Academir Charter School Middle

8th Grade 2014 Summer Math Packet

x

Math

111111111111 Academir Charter School Middle is committed to fostering continuous student learning, development and achievement. This commitment extends through the summer as to prevent the decay of learning. Research indicates that the summer break may have detrimental learning effects for many students. On average, all students lose skills, particularly in mathematics. It is crucial that parents are diligent during the summer to ensure their students' educational growth and success. Given the connection between the importance of sustaining academic skills, summer learning loss and parental involvement, it is reasonable to assume that structured academic activities during the summer would help mitigate this loss and may even produce gains. In an effort to promote and reinforce mathematics learning and retention during the summer months, Academir Charter School Middle is implementing a Summer Math Packet for all grade levels. The Summer Math Packet is a series of worksheets designed to review basic math skills and keep students thinking mathematically over the summer break. The skills addressed in the packet are important to your student moving to the next level of mathematics. Please have your student follow all directions in the packet and have him/her show all work on an organized and labeled notebook paper, as needed. The students are to bring the completed packet and notebook paper showcasing their calculations of problems within the packet with them on the first day of school. Academir teachers will review the packet during the first few days, which will be followed with a diagnostic assessment to gauge students mathematical knowledge and level.

Fractions: Adding and Subtracting Fractions are found on all standardized tests and in all types of applications. Fractions are how the world deals with parts of a whole. Fractions may be proper fractions, improper fractions, or mixed numbers. The power of the number one is the secret of fractions. The number one may take on many different appearances. The equivalent fractions of one have the same numerator and denominator. So there is a series of fractions all equal to one that looks like: = =2 = .. 8 =.

g

To add or subtract fractions, you must have common denominators. The numerators are what you add or subtract. Therefore, the denominators are just labels that must be the same. After having common denominators, add (or subtract) the numerators. That answer is this number over the common denominator. Examples:

To add:

3xa 3 7" 4 ^ 3

Q

2

To subtract:

,

12 17 „5 12 = 1 12

—

),

6x

12

4x

12 7 12

Find the sum or difference and reduce to lowest terms. Write the answer on the line.

=

9. 10i

6-2=

—

10. lag —

3i =

=

2.3+2 =

6.

3. -6 + =

7. 31- — 1/ =

11. 253 —214=

4.24+ =

8. — =

12. 3-15 —

=

Fractions: Multiplying and Dividing (— One way of thinking about multiplication is that the answer is the area of a rectangle. When fractions are multiplied, the answer is part of a rectangle with sides of the two factors. (Picture a square 1 by 1, with the shading of 2 of 34.) When multiplying 2 by 4, the answer is a a 3x3 Example: 4 X 8 - 4 x 8 = 32 To divide fractions, take the reciprocal of the divisor and multiply. This algorithm will give you an answer, but what does it mean? There are several ways to look at the answer. A square pizza is cut into 4 rows and 4 columns, so one piece would be 1 16- of that pizza. 34 = 4 ; there was only 4 of a pizza and you eat a quarter of it. You had 3 pieces.

g.

Example:

3 2 3 3 a 8 = 3 = 8 X 2= 16

With mixed numbers, change them to improper numbers and follow the Order of Operations. Examples:

42x13=3x1=f= 6 1 2-15 = 1i=

Find the product or quotient and reduce to lowest terms. Write the answer on the line.

1.2x4=

5 .3÷g=

2.3x 14=

9.14x5=

10.15-÷4=

3.5x4=

7.12x3=

11. 14. ÷

4. 4=2=

8.23x 11=

12. 8i ÷ 1 14 =

13. What is the area of a poster that is 12 feet by 22 feet?

=

14. What is the area of a desktop that is

22 feet by 5 feet?

Decimals Decimal operations are special kinds of fractions. The place value is critical, and basic whole number operations are fundamental in understanding decimals. With addition and subtraction, the alignment is important. Examples: A. 386.37 + 86.305 =

C.

B. 28.06 - 5.802 =

386.370

28.060

+ 86.305

- 5.802

472.675

22.258

4.81 x 2.3 = 4.81

D. 10.53 + 3.9 = 2.7

x 2.3 9620

3.9)10.53 . =73 273

11.063

- 273

1443

0 Determine the correct answer to the following problems. Write the answer on the line.

1. 2.7 + 8.7 =

S. 8.45 - 2.795 =

2. 18.3 + 2.55 =

6. 15 - 9.372 =

10. 21.6 4- 0.3 =

3. 73.67 + 42.072 =

7. 5.3 x 0.48 =

11. 0.95 + 2.5 =

4. 3.7 - 2.3 =

8. 18 x 2.85 =

12. 39.15 + 0.06 =

3

9. 28.6 x 3.07 =

Scientific Notation (

Scientists and engineers work with very large and very small numbers. Since these numbers take up a lot of space, a shorthand way of writing these numbers is needed. This method is called scientific notation. Scientific notation is based on the fact that multiplying or dividing by 10 moves the decimal point right and left. Another way of saying this is that scientific notation is based on the powers of ten. So a large number such as 37,500 would be expressed as 3.75 x 10 4 . The numerical part is a number between 1 and 10, times the number of places the decimal point moves. If the number is larger than 10, then the exponent will be positive. If the number is smaller than 1, then the exponent will be negative. Examples: Express the following in scientific notation:

