2. Write the decimal 4.0038 as a fraction.

4.0038

38 10000 19 4 5000

4

4.

3 2000 31 200 31 2000 3 20

Simplify when possible. 7

8

Comparing Decimals Comparing Two Numbers in Decimal Notation

Section 3.2

1. Start at the left and compare corresponding digits. If the digits are the same, move one place to the right. 2. When two digits are different, the large number is the one with the larger digit.

Comparing, Ordering, and Rounding Decimals

The numbers in the tenths place are both 3. The numbers in the hundredths place are different.

1.345

1.353

Since 4 < 5, we know that 1.345 < 1.353. 9

CQ3-03. Place the set of numbers 0.709, 0.71, 0.79, and 0.079 in the proper order from smallest to largest.

Ordering Decimals Example: Place the following four decimals in order from smallest to largest.

1.

786.848 786.800 786.480

10

0.079,0.709,0.71,0.79

2. 0.079,0.79,0.709,0.71 Add zeros to make the comparison easier.

3.

0.79,0.709,0.71,0.079

4.

0.79,0.079,0.709,0.71

786.484

Rearrange with the smallest first. 786.48, 786.484, 786.8, 786.848 11

12

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MAT001 – Chapter 3 - Decimals

Rounding Decimals

Rounding Decimals

Rounding Decimals 1. Find the decimal place (units, tenths, hundredths, and so on) for which rounding off is required. 2. If the first digit to the right of the given place value is less than 5, drop it and all digits to the right of it. 3. If the first digit to the right of the given place value is 5 or greater, increase the number in the given place value by one. Drop all digits to the right of this place. 346.74 8

Round the 4 up to 5.

346.74 2

hundredths place

Do not round the 4 up.

Example: 1. Round 198.438 to the nearest tenths. 198.4

198.438

2. Round 45.29502 to the nearest hundredths. 45.30

45.29502

hundredths place

346.74

346.75

13

14

CQ3-04. Round 57.30945 to the nearest thousandth.

Section 3.3

1. 57.3095

Adding and Subtracting Decimals

2. 57.309 3. 57.31

4.

57.3 15

Adding Decimals

Adding Decimals

The addition of decimals is related to the addition of fractions.

3 4 10 10

7 10

Example: Add 718.97 + 496.5.

0.3 0.4 0.7

718.97 + 496.5 0 1215.47

Adding Decimals 1.

2. 3.

16

Write the numbers to be added vertically and line up the decimal points. Extra zeros may be added to the right of the decimal points if needed. Add all the digits with the same place value, starting with the right column and moving to the left. Place the decimal point of the sum in line with the decimal points of the numbers added.

Write the numbers vertically and line up the decimal points. Add zeros where necessary for place holders.

This value is the same as 1215 47 . 100

17

18

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MAT001 – Chapter 3 - Decimals

CQ3-05. Add:

1

The subtraction of decimals is related to the subtraction of fractions. 4 3 1

1.

0.292

2.

0.382

3.

16.123

4. 2

5

4

5

2

10

3

10

5.4 2.3 3.1

10

Subtracting Decimals 1.

0%

0%

2

3

1

6

Write the numbers to be subtracted vertically and line up the decimal points. Extra zeros may be added to the right of the decimal points if needed. Subtract all the digits with the same place value, starting with the right column and moving to the left. Borrow when necessary. Place the decimal point of the difference in line with the decimal points of the two numbers being subtracted.

2. 0%

7.123 3

Subtracting Decimals

9 + 6.9 + 0.223

7

8

9

10

11

12

13

14

0%

3.

4

15

19

20

Subtracting Decimals

Subtracting Decimals

Example: Subtract 243.967 – 84.2.

Example: Subtract 8 – 1.623.

13

9 9 10 7 10 10 10

1 3 13

243.967 – 84.200 159.767

8. 000 – 1.623 6.377

Write the numbers vertically and line up the decimal points Add zeros where necessary for place holders.

Write the numbers vertically and line up the decimal points. Add zeros where necessary for place holders.

This value is the same as 159 767 . 1000

21

CQ3-06. Subtract: 6.004 – 0.4 1.

CQ3-07. Simplify: 6 – 0.881 + 4.3

2.004

2.

7.78

2.

2.381

3.

2.219

5.604

4.

0%

5.964

0%

0%

2

3

1

2

1. 6

3.

1

22

3

4

5

6

7

8

9

10

11

12

13

14

15

0%

0%

4.

