Log College Middle School

8th Grade

Student _________________________

Period ______

Revised: 7/2015

TABLE OF CONTENTS

Some English Equivalents for Math Terms & Useful Definitions…………… 1 – 2

Verbal Models & Related Information ….……………………………………... 3 – 5

Math Vocabulary: Examples & Practice. …..……………………………….. 6 – 14

Verbal Model Format: Directions & Examples ………...….……………… 15 – 19

Verbal Model Format: Problems ……..…………………………..…..…….. 20 – 41

Review Problems ………………………………………………..… 42 – 45 & 49, 50

Pythagorean Theorem ………….……………………………………………….. 46

System of Equations Problems …………………………………………………. 51

Multiplication Table ……………………………………………………... Back Cover

Special Note

This workbook is designed for 8th grade math students at Log College Middle School. Any

person who wishes to copy any part of this book must have written permission from Mrs. Gismondi,

who created, designed and compiled the work.

Common Core Math Standards

Mathematics is a technical language that requires precision and attention to details. Changing

one word can change the entire meaning of a statement or problem. Modeling helps the student

translate a problem from words into an algebraic model that can be manipulated into multiple

representations to arrive at a solution. This workbook identifies nine different algebraic models and

demonstrates how to translate an English sentence to its equivalent algebraic model. If a student can

identify which model the problem represents, then there is a greater likelihood that the student will

persevere to find a solution. Therefore, this workbook encompasses all the Standards for

Mathematical Practices of the Mathematics Common Core Standards. Also addressed in this

workbook are the Standards for Mathematical Concepts dealing with Algebra, Geometry, and Ratio

and Proportional Relationships.

Addition

add

addend

altogether

bigger than

greater than

in all

increased by

larger than

longer than

more

more than

older than

plus

sum

taller than

tally

together

the sum of

the tally of

the total of

total

Subtraction –

amount of increase

decreased by

deduct

deducted from

difference

diminished by

fewer than

how many fewer

how many more

how much greater

less

less than

minus

shorter than

smaller than

subtract

subtracted from

take away

the difference between

younger than

Multiplication

of use

as many as

as much as

double use 2●

factor

multiply

product

times

the product of

thrice use 3●

triple use 3●

twice use 2●

Division

(any fraction)

average

divided by

divided into

dividend

divisor

out of

quotient

the average of

the quotient of

Variable

number

the number of ...

unknown

h, x, y, n

Exponent

base power

cubed (....)3

squared (....)2

the square of (....)2

the square root of

Equality =

any verb

equals

is

is the answer to

is the same as

Inequality

does not exceed

is at least

is at most

is between x

is greater than

is greater than or equal to

is larger than

is less than

is less than or equal to

is more than

is no less than

is no more than

is not equal to

is not quite is smaller than

Parentheses ( )

groups of …

the quantity of …

two operations occur together

without a numeral

between them

Separate Values, Terms

and

, (comma)

Proportion Notation

is to /

as =

+ •or∗

ab

ab •

/ or ÷

ab

/← →⎯ ÷← →⎯⎯

...radicandindex

> ≥ ≤ < ≠

≤≥≤< <

>≥

><

≤>≥≤≠

<<

SOME ENGLISH EQUIVALENTS FOR MATH TERMS

means switch the order of the terms

Rate

a piece rate

average per

each percent %

for one speed

for each

2

Positive Numbers

above

deposit of …

east

forward

gain of …

increase of …

north

profit of …

up

Negative Numbers

“the opposite of …”

backward

below

decrease of …

down

drop

loss of …

south

west

withdrawal of …

Additive Inverse –

“The opposite of …”

Absolute Value

The distance a number is from zero

= a positive number

Reciprocal

The numerator and denominator

of a fraction are flipped.

Also called the multiplicative inverse

of a number.

Coefficient

Number by which the variable

is multiplied.

Constant

Number without a variable

Natural Numbers

{1, 2, 3, …}

Counting numbers

Whole Numbers

{0, 1, 2, 3, …}

Integers

{0, 1, 2, 3, …}

Includes all whole numbers and

their opposites

Rational Numbers

Any number that can be written

as where a and b are Integers

and b 0.

Includes all integers,

fractions, terminating

decimals, repeating decimals,

percents and perfect roots.

Irrational Numbers

Any number that is not Rational.

Includes non-terminating

non-repeating decimals, and non-

perfect roots.

Real Numbers

Includes all Rational and Irrational

Numbers

. . .

. . .

± ± ±

ba

≠

± ±±

±

Additive Inverse: a + (– a) = 0

Multiplicative Inverse: a = 1

Commutative Property: a + b = b + a

a b = b a

Associative Property: (a + b) + c = a + (b + c)

(a b) c = a (b c)

Identity Property: a + 0 = a AND a 1 = a

Distributive Property: a (b + c) = a b + a c

Multiplicative Property of Zero: a 0 = 0

Additive Property of Equality:

If a = b, then a + c = b + c.

Multiplicative Property of Equality:

If a = b, then a c = b c

Transformation Type Rule

TranslationMoverightaunits Addatoeachx-coordinate.Moveleftaunits Subtractafromeachx-coordinate.Moveupbunits Addbtoeachy-coordinate.Movedownbunits Subtractbfromeachy-coordinate.

Reflection Acrossthey-axis Multipleeachx-coordinateby-1Acrossthex-axis Multiplyeachy-coordinateby-1

Rotation

180° Multiplybothcoordinatesby-1.

90°clockwiseMultiplyeachx-coordinateby-1,andthenswitchthex-coordinatewiththey-coordinate.

90° counterclockwise

Multiplyeachy-coordinateby-1,andthenswitchthex-coordinatewiththey-coordinate.

Dilation ScaleFactorMultiplyeachcoordinatebythescalefactor.