37,854 = 3.7854 x 104

0.00216 = 2.16 x 10 -3

Express the following in standard notation: 5.2 x 104 = 52,000 7.84 x 10-3 = 0.00784 Express each of the following in scientific notation. Write the answer on the line. 1. 54,300 =

S. 0.00033 =

2. 703,000 =

6. 0.0628 =

3. 753 =

7. 0.00000035 =

4. 15,000,000 =

8. 0.0205 =

Express each of the following in standard notation. Write the answer on the line.

9. 1.56 x 104 =

12. 7.89 x 10-4 =

10. 6.033 x 103 =

13. 2.46 x 10-2 =

11. 8.566 x 106 =

14.4.79 x 10-6 =

ii

Using Variables A variable is a letter or symbol that stands for a number. This number may change or may be a single number. When one of more variables are placed together, like a x b, this means a times b. A number with variables is referred to as a coefficient. You will find that variables will be used as a translation of sentences into algebraic exp ressions. The basic phrases are: The sum of 7 and 3 is 10. + (addition) as in 7 + 3 = 10 The difference of 7 and 3 is 4. - (subtraction) as in 7-3 = 4 x or • (multiplication) The product of 7 and 3 is 21. as in 7 x 3 =21 For / (division)

The quotient of 12 and 3 is 4. The quotient of 3 divided into 12 is 4. The quotient of 12 divided by 3 is 4.

Example: Write an expression for the following: A. the sum of a number and 12 x+ 12

as in 33-1f or 12 =4 4 3 as in 311.as in 12 =4 3

B. the product of 5 and a number 5xx

Write the following expressions using n as the number.

1. a sum of a number and 5

4. seven less than a number

2. the difference between 15 and a number

5. the quotient of a number and 6 6. eight times a number

3. a number increased by 15 Example: Write equations and solve for the variable using n as the variable. A. Three more than a number is 10. n + 3 = 10 n + 3 - 3 = 10 - 3 n=7

B. The product of 7 and a number is 21. 7 xn= 21 7 xn+ 7 =21 4-7 n= 3

Write the following expressions using n as the number.

7. Four less than a number is 6.

10. Seven times a number is 56.

8. The sum of -8 and a number is 17.

11. The product of 3 and a number less 4 is 8.

9. The quotient of 28 and a number is 4. 12. A quarter of a number plus 3 is 10.

5

Exponents When working with variables, you will find that you need to understand exponents. Exponents are mathematical expressions for any real number b and any positive integer n, where b is the base and n is the exponent. 34 = 3 x 3 x 3 x 3 = 81. bn =bxbxbx...xb 3 is used as a factor four times. A zero exponent:

b° = 1, if b is any nonzero real number. 5 ° = 1

Negative exponent: Properties of Exponents Examples: Product Rule:

53 = 5 = 115

=

32 x 34

b""

bon x

bm

Quotient Rule:

b"

brn = b(n -m)

Power Rule:

(ab)" = a" x b"

36

or

57 ÷ 53 = 57-3 = 54

(3x)4

34x4 = 81 x4

Simplify the following expressions. 1. 23 x 22=

5. 73 x 73=

2. 32 x 34=

6. n2 x n5=

3. 52 x 53=

7. n4 x n6=

4. (-2)3 x (-2)2=

8. (3)0 4.

Evaluate each expression when x = -2 and y = 4.

9. x=

=

13.y=

10. (xy) 2=

14. (x + y)3=

11. x4 x )e=

15. (-54 3=

12. y3 x

16. (-4y)x=

Order of Operations In problems with more than one operation involved, you will need to determine which order of operations will be performed. The rule for order of operations has been established. Order of Operations

1. Perform operations inside parentheses or grouping symbols. 2. Simplify terms with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Example:

Simplify (9 + 7) — 3(4 x 5 — 3 x 6) 2 (9 + 7) — 3(4 x 5 — 3 x 6) 2 _ 16 — 3(4 x 5 — 3 x 6) —2 _ 16 — 3(20 — 18) —2 _ 16 — 3 x2 —2 =8—6 =2

Work inside the parentheses first. Multiply and divide within parentheses. Complete subtraction within parentheses. Divide and multiply, left to right. Subtract.

Simplify each expression.