4

23

1

2

9.419 3

4

5

6

0%

0%

2

3

1

7

8

9

10

11

12

13

14

15

0% 4

24

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MAT001 – Chapter 3 - Decimals

CQ3-08. Mary Ann follows her investments in the stock

CQ3-09. Boston had five snowstorms in one month, whose

market daily. Over a five-day period, she gained $98.87, gained $14.24, lost $88.18, gained $23.44, and lost $49. How much did she lose overall?

snowfall amounts were: 16.3 inches, 10.2 inches, 4.25 inches, 22.3 inches, and 8.9 inches. The normal amount of snow for the month was 22.35 inches. How many more inches of snow than normal fell that month?

1.

$22.01

1.

29.6 inches

2.

$100.37

2.

39.6 inches

3.

$0.63

3.

49.6 inches

4.

$50.49

4.

61.95 inches

0%

0%

0%

2

3

1

0%

0% 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0%

0%

2

3

0%

4

25

1

2

3

4

5

6

7

8

9

10

11

12

13

14

4

15

26

Multiplying Decimals

Section 3.4

The multiplication of decimals is related to the multiplication of fractions. 3 4 12

10 100

Multiplying Decimals

1000

0.3 0.04 0.012 one decimal place

two decimal places

Multiplying Decimals 1. 2. 3.

Multiply the numbers just as you would multiply whole numbers. Find the sum of the decimal places in the two factors. Place the decimal point in the product so that the product has the same number of decimal places as the sum in step 2. You may need to write zeros to the left of the number found in step 1.

27

Example: Multiply 0.043

0.4.

0.17 0.4 0.068

28

Multiplying Decimals

Multiplying Decimals Example: Multiply 0.17

three decimal places

2 decimal places 1 decimal place

0.012.

0.043

3 decimal places

0.012

3 decimal places

86 43 0.000 516 5 decimal places in product (3 + 3 = 6)

3 decimal places in product (2 + 1 = 3)

29

Note the we need to insert three zeros before the 516.

30

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MAT001 – Chapter 3 - Decimals

1. 1.9575

1.

$96.54

2. 19.5777

2.

$92.51

3.

$99.95

4.

$88.00

3. 0.19575

0%

0%

0%

2

3

1

4. 1

2

CQ3-11. Harold purchased 22.4 gallons of gasoline at a price of $4.13 per gallon. How much did he pay? Round to the nearest cent.

7.251 0.027

CQ3-10. Multiply:

0%

0%

4

0.195777 3

4

5

6

7

8

9

10

11

12

13

14

15

31

1

2

3

4

5

6

7

1

8

9

10

11

12

13

14

0%

0%

2

3

0% 4

15

32

Multiplying by a Power of 10

Multiplying by a Power of 10

Multiplication of a Decimal by a Power of 10 Observe the following pattern. 101 = 0.025

0.025

10 = 0.25 two zeros

102 = 0.025

0.025

Decimal point moved two places to the right.

103 = 0.025

Example: Multiply 4.59

100 = 2.5 three zeros

0.025

To multiply a decimal by a power of 10, move the decimal point to the right the same number of places as the number of zeros in the power of 10.

Decimal point moved one place to the right.

one zero

4.59

Decimal point moved three places to the right.

1000 = 25

10,000.

10,000 = 45,900. = 45,900 four zeros

Decimal point moved four places to the right.

33

2.04518 103

CQ3-12. Multiply:

Section 3.5

1. 204.518 2. 0.00204518 3.

Dividing Decimals

2045.18

4. 0.0204518

0%

0%

0%

2

3

1

1

34

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0% 4

35

36

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MAT001 – Chapter 3 - Decimals

Dividing by a Whole Number

Dividing by a Whole Number When dividing a decimal by a whole number, place the decimal point for the quotient directly above the decimal point in the dividend.

2.16 . 8 17.28 16 12 8 48 48 0

Example: Divide 0.104

The decimal points are aligned, one above the other.

4. Note the zero after the decimal point.

0.026 4 0.104 8 24 24 0

Divide as you normally divide whole numbers.

37

38

Dividing by a Decimal

Dividing by a Whole Number Example: Divide 263.82

When dividing a decimal by another decimal, the division problem is converted into an equivalent problem that has a whole number as a divisor.

7 and round to the nearest hundredths.

Dividing by a Decimal

Carry out the division to the thousandths place, and then round.