ARITHMETIC PROPERTIES

3

Leg2

TRIANGLE: Perimeter = side1 + side

2 + side

3

Area = base ● height

180 = Angle1 + Angle

2 + Angle

3

TRAPEZOID: Perimeter = side1 + side

2 + side

3 + side

4

Area = height ( base 1 + base

2 )

PARALLELOGRAM: Perimeter = side1 + side

2 + side

3 + side

4

Area = base ● height

RECTANGLE: Perimeter = 2 length + 2 width

Area = length ● width

SQUARE: Perimeter = 4 side

Area = side2

CIRCLE: Circumference = 2 radius use calculator or 3.14

Area = radius2

use calculator or 3.14

CUBE: Surface Area = 6 side2

Volume = side3

PRISM: Surface Area = 2 length ● width + 2 length ● height + 2 width ● height

Volume = length ● width ● height

CYLINDER: Surface Area = 2 radius2 + 2 radius ● height

Volume = radius2 ● height

CONE: Surface Area = radius2 + radius ● slant height

Volume = radius2 ● height

PYRAMID: Surface Area = Area of base + slant height ● perimeter of base

Volume = height ● area of base

SPHERE: Surface Area = 4 radius2

Volume = radius3

PYTHAGOREAN THEOREM:

For every right triangle, Hypotenuse2 = leg12 + leg2

2 Leg1

ANGLE MEASURE To find the Sum or number of sides in a polygon, use: Sum of angles = 180 (n – 2) Of ANY POLYGON To find one angle of any polygon with n sides, use: 180 (n – 2) = a1 + a2 + … + an To find one angle of a regular polygon with n sides, use: Angle1 = 180 (n – 2) / n

12

12

•

π ππ π

π ππ

π π1

3π

12

1

3

π4

3π

2 2 2c a b= +

Hypotenuse = c

Verbal Models

4

DISTANCE = rate ● time

PREDICTED VALUE = Rate per Unit ● Number of Units

PREDICTED VALUE = Original Value + Rate per Unit ● Number of Units

PROPORTIONS: isof

= percent100 OR

Pr edicted Value1

Number of units1

=Pr edicted Value2

Number of units2

SLOPE = m =

ΔyΔx

=y1 − y2x1 − x2

for points (x1, y1) and (x2, y2)

SUM = first + second

TEMPERATURE (choose one): FAHRENHEIT = Celsius + 32 OR CELSIUS = ( Fahrenheit – 32 )

MEASUREMENT LABELS FOR WORD PROBLEMS

Measurement Labels

Perimeter, Circumference, Distance, Base, Height, Length, Radius, Side, Width

mm, cm, m, km, inches, feet, yards, miles

Area, Surface Area mm2, cm , m , km , inches , feet , yards , miles

Volume cm , m , km , inches , feet , yards , miles

Rate rate per unit, mph, mpg, cost per unit, % (percent)

Time seconds, minutes, hours, days, months, years

am • an = am + n

am

an = am−n

a−1 = 1

a1

(reciprocal) (square root) (cube root)

95

59

2 2 2 2 2 2 2

3 3 3 3 3 3 3

am( )n = amn

a0 = 1

a12 = a a

13 = a3

Customary Measures MEASUREMENT TABLES Metric Measures

1 mile = 5280 feet (ft)

1 yard (yd) = 3 feet

1 foot = 12 inches (in.)

1 day = 24 hours (hr)

1 hour = 60 minutes (min)

1 minute = 60 seconds (sec)

1 ton (T) = 2000 pounds (Lb)

1 pound = 16 ounces (oz.)

1 gallon (gal) = 4 quarts (qt)

1 quart = 2 pints (pt)

1 pint = 2 cups ( c )

1 cup = 8 fluid ounces

1 millimeter = 0.001 meter

1 centimeter = 0.01 meter

1 decimeter = 0.1 meter

1 meter (m)

1 decameter = 10 meters

1 hectometer = 100 meter

1 kilometer = 1000 meters

Substitute meter with liter or gram and the

numerical values

remain the same.

FORMS OF LINEAR EQUATIONS

RULES FOR EXPONENTS

Slope Intercept Form: y = mx + b where m = slope and b = y-intercept. Standard Form: Ax + By = C where A, B, C are integers and A ≥ Point-Slope Form: (x – x1) = m(y – y1) where m = slope and point is (x1, y1).

5

a bc

d

a b

a b c

d

e f g

h

a b

Vertical Angles Measure of Angle a = Measure of Angle

b

Measure of Angle c = Measure of Angle

d

Complementary Angles Supplementary

Angles

Measure of Angle a + Measure of Angle

b = 90

o

Measure of Angle a + Measure of Angle

b = 180

o

Parallel Lines cut by a Transversal Measure Angle

a = Measure Angle

b = Measure Angle

e = Measure Angle f

Measure Angle c = Measure Angle

d = Measure Angle

g = Measure Angle

h

Adjacent angles are supplementary angles (share a vertex and a side).

Alternate Interior angles are congruent. m d m g and m b m e

Alternate Exterior angles are congruent. m a m f and m c m h

Corresponding angles are congruent. m a m e, m c m g, m d m h,

and m b m f

Isosceles Triangle

Two sides called legs are equal. Leg1 = Leg2

The other side is called the base.

The base angles opposite the legs are equal. �base1 = �base2

The other angle is called the vertex angle.

≅ ≅

≅ ≅

≅ ≅ ≅ ≅

secant

tangent

diameter Central �

radius chord

arc Inscribed �

Parts of a Circle

Diameter = 2 radius

Radius = diameter / 2

Central �= intercepted arc

Inscribed � = 2 ● intercepted arc

Tangent intercepts circle in one point.

Secant intercepts a circle in two points.

6

MATH VOCABULARY

Mathematics is a foreign language that uses symbols to express ideas and concepts. Since we speak

English, we must learn to translate back and forth between the English we speak and think and the math

symbols we use to express those thoughts.

There is only one way to express the English word seven as a math symbol, 7. However, the math

symbol “+” can be expressed in English in many different ways. Look at page 1 of this workbook to find 20

different English words or phrases that can be used to represent the math symbol “+”. This unit discusses

how to translate between English phrases and their math symbols.

When you complete this unit and the Algebraic Models unit, you will be able to translate most word

problems so you can solve them correctly.

TO TRANSLATE FROM ENGLISH TO MATH:

1. Underline each English word or phrase that matches with one found on page 1,

SOME ENGLISH EQUIVALENTS FOR MATH TERMS.

2. Underneath each phrase place the math symbol that corresponds to the phrase. This is called

the translation.

3. On the right, rewrite the math symbols so they make sense. This is called the interpretation.

The following are seven examples that demonstrate how to translate and interpret English words or phrases

into their equivalent math symbols.