1. 6 + 2 x 8 =

10. (3 + 5)5 — 20 + (2 + 5) =

2.18 / 2 —7=

11. 3 x 9 —2 x 8 + 5 x 6 =

3.15 + 9/ 3 =

12. 3(1 + 6 + 2(5 + 2)) =

4. 5 — 32 / 16 =

13. 12 x 4 — 24 =

5.7 + 3 x 5— 11 =

14. (22 + 32) x (1 + 1) 2 =

6. 14 — 3(15 — 8) =

15. 2(3(9 — 5)) =

7. 8 x4+ 3 x5 =

16. 2 + 5 x 8 —15 =

8.24 / 4— 18 / 9 =

17. 2 x 5 + 15 —3 =

9. 4(4 + 5) + 7(7 — 5) =

18. 62 + 6 + 12=

Real Numbers You should review the properties of the classification of real numbers. Example: Look at the numbers -3.8,1 , 0, -7C, Thrj, 57, - 14. Natural: Whole: Integers: Rational: Irrational: Real:

57 0, 57 0, 57, 14 0, 57, 14, -

-

1-

-1

0, 57, 14, 3.8, Tr, -

-

-

-453

numbers used to count natural numbers and zero whole numbers and their opposites integers and terminating and nonrepeating decimals infinite, nonrepeating decimals rational and irrational numbers

Name the set(s) of numbers to which each number belongs. 7

1. 9

12 5

2. 41.25

8. -■1100

3. 52

9.

4. 0

10. 6.25

5. V

11. 271

6. -40

12. 0.121221222...

Give an example of each kind of number and where you would use it.

13. A whole number 14. A negative integer 15. A rational number 16. A positive fractional number

The different number systems were invented to solve problems that could not be solved in their current number system. Identify a number system that would be needed to solve the equations below. d. x2 = 15 b. x+ 12 = 5 c. 8x = 4 a. x + 3 = 7

B

Adding and Subtracting Integers Review the rules for adding integers. • To add two numbers with the same sign, add their absolute values. The sum has the same sign as the numbers. • To add two numbers with opposite signs, subtract their absolute values. The sum has the same sign as the number with the greater absolute value. Review the rules for subtracting integers. • To subtract numbers, rewrite the problem to add the opposite of the num bers. • Follow the rules for adding integers. Example B:

Example A: 9+ 4

12 — 21 12 + (-21)

-

9 — 4 Find the difference of the their absolute values. = 5 Difference -5 Since -9 has the greater absolute value, the answer takes the negative sign.

= 21 — 12 =9 -9

Rewrite the problem to add the opposite of ( +21). The difference of their absolute values Subtract Since (-21) has the greater absolute value, the answer takes the negative sign.

Simplify each expression. Check the sign of your answer.

1. 5 + (-4)

4. 15 + (-6)

7. 23 — -7

10. 5.6 — (-8.7)

2. 12 + 7

5. 4 + (- 17)

8. 7 — 9

11. 12.8 — 19.3

3. - 15 + 6

6. 17 — (-9)

9. 5.1 —(-4.8)

12. 6.7 — 8.3

-

-

-

-

-

-

Evaluate each expression for a = 6 and b = -4. Check the sign of your answer. 13. a + b

15. a + (-b)

17. a— b

19. a— (-b)

14. -a+ b

16. (-a)+ ( -b)

18. -a — b

20. -a — (-b)

Multiplying and Dividing Integers You should review the rules for multiplying and dividing integers. • The product or quotient of two numbers with the same signs is always positive. • The product or quotient of two numbers with different signs is always negative. Simplify each expression. Check the sign of your answer.

1. 3 x 4

8. 10 + 1/4

2. 5 x 6

9. 3 x 5.2

-

-

-

3. 36 + 12

10. (-0.6) x 28

4. 84 + -7

11. 11.2 + (-0.6)

5. 82.4 + 8

12. 0 + -98

6. 11.2 + -4

13. 4 x (17 4- -6)

7. 18 + .5

14. 6 x (- 12 + - 13)

-

-

-

-

-

-

Evaluate each expression for a = -25 and b = 5. Check the sign of your answer. 1S. a x b

17. a + (-b)

16. -a x b

18. -a + (-b)

Evaluate each expression for t = -4. Check the sign of your answer.

19. 28 x t

21. 28 + (1)

20. - 12 x t

22. - 12 +

to

It)

10

Scientific Notation To write a number in scientific notation, use the following steps: • Move the decimal to the right of the first integer that is not a zero. • If the original number is greater than 1, multiply by 10", where n represents the number of places the decimal was moved to the left. • If the original number is less than 1, multiply by 10', where n represents the number of places the decimal was moved to the right. Examples:

Write each number in scientific notation. A. 8,750,000 standard form Move the decimal to the left six places. 8.75 x 10 6 Drop all insignificant 0's. Multiply by the power of 10. standard form B. 0.000078 Move the decimal to the right five places. 7.8 x 10-5 Drop all insignificant 0's. Multiply by the power of 10.

Write each number in scientific notation.