37.688 7 263.82 0 21 53 49 48 42 62 56 Note that the remainder is not 60

An extra zero was added to carry out the division to the required place.

37.688 is rounded to 37.69 (the nearest hundredths).

zero.

1.

Make the divisor a whole number by moving the decimal point to the right. Mark that position with a caret ( ). Count the number of places the decimal point moved.

2.

Move the decimal point in the dividend to the right the same number of places. Mark that position with a caret.

3.

Place the decimal point of your answer directly above the caret marking the point of the dividend.

4.

Divide as with whole numbers.

39

Dividing by a Decimal Example: Divide 4.209

2.3 . 1.83 4.20 9

Divide as usual.

CQ3-13. Divide and round your answer to the nearest tenth: 9.67 0.34

1.83.

1.83. 4.20.9 Place the decimal point of the answer directly above the caret.

40

366 549 549 0

1.

32.3

2.

28.4

3.

22.5

Move each decimal point to the right two places.

Mark the new position by a caret ( ).

0%

4. 41

1

2

28.7 3

4

5

6

0%

0%

2

3

1

7

8

9

10

11

12

13

14

15

0% 4

42

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MAT001 – Chapter 3 - Decimals

CQ3-15. A butcher has 66 pounds of meat. He wants to pack the meat into packages weighing 0.75 pounds each. How many packages can he make? 1. 49.5 packages

CQ3-14. Find the value of x:

0.15x 8.25 1.

55

2.

0.55

3.

0.1375

2. 80 packages 3. 86 packages 0%

75

4. 1

2

3

4

0%

0%

2

3

1

5

6

7

8

9

10

11

12

13

14

0%

0%

4. 88 packages

4

15

43

1

2

3

4

5

6

7

8

9

0%

0%

2

3

1

10

11

12

13

14

0% 4

15

44

Equivalent Fractions and Decimals A number can be expressed in two equivalent forms: as a fraction and as a decimal.

Section 3.6

Fraction

Converting Fractions to Decimals and the Order of Operations

Decimal

1 4

3.25

three and one-fourth

three and twenty-five hundredths

3

Same quantity, difference appearance

Common equivalent fractions and decimals 1 2

1 4

0.5

1 5

0.25

0.2

1 10

0.1

45

Converting a Fraction to a Decimal

46

Terminating and Repeating Decimals 3 8

Common equivalent fractions and decimals 1 2

0.5

1 4

0.25

1 5

0.2

1 10

0.1

Converting a Fraction to an Equivalent Decimal Divide the denominator into the numerator until a) the remainder becomes zero, or b) the remainder repeats itself, or c) the desire number of decimal places is achieved. 47

1 16

0.375

0.0625

Terminating decimals (The remainder is zero when converting the fraction into a decimal.) repeating digit

1 3

0.333

repeating group of digits

0. 3

13 22

0.59090

0.5 90

Repeating decimals (When converting, the remainder is a digit or group of digits that repeats.) 48

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MAT001 – Chapter 3 - Decimals

Converting a Fraction to a Decimal Example: Write 5 as an equivalent decimal. 18 0.277 5 18 5.000 18 36 140 repeating 126 remainders 140

1.

0.9375

2.

0.8375

3.

0.3125

4.

1.066 6

0.27

2

3

4

5

6

7

0.6

2.

1.6

3.

0.625

4.

0.5125

1

2

3

4

3.

1

2

3

4

5

6

0%

0%

2

3

1

8

9

10

11

12

13

9

10

11

12

13

14

4

15

50

14

0%

0.75 ___ > 0.7

4

15

51

52

Order of Operations Order of Operations Do first

Do last

1. 2. 3. 4.

Perform operations inside any parentheses. Simplify any expressions with exponents. Multiply or divide from left to right. Add or subtract from left to right.

Example: Evaluate 9.6 + 3.6 – (0.4)2. 0%

0%

0%

2

3

1

7

8

0%

Change the fraction into a decimal for easier comparison.

22 3 7

4.

7

3

3 ____ 0.7 4

22 3.14 7 22 3.14 7 22 3.14 7

2.

6

0%

2

Example: Fill in the blank with one of the symbols .

CQ3-18. Select the correct statement. 1.

5

0%

1

Ordering Fractions and Decimals

15 as a decimal. 16

0%

1

1.