EXAMPLE 1: Six more than eight

Step 1 Underline: Six more than eight

Step 2 Translate: 6 + 8 Step 3 Interpret: 6 + 8

EXAMPLE 2: Twelve times a number

Step 1 Underline: Twelve times a number

Step 2 Translate: 12 • n Step 3 Interpret: 12n

Notice that the multiplication symbol is not shown in the Interpretation. When two symbols occur together

without a + − ÷, , between them, mathematicians know that the multiplication symbol is there. So, do not use a multiplication symbol unless it is between two numbers called constants.

7

EXAMPLE 3: Five minus twice a number

Step 1 Underline: Five minus twice a number

Step 2 Translate: 5 – 2 • n Step 3 Interpret: 5 – 2n

EXAMPLE 4: Seven less than six times a number

Step 1 Underline: Seven less than six times a number

Step 2 Translate: 7 −← →⎯ 6 • n

Step 3 Interpret: 7 −← →⎯ 6 n

Answer: 6n – 7

Notice that “less than” means to “subtract and switch the order of the numbers”. That means you must use

the subtraction symbol with the arrow above it −← →⎯ . To interpret this expression, you must first rewrite

without the multiplication symbol • and then switch the order of the 7 and 6n. The correct answer is 6n

– 7.

EXAMPLE 5: The sum of four times a number and nine

Step 1 Underline: The sum of four times a number and nine

Step 2 Translate: + 4 • n , 9

Step 3 Interpret: + 4n , 9

Answer: 4n + 9

Notice that the addition symbol came first, which does not make any sense. When the sentence or phrase begins with + − • ÷, , , or symbols, you replace the comma with that symbol and rewrite the phrase.

EXAMPLE 6: Twice the sum of seven times a number and three

Step 1 Underline: Twice the sum of seven times a number and three

Step 2 Translate: 2• + 7 • n , 3

2• ( + 7 • n , 3 )

Step 3 Interpret: 2 ( 7n + 3 )

Notice on the first line of the translation that two operations, • +and , occur together without a number between them. This is not allowed so start parentheses between the symbols to separate them and then place the second bracket at the end of the phrase. When you translate these problems, you should only

write the second line of the translation for your work.

EXAMPLE 7: Nine times the sum of seven times a number and eight

Step 1 Underline: Nine times the sum of seven times a number and eight

Step 2 Translate: 9 • ( + 7 • n , 8 )

Step 3 Interpret on Right of problem: 9( 7n + 8 )

8

Vocabulary Notes from Power Point Six plus seven

Six increased by seven

Six more than seven

The sum of six and seven

The total of six and seven

Eleven minus eight

Eleven subtract eight

Eleven take away eight

The difference between eleven and eight

Eight less than eleven

Eight subtracted from eleven

QUESTIONS

1. When should subtract with arrows be used? 2. What popular subtraction phrase may not be used with math symbols?

9

Vocabulary Notes from Power Point

Ex 1: a number increased by five

Ex 2: twelve times a number

Ex 3: twice a number greater than fifteen

Ex 4: the sum of five times a number and eleven

Ex 5: thirty-five less than nine times a number

Ex 6: the quotient of a number and four

Ex 7: twice the difference between nine times a number and two

Ex 8: six times the total of a sweater and a blouse

Ex 9: seven inches taller than Alex

Ex 10: three times the total of two numbers

Ex 11: seventeen less than the number of books

Ex 12: eleven times the number of doctors less four

10

Translate and interpret the following into mathematical symbols.

1. the sum of six and a number

2. twelve less than a number

3. the product of a number and eleven

4. eighty-four divided by a number

5. twenty decreased by twice a number

6. eleven greater than seven times a number

7. the quotient of a number and sixteen

8. twice the sum of four and a number

9. the total of two numbers

10. seven times a number fewer than forty-five

11. the tally of twelve times a number and seventeen

12. a number deducted from another number

13. triple the total of seven and a number

14. the quotient of two numbers

15. eighty-one minus thrice a number

16. seven taller than George

17. five times a number less sixteen

18. six times the difference between five and four times a number

19. three times John subtracted from one hundred

11

Translate and interpret the following into mathematical symbols.

1. seven times the sum of nine and a number

2. eight times the difference between six and a number

3. nine times the total of five and a number

4. twice the sum of seven and a number

5. ten times the difference between two numbers

6. eleven times the sum of two pianos

7. twenty times the sum of three numbers

8. fourteen times the difference between eleven and eight times a number

9. twice the total of the oranges and the apples

10. fifteen times the sum of seven times a number and four

11. six times the difference between six and five times a number

12. nine times the total of eight times a number and seven

13. twenty-one times the difference between two and a number

14. seven times the quotient of two numbers

15. nine times the total of twice a number and five

16. eleven times the difference between three times a number and six

17. fourteen times the sum of nine and seven times a number

18. twelve times the difference between eight and twice a number

19. twice the sum of five times a number and sixteen

12

Translate and interpret the following into mathematical symbols.

1. four times the sum of seven and a number

2. eight times the difference between a number and nine

3. five times the total of two scores

4. six times the quantity of base1

plus base2

5. twice the quantity of one plus George

6. five times the quantity of a number increased by six

7. seven more than twice the number of doctors

8. the quotient of the number of miles and fourteen

9. sixteen less the number of cars

10. sixteen less than the number of cars

11. twelve fewer than three times the number of accidents

12. eight less than fifteen times the number of feet

13. twenty-five more than triple the number of years

14. the product of thirty-five and the number of packages

15. the number of boys increased by the number of girls

16. forty-five plus the number of club members

17. twice the number of hours diminished by five

18. the number of points less than twenty

19. the number of baseball players times five

13

Translate and interpret the following into mathematical symbols.

1. the number of kilometers per hour increased by five

2. fifteen less than the number of scuba divers

3. the quotient of $170 and the number of payments

4. subtract the number of children from seven hundred fifty

5. twice the number of artists divided by fifteen

6. five times the number of dollars

7. the product of fourteen and the number of lemons

8. twelve less than twice the number of mice

9. two more than the number of peaches

10, the number of months less than one hundred twenty

11. twice the sum of seven and the number of cows

12. eight times the number of elephants

13. seven times the number of snakes

14. nine times the number of lions

15. the total of eight times the number of elephants, seven times the snakes, and nine times the lions

16. seventeen greater than twice the number of guards

17. six less than five times the number of donuts

18. eight fewer than twice the number of cattle

19. twice the sum of four times a number and sixteen

14

Translate and interpret the following into mathematical symbols.