1. 560,000

4. 783 million

7. 2,310,000,000

2. 6,250,000

S. 0.0000623

8. 40,000

3. 165 billion

6. 0.00378

9. 0.00305

Write each number in standard notation. 10. 2.67 x 107

13. 4.5 x 10-3

16. 7.051 x 10-6

11. 4.5 x 103

14. 4.08 x 10 -2

17. 9.93 x 102

12. 5.003 x 10 6

15. 1.67 x 104

18. 2.68 x 10 10

UI

(AE -xL)Kz *sL

(17 — nL)_ '01

(E — x17)_ '6

L(9 —n) 111

toe - xo

3(9 — xE) 'EL

(t7 +

(i7 — x9_)_ 'ZL

+

*ti

*L

(9 -x9 p/i '9

'L L

Ipadom aApnqpria alp 6uisn uoissaJdxa 113E8 AMICIUMS

•uopsaJdxo uogoemns 9 — x0Z_ (9 + x0z_ 'Ai!ldw!S •Buyildwnw Aq 6 pue o; g_ einqpism (E)(9_) + (KOS_ (E x17)9_ '8 OZ +x8 'Ai!ldw!S •6u!Alc1Rinw Aq g pue einqpism (9)t? (()t7 (g +n)v•If •Amidws :saidurexi 6u!Aid!linw Aq seseinueied ay; appu! wiej Lime seseinualed NI to ;um ui Jeqwnu eqj einqpisflo not 'esimemi •Houed e enteoal lsnw wooisseio aye ui auofuen3 •ssup ay} o; siioued no 6upsed se Auedoid enimmazsla eq jo Num ueo noA .

Apadaid aionqinsia

INMNNENIIMMINNIINNIMINUNINF AMMINNIMMUMMMIMMIMMINMENNMIMM

Graphing Data on Coordinate Planes The coordinates of a point give it location on a coordinate plane. To locate a point on the plane, you need to give an order pair (x, A. Starting at the origin, (0, 0), you move x units to the right or left along the x-axis and then y units up or down parallel to the y-axis. Example:

Given the coordinates of points A, B, C, and D: Point A is 2 units right of origin and 2 units up. The coordinates of A are (2, 2). Point B is 2 units left of origin and 3 units up. The coordinates of B are (-2, 3). Point C is 3 units left and 0 units up. The coordinates of C are ( -3, 0). Point D is 3 units right and 2 units down. The coordinates of D are (3, -2).

Name the coordinates of each point on the coordinate plane.

1. G

6. M

2. H

7. N

3. J

8. R

4. K

9. P

5. L

10. Q

MIIIMMINREA 'MENEM= NUEMM EN. MX IN WINNI NE WINN NNE NE. ME J ..111

Graph the points on the same coordinate plane.

11. S (3, 4)

14. V (-4, 5)

12. T (0, 5)

15. W (-4, 5)

13. U (-3, -4)

16. Z (3, -3)

-

I

s

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111

Relations and Functions A function is a relation that assigns exactly one value in the range to each value in the domain. The domain includes all of the x values in the function. A function rule may be given as an equation. The function f(x) = 2x + 3 will take any value of x and change it into 2x + 3. You read this as "f of x equals two x plus three." When x = 5, the function of 5 would be 2 • 5 + 3 or 13, or f(5) = 2.5 + 3. Evaluating a function means finding a value in the range for a given value from the domain. Example: Find the domain and range of this relation. {( -4,3), (-2,-5), (0,0), (1,4)} Domain: -4, -2, 0, 1 Range: 3, -5, 0, 4

Example: Evaluate f(x) = 2x + 3 for x = 0, 1, and 2 f(0) = 2(0) + 3 = 3 f(1) = 2(1) + 3 = 5 f(2) = 2(2) + 3 = 7

Find the domain and range of each relation. 1. {( -2,3), (0,1), (2,5), (5,7)} 2. {(7, -5), (3, -2), (2,1), (3,2)} 3. {(1,0), (2,-3), (3,1), (4, -3)} Evaluate each function rule for x = -4 4. f(x) = 3x

7. g(x) = -3x— 4

5. f(x) = x— 5

8. g(x) = x+2x + 5

6. f(x) = 2x + 8 Find the range of each function with the given domain. 11. h(x) =

9. f(x) = 5x, { -2, 0, 2} f(-2) =

f(0) =

f(2) =

g(0) =

h(0) =

h(2) =

12. f(x) = 3) 2 — 15; {-3, -2, 2}

10. g(x) = 2x2 — 3; {-3, 0, 3} g(3) =

h(-2) =

+ x — 2; {-2, 0, 2}

g(3) =

f(-3) =

f(-2) =

f(2) =

111111111111111111111111111111111111111111111111111111111111111111111111111111111H1111111111111111111lil 11111111111111111

Function Rules, Tables, and Graphs You may find that function rules will be easier to see if you model them with a table and a graph. Example:

Graph the function y = 2x + 3 First: Pick at least four values for x. Choose some negative values for x. Place these values into a table.

x -3 -1

y = 3x + 2 y = 3(-3) + 2 = -7

I)

y = 3(0) + 2 = 2

2

y = 3(2) + 2 = 8

Y=3(4)+2=-1

(x, y) (-3, ..7) (-1, -1) (0, 2) (2, 8)

Second: Evaluate the function to find the value of y for each value of x. Third: Plot the ordered pairs to graph the data.