0%

5 18

49

CQ3-17. Write

5 as a decimal. 8

CQ3-16. Write

8

9

10

11

12

13

14

15

0% 4

53

= 9.6 + 3.6 – 0.16

Exponents

= 13.2 – 0.16

Addition

= 13.04

Subtraction 54

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MAT001 – Chapter 3 - Decimals

Order of Operations

CQ3-19. Perform the operations in the proper order: 2

(0.24 )

Example: Evaluate (2.4)2 + 3.6 (1.2 – 0.7). (2.4)2 + 3.6

1.

0.068

2.

0.012

3.

0.3

4.

0.392

0.16 0.32 0.1

(1.2 – 0.7) = (2.4)2 + 3.6 = 5.76 + 3.6

0.5 0.5

Parentheses Exponents

= 5.76 + 7.2

Division

= 12.96

Addition

0%

55

1

2

3

4

5

6

0%

0%

2

3

1

7

8

9

10

11

12

13

14

0% 4

15

56

Problem Solving Steps 1. Understand the problem.

Section 3.7

a) Read the problem carefully. b) Draw a picture if this is helpful. c) Fill in the Mathematics Blueprint so that you have the facts and a method of proceeding in this situation.

Solving Applied Problems Using Decimals

2. Solve and state the answer. a) Perform the calculations. b) State the answer, including the unit of measure.

3. Check. a) Estimate the answer. b) Compare the exact answer with the estimate to see if your answer is reasonable. 57

Mathematics Blueprint

Mathematics Blueprint #1 Example:

The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do.

Kelsey is making holiday cookies, which she will give as gifts to her friends and co-workers. She has made 34.5 pounds of cookies, which need to be divided up into equal packages containing 0.75 pound of cookies. How may packages can Kelsey make?

Mathematics Blueprint for Problem Solving Gather the Facts

What Am I Asked to Do?

How Do I Proceed?

58

Key Points to Remember

Mathematics Blueprint for Problem Solving Gather the Facts

What Am I Asked to Do?

Find the number There are 34.5 of packages lbs of cookies and each package Kelsey can make. is 0.75 lb.

59

How Do I Proceed?

Key Points to Remember

Divide 34.5 by 0.75.

Move the decimal point over two places before dividing.

Example continues. 60

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MAT001 – Chapter 3 - Decimals

Mathematics Blueprint #1

Mathematics Blueprint #2 Example:

Kelsey is making holiday cookies, which she will give as gifts to her friends and co-workers. She has made 34.5 pounds of cookies, which need to be divided up into equal packages containing 0.75 pound of cookies. How may packages can Kelsey make?

34.5

0.75

0.75 34.5

David bought apples and pears at the grocery store for a fruit salad. At the checkout counter, the apples weighed 2.7 pounds and the pears weighed 1.8 pounds. If the apples cost $1.29 per pound and the pears cost $1.49 per pound, how much did David spend on fruit? (Round your answer to the nearest cent.)

the number of packages 46 0.75 34.50 300 Kelsey can make 450 46 packages. 450 0

Mathematics Blueprint for Problem Solving Gather the Facts

What Am I Asked to Do?

How Do I Proceed?

Key Points to Remember

2.7 lbs of apples costing $1.29/lb.; 1.8 lbs of pears costing $1.49/lb.

Find the total cost Apples: multiply David spent on 2.7 by 1.29; the fruit. Pears: multiply 1.8 by 1.49; Add the two products.

Multiplication comes before addition.

Example continues. 62

61

CQ3-20. Dennis and Rob rented a van and drove from Boulder,

Mathematics Blueprint #2

CO to Palo Alto, CA. The cost was $300 for the first two weeks and then $50 per day after that. Also, they were not charged a mileage fee for the first 2,000 miles, but had to pay 12 cents a mile for every mile after that. They took their trip, and it took 19 days. Also, they drove 2,350 miles total. What was the total cost of the van rental?

David bought apples and pears at the grocery store for a fruit salad. At the checkout counter, the apples weighed 2.7 pounds and the pears weighed 1.8 pounds. If the apples cost $1.29 per pound and the pears cost $1.49 per pound, how much did David spend on fruit? (Round your answer to the nearest cent.)

Apples: 1.29 2.7 903 2580 3.483

Pears:

1.49 1.8 1192 1490 2.682

3.483 2.682 6.165 David spent $6.17 on the fruit.

1.

$592

2.

$832

3.

$550

0%

0%

0%

2

3

1

4. 63

1

2

0% 4

$342 3

4

5

6

7

8

9

10

11

12

13

14

15

64

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