1. the sum of eight and five times a number

2. the product of four and a number

3. the difference between six and seven times a number

4. the quotient of a number and nine

5. five times the sum of twelve and a number

6. seven less than nine times a number

7. four increased by six times a number

8. eleven times the total of twice a number and five

9. five times a number more than seven

10. twice a number subtracted from three

11. fourteen times a number less than twelve

12. the difference between two numbers

13. four decreased by six times a number

14. eighty times a number diminished by fifty

15. the product of a number and five

16. the difference between twice a number and sixty

17. thrice the total of twice a number and seven

18. twenty times the sum of nine and eleven times a number

19. twenty times the sum of eleven times a number and nine

15

VERBAL MODEL FORMAT

Word problems are easier to solve if you can first identify all the information and then show the relationships

between the given information using a combination of English words and math symbols. The Verbal Model Format is a

way to help you organize your information.

VERBAL MODEL FORMAT FOR SOLVING WORD PROBLEMS

1. Write one word or short phrase to describe what must be found. Assign a variable to it.

This is included in the DATA.

2. Use one word or a short phrase to describe each piece of information given in the problem. Place a math

expression/ number beside it on the right. This is included in the DATA.

3. Use the above words/ phrases to write an expression in English that shows the relationship between them.

Use +, −, •, /, ...( ), >, < and = to separate words.

This is the VERBAL MODEL.

4. Substitute equivalent math expressions/numbers under words/phrases in the verbal model. This is the ALGEBRAIC MODEL.

5. Solve equation for a math answer.

6. Answer the question in an English sentence.

MODEL 1: NUMBER

Eight more than three times a number is twenty-six. Find the number.

Data: number = n = 6 Verbal Model: 8 + 3number = 26

Algebraic Model: 8 + 3n = 26

–8 = – 8

The number is 6. ⅓ � 3n = 18 � ⅓ n = 6

Notice that this problem looks like the math phrases we have been translating. There are two differences: this is a

sentence with a verb; and, we are asked to find the number.

There is only one piece of information (DATA) in this problem -- identifying the variable. The Verbal Model is the

interpretation with the data written in words. The Algebraic Model is the interpretation using only math symbols.

MODEL 2: SENTENCE

Julie has $15. This is $7 more than twice the amount of Charlie’s money. How much money does Charlie have?

Data: Julie = 15 = 7 + c Verbal Model: 15 = 7 + Charlie

Charlie = c = 8 Algebraic Model: 15 = 7 + c

Charlie has $8. – 7 = –7

8 = c

Notice in this problem, the word this referred to $15 from the first sentence. There is more information about Julie in the

second sentence so everything after This must be written on the same line as Julie in the Data. The second piece of data

is identifying Charlie as the variable c. In this type of problem, the Algebraic Model comes directly from the line in the

Data that has two = signs in it. The Verbal Model is obtained by substituting the words for the variable.

16

MODEL 3: THE SUM OF TWO NUMBERS

One number is eight more than three times a second number. Their sum is 72.

Find the numbers.

first number = 8 + 3s = 8 + 3 � 16 = 56 first + second = sum

second number = s = 16 (8 + 3s) + s = 72

sum = 72 4s + 8 = 72

. – 8 = – 8

The numbers are 56 and 16. ¼ � 4s = 64 � ¼

s = 16

MODEL 4: FORMULAS A formula is another kind of Verbal Model. To find the appropriate formula, look on page 3 of

this workbook. The Data identifies all the English words in the formula and assigns math symbols to each.

EXAMPLE 1: VOLUME

A prism has a length of 17 inches, a width of 14 inches, and a height of 18 inches. Find its volume.

Data: length = 17 Verbal Model: Volume = length • width • height

width = 14 Algebraic Model: v = 17 • 14 • 18

height = 18 v = 4284

Volume = v = 4284

The Volume of the prism is 4284 square inches.

EXAMPLE 2: FORMULAS WITH ALGEBRAIC EXPRESSIONS

Find the length of a rectangle if its width is 17 feet more than twice its length and its perimeter is 142 feet.

length = L = 18 Perimeter = 2 length + 2 width

width = 17 + 2L = 17 + 2�18 = 53 142 = 2 L + 2 (17 + 2L)

Perimeter = 142 142 = 2L + 34 + 4L

142 = 6L + 34

Its length is 18 feet. - 34 = - 34

Its width is 53 feet. ⅙ � 108 = 6L � ⅙

18 = L

In solving problems using the Verbal Model Format, you must keep all = signs under each other in a straight

column. You must follow Order of Operations when simplifying the work. Finally, after you find the mathematical answer,

return to the DATA and place the answer beside the variable. Use this information to write your sentence that answers

the problem.

Verbal Model 5: Predicted Value = rate per unit • number of units

Rate of change per unit is an important concept in these new models. Some key words that indicate rate of change in a word problem are: a piece average each for each for one per percent % rate speed

However, rate can never stand alone. It must always be connected to another value that states the number of units that

are either given or desired. The units must be the same for the rate per unit and the number of units. When one piece of

information is given, the other is implied.

Rate per Number of

Example 1: A car is traveling 45 miles per hour. Example 2: John bought 8 cantaloupes.

Rate per mile = 45 Rate per cantaloupe = c

Number of miles = m Number of cantaloupes = 8

Notice the Verbal Model includes

only the words used in the data.

It is important to keep

parentheses around any piece of

data that contains an addition or

subtraction sign. Line up your

data under the words it

represents. Keep = signs under

= signs in the verbal and

algebraic models.

17

Note: The words after rate of … and after number of … must be the same. We have discussed rate per unit and the number of units. Predicted value is the product of those two terms. It is

the distance you will travel or the total cost you will pay.

It is important that you underline the information given in the problem. It is also helpful to identify each placing the

parts of the model near the underlined words.

Example 3:

Number of rate per gallon PC

Julian purchased 9.8 gallons of gas at a rate of $1.539 per gallon. How much did he pay the cashier?

Rate per gallon = 1.539 Predicted Cost = rate per gallon • number of gallons

Number of gallons = 9.8 c = 1.539 • 9.8

Predicted Cost = c = 15.08 c = 15.08

Julian paid the cashier $15.08.