Use a table to graph each function. Choose at least four values for x. Some x values need to be negative.

1. y= 3x+ 1 2. y = -x + 2 3. y= x— 3 4. y = -2x + 5

5. y= .)e —3 6. y= 2)t — 3x-5

U111111111111111111111111UI1i111111IIUUIIIIIIIIIiIIIIIIIIIIIIIIIIIIIli11111111111111ilOMB UIIilllltllllltiilli U

Writing Function Rules You can write functions (which are equation rules from situations) by analyzing the sentence phrase by phrase. Example: Write a function for the following situation. The width of a sheet of plywood is one half the length. Let / represent the length and w represent the width. Then the function of width is 1/2/ and w = 1/2/

Write a function for each of the following situations.

1. The distance a car travels is rate times time. 2. The area of a rectangle is length times width. 3. Your grade is the sum of the percentages on three tests divided by three. 4. Gas mileage is miles driven divided by the number of gallons. S. Scientific work is determined by the product of force and distance.

6. The volume of a rectangular prism is the length times the width times the height.

Use the functions above to determine the following of each.

7. What is the distance traveled by a car when its rate is 60 miles per hour and it is driven for 2 1/2 hours? 8. What is the area of a rectangle when the length is 12 feet and the width is 6.5 feet? 9. What is your percentage grade when three tests were 95%, 92%, and 83%? 10. What is the gas mileage if you drive 258 miles on 12 gallons of gas? 11. How much work was accomplished if a horse pulled 550 pounds for 6 feet? 12. What is the volume of a rectangular prism that is 10 feet by 12 feet by 8 feet?

1111111111111111 1111 MN 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

Solving One-step Equations To solve an equation, you need to isolate the variable having a coefficient of 1 on one side of the equation. By using the inverse operation, you are able to next remove terms and the coefficient to the other side of the equation. For any number a, b, and c If a = b, then a+ c= b+ c • Addition Property of Equality: If a= b, then a—c= b— c • Subtraction Property of Equality: If a= b, then axc= bxc, with c O. • Multiplication Property of Equality: If a = b, then a/c = b/c, with c O. • Division Property of Equality: Example: Solve each equation. A. x+ 12 = 7 x+ 12 — 12 = 7 — 12 Subtract 12 from both sides. Simplify. x=

B. 5x= 35 5x_ 35 5 5 x=7

Solve each equation.

1. x + 15 = 45

10.3 =20

2. y + 42 = -89

11. 1= 99

3. 125 + x= -56

12. 4 y= 9

4. m — 31 = -15

13. -32x. 14

5. -28 = k+ 5

14. a+ 23 =

6. x— 39 = 18

15. 11 = 2

7. 13x= 52

16. -52m= -22

8. 7y = 63

17. 13

9. 6m = -42

18. -25 = y — 6

.-g

Divide both sides by 5. Simplify.

11111110N11111111111111111.1111111111111111111111111111111.111111111111111111111111111111111111M11111111111111111111111101111111

Solving No-step Equations You use order of operations to determine if you do multiplication before addition. When solving two-step equations, you must reverse the order of operations and perform the inverse operation. The goal is to isolate the variable to one side of the equation. Example: Solve the equation. Reverse the order of operation of adding 5. 3x + 5 = 11 3x + 5 - 5 = 11 -5 Subtract 5. Simplify. 3x= 6 3x 6 Divide by 3. 3-3 Simplify. x=2 Solve each equation. 9. 4x + 9 =

1. 3x- 5 = 13 2. 2/} + 3 = 9

10. 4(x + 3) = 12

3. 4x + 7 = 21

Mix+ 4 = 6

4. 3x+ 4 = 10

12. (x + -5) = 8 4

S. 5x-4 = 16

13. 1.5(x - 3) = 7.5

6. 7x + 16 = 30

14.

7. 4ax - 4 = 16

+ 15 = 18

15. 1.2x - 0.6 =

8. 3x - 8 = 16

16. 15x + 12 = 72

18

Pro Portions You know that ratios make comparisons. Two ratios that make the same comparison are called equivalent ratios. A statement that shows two ratios as equal is called a proportion. To check to see if the ratios are equal, cross multiply. If the cross products are equal, they form a proportion. l- -[ 2) 3 x 2 = 6

equivalent—when two things are equal 2 basketballs

1 basketball

6 basketballs

3 basketballs

♦

Equivalent _1 ratios are proportions.

1x6=6

Because 6 = 6 the ratios are a proportion.