Example 4:

Number of Pred. Tip

Sarah is a waitress at the local restaurant. If a check was $37.54 and she receives a $5.63 tip, how much should she

expect for the tip?

PT

Number of dollars = 37.54 Predicted tip = rate per dollar • number of dollars

Rate per dollar = r =15%

1

37.54 5.63 = r • 37.54

1

37.54

Predicted tip = 5.63 0.15 = r

Sarah received a 15% tip. 15% = r

Verbal Model 6: PROPORTIONS

A proportion is a statement of equality between two ratios or fractions. They expand the use of Model 5

(Predicted Value = rate per unit • the number of units). Proportions compare the predicted value and number of units of

one thing to the predicted value and number of units of a second thing. Each of these varies by the same rate. It is

easier and more efficient to create a ratio of the two things being compared with predicted value in the numerator and

“number of” in the denominator. Money will usually appear in the numerator and time will usually appear in the

denominator. Remember, the key to a proportion is the phrase “the same rate”.

The verbal model used for proportions is:

Predicted Value1 = Predicted Value2 Number of Units1 Number of Units2

Example 1: Eight is to fifteen as what number is to twenty-four.

8 / 15 = n / 24

This is the way mathematicians write ratios. “Is to” means “/” and “as” means “=”. Therefore, a direct translation

looks like the line under the problem. To solve we write the fractions in the vertical format shown below.

Number = n = 12.8

8

15= number

24 This is the verbal model.

The number is 12.8. 24 .

8

15= n

24. 24 This is the algebraic model.

12.8 = n

18

PV no. of = rates no. of Example 2: Paul can type 115 words in 2 minutes. If he continues to type at the same rate, how long will it take him to

type 1000 words?

PV

wordsminutes=m=17.39

115

2 =

1000

m Paul will take 17.39 minutes to type 1000 words.

1

115 . 115m = 2000 .

1

115

m = 17.39

In Example 2, the verbal model and the data are combined into one ratio: words

minutes=m .

The first sentence creates the first ratio and the second sentence creates the second ratio. Terms with the label “words”

go on the top while terms labeled minutes (time) go in the denominator. Place an equal sign between the two ratios.

Example 3:

Nine is what percent of eleven?

Four is 12% of what number?

What number is 15% of 35?

Verbal Model 7: Predicted Value = Original Value + rate per unit • number of units

This model is very similar to Model 5 except that there is another quantity involved. This other quantity is called

the original value , which means the value that would still exist even if the number of units were zero. Below are two

examples of this model.

OV rate per mile

Example 1: A traveling salesman is reimbursed for his meals and for $.32 for each mile he drives. One day, the

salesman traveled 506 miles and paid $7.62 for lunch. How much money will he be reimbursed?

number of OV PV

Original cost (meals) = 7.62 Predicted Amt = Original Cost + rate per mile • no. of miles

Rate per mile = .32 r = 7.62 + .32 • 506

Number of miles = 506 r = 7.62 + 161.92

Predicted Amount = r = 169.54 r = 169.54

The salesman should receive $169.54.

OC & number of rate per dollar PC

Example 2: The dinner check came to $72.38 and you are going to leave a 15% tip. What is the final cost of the dinner?

Note: In this problem, the tip is based on the cost of the dinner. After the tip is determined, it must be added to the cost of

the dinner to determine the final cost. This can be done in two steps or combine the steps using Model 7.

Predicted Cost = c = 83.24 Predicted = Original + rate per • number of

Original Cost = 72.38 Cost Cost dollar dollars

Rate per dollar = 15% c = 72.38 + 15% • 72.38

Number of dollars = 72.38 c = 72.38 + 10.86

c = 83.24

The final cost of the dinner is $83.24

Use the verbal model:

100is percentof

=

19

Example 3: A vacuum cleaner sold for $269.96 during a 25% off sale. What was the original cost of the vacuum?

Predicted Cost = 269.96 Predicted = Original – rate per • number of

Original Cost = c = 359.95 Cost Cost dollar dollars

Rate per dollar = 25% = .25 269.96 = c – 0.25c

Number of dollars = c = 359.95

1

.75. 269.96 = 0.75c .

1

.75

359.95 = c

The original cost was $359.95.

APPLICATIONS FOR MODELS 3 TO 7: Consecutive Integers

This is the data for consecutive integers problems. You must use the correct data so be very careful to identify which kind

of consecutive integers you are discussing. Consecutive integers can only have answers that are whole numbers with a

positive or negative sign. Consecutive integers may not be fractions or decimals.

Consecutive Integers Consecutive Even Integers Consecutive Odd Integers first integer = f first even integer = f first odd integer = f

second = f + 1 second = f + 2 second = f + 2

third = f + 2 third = f + 4 third = f + 4

fourth = f + 3 fourth = f + 6 fourth = f + 6

Example 1: The sum of two consecutive integers is 33. Find the integers.

first integer = f = 16 first + second = sum

second = f + 1 = 16 + 1 = 17 f + ( f + 1 ) = 33

Sum = 33 2f + 1 = 33

– 1 = – 1

The numbers are 16 and 17. ½ � 2f = 32�½

f = 16

Example 2: Find three consecutive even integers such that the sum of the second and three times the third is the same

as five more than the third.

first integer = f second + 3 • first = 5 + third

second integer = f + 2 ( f + 2 ) + 3 f = 5 + (f + 4 )

third integer = f + 4 4f + 2 = f + 9

–f = –f

There are no integers. 3f + 2 = 9

Integers cannot be decimals. – 2 = –2

⅓� 3f = 7 � ⅓ f = 2.33

20

Model 1 Notes from Power Point Example 1: The sum of seven times a number and five is the same as ninety.

Find the number

Example 2: Find the number

such that

the number decreased by eighty-five is thirty.

Example 3: Twenty-four is fourteen less than twice a number.

Find the number.

Example 4: Eight times a number is the answer to ten times that number increased by four.

Find the number.

21

Use the verbal model format to set-up the following. Verbal Model 1

1. Sixteen more than a number is eleven. Find the number.

2. Eight more than a number is three. Find the number.

3. Five is six less than a number. Find the number.

4. The sum of twelve and a number is sixteen. Find the number.

5. A number less than thirty-five is twenty. Find the number.

6. Three times a number is fifty-one. Find the number

7. 84 is the same as twice a number. Find the number.

8. Sixty-six is half a number. Find the number.

9. One more than three times a number is 7. Find the number.

10. The sum of nine times a number and eight is 44. Find the number.

11. The difference between twice a number and eleven is seventeen. Find the number.

12. The difference between a number and five is twenty-three. Find the number.

13. Seven more than three times a number is 79. Find the number.

14. Twice the sum of six times a number and nine is forty-five. Find the number.

15. Eight times the difference between seventeen and five times a number is the same as four times that

number. Find the number.

16. Find the number such that twelve times the number minus six is the same as three times the same

number subtracted from twenty-four.

17. Find the number such that twice the difference between three times a number and eleven is seven.

22

Model 2 Notes from Power Point Example 1:

Sally has three more than twice Mike.

If Sally has 82,

how many does Mike have?

Example 2:

Claire is three years older than John.

How old is John,

if Claire is 45 years old?

Example 3:

Margie has $85

which is nine less than twice Gloria.

How much does Gloria have?

Example 4:

The painting is 150 years old.

This is twice as old as the sculpture.

How old is the sculpture?

23

Verbal Model 2

Use the verbal model format to set-up the following.

1. John weighs 135 pounds, which is 16 more than Sue. How much does Sue weigh?

2. Alex has $155 in the bank. This is 11 more than six times the amount in George’s bank account.

How much is in George’s account?

3. A player scored seven more points than twice his closest competitor. If the player scored 21 points,

how many points did his competitor score?

4. Albert is 16 years older than three times his nephew’s age. If Albert is 46 years old, how old is his nephew?

5. Cathy rowed the boat 29 minutes longer than three times Karen’s time. If Cathy rowed for 110 minutes,

how long did Karen row?

6. Juan’s age is 65 years less than twice his mother’s age. If Juan is 21 years old, how old is his mother?

7. Mae’s house is 150 years old. This is 45 years younger than the age of Albert’s house. How old is Albert’s

house?

8. Rover has lived for 14 years, which is one year less than three times his owner’s age. How old is Rover’s

owner?

9. Patti has fifty-seven pieces. This is three times Charley’s amount. How many pieces does Charley have?

10. Stuart is six years older than twice Harriett’s age. If Stuart is 31 years old, how old is Harriett?

11. The painting is 55 years old, which is nineteen years older than twice the age of the photograph. How old

is the photograph?

12. Frank is thirty-four years old, which is six years less than eight times Steve’s age. How old is Steve?

13. One number is seventeen less than nine times a second number. If the first number is 82, how much is the

second number?

14. The first number is eight times the sum of the second number and two. If the first number is 36, find the

other number.

15. Lottie has $86, which is $19 more than three times Karen’s amount. How much does Karen have?

24

Model 3 Notes from Power Point Example 1:

The first number is seven less than the second.

Their sum is 453.

Find the numbers.

Example 2:

Lyle has twice as much money as George.

Together they have $526.

How much does George have?

Example 3:

One number is six greater than twice

the second number.

Find the numbers,

if their sum is 90.

Example 4:

Together, a top, pants and sneakers cost $109.94.

How much did each cost,

if the top cost $10 less than the pants and

the sneakers cost $40 more than the top.

25

Use the verbal model format to set-up the following. Verbal Model 3

1. Teresina has fifteen more than four times Karin. Their total is 94. How much does each girl have?

2. Meredith has twenty-five more than twice Jill. Their total is sixty-three. How many does each have?

3. Greg has forty less than twice Jenn. Together they have 37. How many does each have?

4. Andy’s weight is 5 kg less than twice his brother’s. Together they weigh 100 kg. What are their weights?

5. Andrew made four more cookies than twice the number of John’s cookies. Together they made 800

cookies. How many cookies did each make?

6. A cable, 84-m long, is cut so that one piece is 18 m longer than the other. Find the length of each piece.

7. A bottle filled with liquid weighs 9.6 kilograms. If the liquid by itself weighs 5 times as much as the bottle,

what is the weight of the bottle?

8. Greg has thirty-six more than five times Teresina. Their total is 254. How much does each have?

9. The second of two numbers is eleven less than nine times the first. Their total is 567. Find each number.

10. The sum of two numbers is 84. The first is 9 more than 4 times the second. Find the numbers.

11. The larger of two numbers is 1 less than 8 times the smaller. Their sum is 179. Find the numbers.

12. The total of two numbers is 687. The first number is 65 times the second number. Find each number.

13. The sum of two numbers is 87. The larger is 33 less than twice the smaller. Find the numbers.

14. The first number is twenty-four less than seven times the second. Their total is 126. Find the numbers.

15. The larger number is fifty-nine greater than the smaller. Their total is 115. Find the numbers.

16. The sum of three numbers is 61. The second number is 5 times the first, while the third is 2 less than

the first. Find the numbers.

17. Together a chair, a table and a lamp cost $562. The chair costs 4 times as much as the lamp,

and the table costs $23 less than the chair. Find the cost of each.

18. Harry made four times Mark’s amount. Nathan made seven less than Harry. Together they made 506.

How much did each make?

19. Julia has twenty-six less than Sally. Karen has eighteen more than Sally. Together, all three girls

have 91. How many does each have?

20. Albert has nine more than three times George. Sam has two less than Albert. Together, the

three boys have 317. How many does each boy have?

21. The first number is eleven less than the second number. The third number is twelve more than twice

the second number. Their total is 1125. Find each number.