On a test, Sulu got 6 answers right out of 9. Jim got 8 answers right out of 10. Are the students' scores proportional? STEP

1 Set up the two ratios. Be sure that the numbers in

the ratios are set up in correct order. Suki Jim 6 8 9 10

STEP

2 Find the cross products.

STEP 3

Compare the cross products. 60 # 72, so the ratios are not a proportion. Suld and Jim's scores are not proportional.

(

ON YOUR OWN

This summer 8 out of 12 people took a vacation. Last summer 3 out of 4 people took a vacation. Are these ratios proportional?

Building Skills

Write = or # between the ratios. 1.

2: 3

12 : 18

2.

3 8

4.

6: 8

12 : 16

5.

4: 9

7.

7 2

8.

9: 5

21 6

6 18

-3.

2 5

8 20

12 : 26

6.

3 7.

14 6

27 : 9

9.

2 —

8

20 80

Problem Solving

Use proportions to solve these problems. 10. Jim gets paid $40 for 4 hours of work. Carmela gets paid $60 for 6 hours of work. Do Jim and Carmela get paid at the same rate?

11. Anna saves $3 for every $10 she earns. Patrick saves $5 for every $20 he earns. Do the two save money at the same rate?

12.

Henry completed 6 of the 15 passes he attempted. Giovanni completed 9 of his 21 passes. Did the two quarterbacks complete passes at the same rate?

13.

Jackie can type 50 words in a minute. Her brother can type 200 words in 4 minutes. Are their typing speeds the same?

14.

Keonta can walk half a mile in ten minutes. Abigail walks one mile in 25 minutes. Do Keonta and Abigail walk at the same rate?

15.

Emily answered 8 of 10 questions correctly on a test. On a different test, Yuki answered 15 of 20 questions correctly. Did Emily and Yuki have the same ratio of correct answers?

Using Cross Products You know that you can use cross products to find whether two ratios form a proportion. You can also use cross products to find a missing number in a proportion. It doesn't matter how you set up the ratios as long as the same categories are on top and the same categories are on the bottom. Example

Read the clipping. If Keisha keeps getting hits at this same rate, how many hits will she get in 60 at-bats? STEP

Leads the Way

1 Set up a proportion.

You do not know how many hits Keisha will get in 60 at-bats. Use a letter, such as n, to substitute for the unknown number. 4 hits 60 12 at-bats STEP 2

Teen Slugger Keisha Jones is off to a big start this season with 4 hits in her first 12 at-bats. Her 9th inning round-tripper gave the Tigers their third straight victory.

Cross multiply.

12 X n = 4 X 60 12n = 240 STEP

3 Solve for ti.

Divide both sides by 12 because n is next to 12. 12n = 240 240 12n 12 12 n = 20 4 _ 20 12 — 60 Keisha would get 20 hits in 60 at-bats.

(

ON YOUR OWN

One photo is 6 inches long and 4 inches wide. You can enlarge the photo so that it will be 18 inches long. How wide will the photo be?

al

Building Skills Solve each proportion. 1.

3 4

12

2.

3 8 _ • 16 — 4

46 n 9

x 5 4. — = 30 6

18 n •-• 30 = 10

6. 40 = 8 5 n

5 4 7. — = — 25 x

3 21 8. = 8 y

9.

—

= 11 20 4

Problem Solving Use proportions to solve these problems. 10.

LaToya gets paid by the hour. During a 20-hour week, LaToya earns $120. If she works only 15 hours, how much does she earn?

11.

Joe is a photographer. For every 15 rolls of film he shoots, 10 are black and white. If Joe uses 57 rolls of film, how many rolls are in black and white?

12.

Luisa swims 35 laps in 25 minutes. How many laps will she swim in 10 minutes?

13.

Darnell walks 140 minutes in 4 days. At this rate, how many minutes will he walk in 6 days?

14.

Dan scored 80 points in 5 basketball games. At this rate, how many points do you expect him to score in 20 games?

15.

Jay buys 60 square feet of tile for $324. Later, he buys another 20 square feet of tile. How much does the extra tile cost?

Q2-

Bit C-Kad-e Ma th 1-{X Ket Ansvve( Racj u r,SL Addoci

Scityihfic Noitch on? , t 1. 5.6 x 105 2. 6.25 x 106 3. 1.65 x 10 11 1 4. 7.83 x 108 5. 6.2 10-5 6. 3.78 x 10-3 7. 2.31 x 109 8. 4.0 x 104 9. 3.05 x 10-3 10. 26,700,000 11. 4,500 12. 5,003,000 13. 0.0045 14. 0.0408 15. 16,700 16. 0.0000751 17. 993 18. 26,800,000,000

.3 S...bt-taci-ii)

1. 5/8 2. 5/6 3. 1 1/6 (7/6) 4. 2 3/4 5. 1/12 6. 1/3 7. 2 1/12 (25/12) 8. 1 3/4 (7/4) 9. 7 1/12 10. 11 1/6 11. 3 7/12 12. 13/24 Writing Answers will vary.