26

Verbal Model 4 Introduction 1

Write the verbal model from page 3 in the workbook that corresponds to each of the problems below:

1. The perimeter of a rectangle is 59 yards. If the length is …

2. Find the missing side of a prism if its volume is 35 cubic feet and …

3. The area of a square is 84 square meters. Find the …

4. The surface area of a sphere is 81 meters2. Find the height if …

5. The volume of a sphere is 169 km3. Find its radius if …

6. The volume of a cube is 36 inches3. Find the missing side if …

7. The length of a rectangle is 4 feet longer than its width. If the area is …

8. The radius of a cone is 19.2 inches. Find its volume.

9. The volume of a prism is 69 km3. Find its height …

10. The surface area of a cube is 45 km2. Its side is …

11. The volume of a cylinder is 80 m3. Its length is …

12. Find the hypotenuse of a right triangle if …

13. Find the sum of the measures of the angles of a pentagon.

27

Verbal Model 4 Introduction 2

Write the verbal model from page 3 that corresponds to each of the problems in the space on the right of the line and write the data to the left of the line:

1. Find the height of a rectangular prism if its volume is …

2. The volume of a prism is 196 cubic meters. Find the …

3. The perimeter of a rectangle is 29 yards. Find the length if …

4. The area of a square is 47.5 km. Find the radius.

5. The volume of a cylinder is 673 m3. Find the …

6. The volume of a cone is 54 square km. Find the …

7. The volume of a sphere is 38.7 cubic inches. Find the radius if …

8. One base edge of a rectangular prism is 945 yards. Find the height if its volume …

9. Find the height of a cone if the volume is 76 miles3 and its …

28

Verbal Model 4 Introduction 3

Show all the steps in the verbal model format. Use the models from pages 3 & 4.

1. The perimeter of a rectangle is 36 inches. The length is 6 inches longer than the width. Find the width.

2. The area of a square is 225 miles2. Find the side.

3. The volume of a cylinder is 87 cm3. Its radius is 5 cm. Find the height.

4. The volume of a prism is 98 cm3. If the length is 6 cm and the width is 10 cm, find the height.

5. Find the length of a rectangular prism if its volume is 620 feet3, its width is 15 ft. and its height is 9 ft.

6. The volume of a rectangular cylinder is 2431 m3. Find its radius, if its height measures 17 m.

7. The volume of a cone is 25.4 km3. Find its radius if its height is 2.3 km?

8. The volume of a cylinder is 84 yards3 and its height is 14 yards. What is its radius?

9. The volume of a sphere is 14.6 cm3. Find its radius.

29

Use the verbal model format to set-up the following. Verbal Model 4

1. A base of a square prism measures 12 feet on a side. Find its height if its volume is 989 feet3.

2. The surface area of a cube is 84 m2. What is the measure of one of its side?

3. The perimeter of a rectangle is 64 cm. Find its dimensions if the width is 13 cm less than its length.

4. The volume of a rectangular prism is 1154 miles3. Two of its sides measure 10 miles and 12 miles. Find

the other side.

5. The volume of a cylinder is 49 yards3. Find the radius if its height is 7 yards.

6. A car travels 135 miles in 6 hours. What is its average speed?

7. A man rides his bike at a rate of 8 miles per hour. How long will it take him to travel 72 miles?

8. John rides his bike 7 miles in 3 hours and 10 minutes. What is his average rate?

9. Sue can run at an average speed of 2 km per hour. How long would it take her to run 9 kilometers?

10. Find the missing height of a cylinder, if its volume is 1449.15 m3 and its radius measures 12.4 m.

11. The area of a square dog pen is 169 square feet. What is the dimension of its sides?

12. A cone has a radius of 5.2 inches. If its volume is 19.7 inches3, find its height?

13. The surface area of a cube is 1734 feet2. Find the length of its side.

14. Find the volume of a cube with a side of 13.53 cm.

15. 76°C is equal to how many degrees Fahrenheit?

16. A sphere has a radius of 17 feet. What is its volume?

17. 18°F is equal to how many degrees Celsius?

30

Use the verbal model format to set-up the following. Verbal Model 4

1. The width of a rectangle is eight more than three times its length. Its perimeter is 608 yd. Find the

dimensions of the rectangle.

2. A cone has a height of 14 inches and a radius of 9 inches. Find its volume.

3. What is the radius of a sphere whose volume is 24429.02 cm3?

4. The volume of a cube is 66 feet3. How long is its side?

5. A prism has a square base that measures 5 inches on a side. Its volume is 145 inches3.

Find the height of the prism.

6. A temperature of 85°C is equal to what temperature in °Fahrenheit?

7. A prism has a square base with a perimeter of 24 in. What is the volume of the prism if its height is 9 in?

8. The surface area of a cube is 1734 ft3. What is the length of its edge?

9. The perimeter of a rectangle is 135 miles. The length is 13 miles longer than twice its width.

Find the length and width of the rectangle.

10. The volume of a cylinder is 904.78 yd3. If the height is 8 yards, how long is its radius?

11. The radius of a cone is 7.2 mm. If its volume is 120 mm3, find its height.

12. A car traveled 425 miles in 8 hours and 10 minutes. What was its average speed?

13. If the volume of a sphere is 4071.5 m3, find its radius.

14. A thermometer measures 25°F. What is the temperature in °Celsius?

15. A ship left port going due north at 16 miles per hour and traveled 217.92 miles. How long was the trip?

16. Find the side of a cube whose volume is 125 cubic meters.

17. Find the length of a prism whose width is 12.3 cm, height is 11.5 cm, and volume is 1273.05 cm3.

31

Use the verbal model format to solve the following completely: Solve Verbal Model 4

1. An angry elephant can run at a speed of about 25 miles per hour. How far can an angry elephant run in

3/4 of an hour?

2. Find the volume of a cone if its radius is 17.58 m and its height is 58 cm.

3. Adam sailed for 50 minutes at an average rate of 8 miles per hour. How far did he go?

4. What is the volume of a cube with a side of 7 feet?

5. A sentry has to walk the perimeter of his home base. The base measures 0.8 miles wide by 1.3 miles long.

How far must the sentry walk if he walks the perimeter twice?

6. Find the volume of a sphere if its radius is 5.6 feet.

7. A container is 11 feet long by 12 feet wide by 15 inches high. What is the volume of the container?

8. Adam is going to tile his bathroom floor that is 12 feet long by 7 feet-6 inches wide. What is the area of the

floor?

9. The dimensions of the tissue box are 5.5” by 4.5” by 4.5”. What is the volume of the box?

10. The base of an isosceles triangle is 35 cm. Each of the legs measures 18 mm. Find the perimeter of the

triangle

11. The sides of a rectangular prism measure 18.4 cm, 16.3 cm and 13.7 mm. Find its volume.

12. Find the side of a cube with a volume of 125 feet3?

13. Find the side of a square with an area of 361 yards2.

14. A car travels at an average speed of 54 mph. How far will it travel in 6 hours?

15. A farm measures 10.3 miles by 12.8 miles. What is the area of the farm?

16. A prism has a square base with a perimeter of 44 in. What is the volume of the prism if its height is 12 in?

17. John’s design had a circle with a radius of 6.3 inches, a triangle with a base of 5.4 inches and a height of

9.2 inches, and a square with a side of 7.3 inches. None of the shapes overlapped. What was the total area

that the three shapes covered in his design?