DISIYIN-hve Pro p& +y p. al. 2x + 10 2. 10x + 15 3. 21 + 49x 4. 63 + 27x 5. 16x + 24y 6. 4x - 2 7. -3x -- 4 8. 2x - 6 9. -4x + 3 10. -12x + 4 11. 5x - 1 12. 6x + 4 13. 6x - 12 14. 14x - 35 15. 14x2 - 6xy

pl. ID\ v,ct, rtaf1p ► i FrCtC t-1 of.) 1. 3/8 2. 1/6 3. 9/20 4. 1 1/2 (3/2) 5. 1 7/9 (16/9) 6. 1 3/5 (8/5) 7. 1 8. 4 9. 3/4 10. 20 11. 7 1/2 12. 7 13. 3 3/4 sq ft 14. 12 1/2 sq ft

.390.00

'Dec i r-r) ,--1A%

Gvaphi

1. 11.4 2. 20.85 3. 115.742 4. 1.4 5. 5.655 6. 5.628 7. 2.544 8. 51.3 9. 87.802 10. 72 11. 0.38 12. 652.5 Writing 13

boar, o(' CoDiCtivackle ch✓ e

1. (3, 1) 2. (-4, 3) 3. (0, -4) 4. (1, -2) 5. (3, -3) 6. (-4, -3) 7. (1, 3) 8. (-2, 1) 9. (-1, -2) 10. (3, 4) 11. 12.

combinations; Answers will vary.

SCientiAic_ Nclecha\ 1. 5.43 x 104 2. 7.03 x 10 5 3. 7.53 x 102 4. 1.5 x 10 7 5. 3.3 x 10-4 6. 6.28 x 10-2 7. 3.5 x 10-7 8. 2.05 x 10-2 9. 15,600 10. 6,033 11. 8,566,000 12. 0.000789 13. 0.0246 14. 0.00000479

Us

fl(A

\i'clo 0.01 € P.

1. n + 5 2. 15 - n 3. n + 15 4. n - 7 5. n/6 6. 8n 7. x - 4 = 6 8. x + -8 = 17 9. 28/x = 4 10. 7x = 56 11. 3x - 4 = 8 12. 1/4x + 3 = 10 Writing Answers may vary (seven dimes plus

13.

14.

eight pennies).

E.)pcent-

w

1. 25 = 32 2. 36 = 729 3. 55 = 3,125 4. -25 = -32 5. 70 = 1 6. n 7 7 . n 2 8. 34x4 = 81x4 9. 4 10. 64 11. 64 12. 1,024 13. 1/16 14. 8

15.

16.

15. 1,000 16. 1/256 Writing 200 x 24 = 3,200

Orcie{ i opefaticil p= 1. 22 2. 2 3. 18 4. 3 5. 11 6. -7 7. 47 8. 4 9. 50 10. 27 11. 41 12. 63 13. 32 14. 52 15. 24 16. 27 17. 22 18. 54

Rectl Noolveirs p= 1. real numbers 2. rational, real numbers 3. rational, real numbers 4. whole, integer, rational, real numbers 5. irrational, real numbers 6. integer, rational, real numbers 7. rational, real numbers 8. whole, integer, rational, real numbers 9. integer, rational, real numbers 10. rational, real numbers 11. irrational, real numbers 12. irrational, real numbers 13. answers will vary 14. answers will vary 15. answers will vary 16. answers will vary

Acid n CAOct 5k.kty -ocimeci to-kler 1. -9 2. 5 3. -9 4. -21 5. -21 6. 26 7. -16 8. -16 9. -0.3 10. 14.3 11. -6.5 12. -15 13. 2 14. -10 15. 10 16. -2 17. 10 18. -2 19. 2 20. -10

NEB Itip fyrnc CVX1

ers p, 10

1. -12 2. -30 3. 4. -12 5. -10.3 6. -2.8 7. 36 8. 40 9. -15.6 10. -16.8 11. 18.67 12. 0 13. -44 14. 150 15. -125 16. 125 17. 5 18. -5 19. -112 20. 48 21. 7 22. -3

Re kcch s anct run GI-10A% p,

P-4

1. domain: (-2, 0, 2, 5) range: (3, 1, 5, -7) 2. domain: (-7, -3, 2, 3) range: (-5, -2, 1, 2) 3. domain: (1, 2, 3, 4) range: (0, -3, 1) 4. -12 5. -9 6. 0 7. 8 8. 13 9. -10, 0, 10 10. 15, -3, 15 11. 0, -2, 4 12. 12, -3, -3 13. 51 14. 60 15. 62

ty)ction Rukt-s/ 1.

x

r

-2 0

-5 4 7

TAbk7-' ,36 G -aptru p. 6 2. -2 4 0 2 0

13

P(0 porhon x

y

-2 0

-5 -3 -2

4.

-2 9 0 5 3

2

5.

x -

6.

y

0 1 2

x

y 9

2

0

-3 -2

2 3

W tri 'hen FUnCtOn RV ins

-6 -3 4

p.

1.d=n2.A.1w3.G.(t1+t2+t3)/3 4. mpg = m/g 5. W = Fd 6. V = 1wh 7. 150 miles 8. 78 sq feet 9. 90% 10. 21.5 mpg 11. 3,300 foot-pounds 12, 960 cubic feet

One-SkeP EctQatitIS p, 1. 30 2. -131 3. -181 4. 16 5. -33 6. 57 7. 4 8. 9 9. 7 10. 60 11. 9 12. 12 13. -4 14. -3 2/3 15. 22 16. 4 17. 65 18. 19 -

-

-

-

&iv oci Tv4a-s0 Evottwev.) 1. 1. 6 2. 24 3. 3.5 4. 2 5. 4 6. 2 7. 26 2/3 8. 8 9. -3 10. 0 11. 3

12. 37 13. 8 14. 4 15. -3 16. 4

r.

x y

1. 2 X 18 = 36; 3 X 12 = 36 Because 36 = 36, the ratios are a proportion. 2. 18 X 3 = 54; 6 X 8 = 48 Because 54 0 48, the ratios are not proportional. 3. 2 X 20 = 40; 8 X 5 = 40 Because 40 = 40, the ratios are a proportion. 4. The ratios are s and R. 6 X 16 = 96; 8 X 12 = 96 Because 96 = 96, the ratios are a proportion. 5. The ratios are 9 and R. 4 X 26 = 104; 9 X 12 = 108 Because 104 0 108, the ratios are not proportional. 6. 3 X 6 = 18; 7 X 14 = 98 Because 18 * 98, the ratios are not proportional. 7. 7 X 6 = 42; 2 X 21 = 42 Because 42 = 42, the ratios are a proportion. 8. The ratios are s and -V-. 9 X 9 = 81; 5 X 27 = 135 Because 81 # 135, the ratios are not proportional. 9. 2 X 80 = 160; 8 X 20 = 160 Because 160 = 160, the ratios are proportional. 10. 40 X 6 = 240; 4 X 60 = 240 Because 240 = 240, the ratios are proportional. They are paid at the same rate. 11. The ratios are 1-3-6- and 3 X 20 = 60; 10 X 5 = 50 Because 60 0 50, the ratios are not proportional. They do not save money at the same rate. 12. The ratios are is and If 6 X 21 = 126; 9 X 15 = 135 Because 126 0 135, the ratios are not proportional. They do not complete passes at the same rate. 13. The ratios are T and -F. 2: 50 X 4 = 200; 1 X 200 = 200 Because 200 = 200, the ratios are proportional. Their typing speeds are the same. 14. The ratios are tc5, and 5. 0.5 X 25 = 12.5; 10 X 1 = 10 Because 12.5 0 10, the ratios are not proportional. They do not walk at the same rate. 15. The ratios are-PT and o. 8 X 20 = 160; 10 X 15 = 150 Because 160 # 150, the ratios are not proportional. They do not have the same ratio of correct answers. -

••

801$ = u 09 09 u 09 081/9 = /409 VZC X OZ = u X 09 OZ _ 09 u pz5

OZ _ SZ —

OZ = x _ 001 xS 001 = xS SZx17=xxS 'L

quiod oz5 = u S _ 0091— u5

S_ SZ — 017 sz = u gg 00Z — 00Z = ug

009`i = us OZX 08 =uXS OZ _ S 171 u 08 samului OIZ = u

S X 017= u X 8 '9 OT _ OC 9 — 81 9= u

0178 iTf 0178 = u 17

9 X 017 1 = U X 17

OC _ OC 081 — u0C = 110C OI X RI = u X ot

9 _ 17 0171 scIri pi = u SZ _ SZ OSC u SZ OSC = uSZ SC X OI = u X SZ OI _ SZ

u

INIA

9 _ 05 S SZ sz = x 9 _ 9 OST OSI = x9

• 1

pue3puri jo silo.' 85 = u

OCXS=xx 9 "t•

SI _ ST OLS uSI OLS = uSI LS X OI = u X SI u OI . LS = H

17

91 Z 8 Z= u 91 _ 91 Z£ u9I ZC = u9I X17=uX9I 'f 6 _9 9 — 17 9= u

06$ = u OZ_ 008'T$ u0Z 008`I$ = u0Z SI X OZT$ = u X OZ

9_9 95 u9 95 = u9

u _ OZI$ ST OZ 17 _ OZ II — S S SS = 17 _ 17 OZZ 7F7 OZZ = OZX II=XXt7 '6 9S _ 8 1Z — 9S= , ( C_C 891 — 891 = 4

tzxs=xxc

*s

6 X17=uX 9 7 Z1 _ 17 6 —T

6= u *17 95 — up 95 = "17 Z1 XC=t1 X17

4 sp(paa ss Q)3 )() ►