18. Find the perimeter of a square if its area is 256 square feet.

19. How many cubes that measure 4 inches on a side can fit into a rectangular box with dimensions of 10

inches by 5 inches by 8 inches?

20. How many balls with a radius of 3 inches can fit into a box with dimensions of 12 feet by 4 feet by 3 feet?

21. A man drove from 12:00 noon to 8:00 PM stopping twice, once for dinner, which took 75 minutes and

another time for a 15-minute snack break. If he drove 313 miles that day, what was his average rate (miles per

hour) on the trip?

32

Directions: Write the data for each using the given information. Verbal Model 5: Identify Rate/Number

1. 5 feet per sec 7. 4 mph over the speed limit

2. 9 miles 8. 16 sweaters

3. $7 each 9. 8 cents per can

4. 19 snakes 10. 6 games

5. 16 elephants 12. 19 fish

6. 12 mph 14. $.25 a piece

33

Model 5 Notes Predicted Value = rate per unit • number of units

Example 1: Al wants to buy six candy bars that cost $.45 a piece. What was the final cost?

Example 2: John bought a stapler for $9.99 but has to pay a 6% sales tax. How much sales tax did he pay?

Example 3: The discount on a sweater marked $29.95 was $7.49. What was the rate of discount?

Example 4: If you leave 15% tip of $3.68, what was the cost of just the meal?

34

Solve the following using the verbal model format. Show all work. Solve Verbal Model 5

1. Emily purchases four CDs that cost $11.95 each. What is the cost of the CDs before sales tax?

2. Margaret paid $10.50 a piece for three t-shirts at Old Navy. What was the total cost?

3. Alison went to Office Max to purchase 8 folders that cost seven cents a piece. How much did she pay for

the folders before sales tax?

4. Larry wants to buy six donuts but only has $3.50. What is the maximum amount of donuts he can buy?

5. A computer game costs $49.99 at the local store that charges a 6% sales tax. How much was the sales

tax on the game?

6. Steve bought an item marked $57.35 and paid $2.87 in sales tax. What was the rate of the sales tax?

7. Jose bought a CD marked $18.99 in The Gallery in Philadelphia. Since Philadelphia charges a 8% sales

tax, how much tax did Jose pay?

8. The “Item of the Week” at Old Navy costs $24.90. How many items could be purchased if the buyer had

$200? (No sales tax was paid on clothing.)

9. The Jones family had a $2 off coupon for an amusement park that was good for an unlimited number of

tickets purchased on the same day. If the Jones family bought six tickets, how much would they save?

10. Sue has a coupon to save on sportswear at the local department store. If she buys jeans marked $52.90,

and saved $13.23, what was the percent off listed on the coupon?

11. Maddie gets a 15% discount on an item marked $42. If she purchases it with the coupon, what will be the

discount?

12. Mr. Patrick was fined $18 for each mile per hour over the speed limit he drove. He was traveling 69 mph in

a 55 mph zone. What was his fine?

13. The company pays its employees $0.57 for each mile they drive to a conference. How many miles away

is the conference if the employee received $94.28?

14. Five friends went to McDonalds’s to have lunch. It took the server 1.6 minutes per customer to take the

order and fill it. How long did it take before the friends could sit down together if the same person served

them all?

15. Joe wants fish for his tank. He chose 2 fish that cost $7.99 each, only one fish that cost $6.25, and 6 that

cost $2.98 each. How much will the 9 fish cost if there is no sales tax?

16. Larry was going 46 mph in a 35 mph zone when the police stopped him. If the fine is $128.50 for each

mph over the speed limit, what was Larry’s fine?

17. When you go to a restaurant, you should leave a 15% tip for the person who waited on you. Calculate the

tip for each of the following checks using the verbal model format.

a. $6.45 b. $48.64 c. $136.21 d. $3.66

35

Model 6 Notes -- IS TO percentis

of 100=

Example 1: Example 4:

Six is to seven as twenty-four is to what number? 14 is 35% of what number?

Example 2: Example 5:

Nine is to what number as one hundred-eight is to sixty? What number is 5% of 60?

Example 3: Example 6:

What number is to fifty as seventeen is to four? 48 is what percent of 64?

36

Model 6 Notes -- Pr edValuenumber of

Example 1:

Mr. Adams drives 330 miles in 6 hours. At this rate, how far will he drive in 8 hours?

Example 2:

If you can purchase 9 kits for $31.50, how much will 15 kits cost at the same rate?

Example 3:

Mike eats 5 pies in 4 minutes. If he continues to eat at the same rate, how long will it take him to eat 23 pies?

37

Solve the following using the verbal model format. Show all work. Solve Verbal Model 6

1. Seven is to nine as fourteen is to what number?

2. Eleven is to thirty-three as what number is to twenty?

3. What number is to forty-nine as three is to seven?

4. Fifty-one is to what number as thirty-four is to two?

5. 52 compared to what number is the same as 4 compared to 14?

6. 16 compared to 80 is the same as 47 compared to what number?

7. At the rate of 3 items for $.10, how many items can you buy for $.50?

8. At the rate of $9.50 for 19 items, how much will 8 items cost?

9. How much will 8 items cost at the rate of 6 items for $9?

10. Five times a number is to fourteen as three is to five. Find the number.

Set up the following using the verbal model format. DO NOT SOLVE.

11. The sum of six and a number is to 56 as the difference between eight and the same number is to 30. Find

the number.

12. The total of twice a number and nine is to eleven as three times the same number is to 55. Find the

number.

13. Find the number such that the sum of four and six times a number is to nine as eleven times that number

is to three.

14. Forty-one is to twelve as the sum of nineteen times a number and eight is to five. Find the number.

Solve the following word problems using the verbal model format:

15. What number is 24% of 52? 16. What number is 87% of 41?

17. 90 is 45% of what number? 18. 260 is 65% of what number?

19. 36 is what percent of 45? 20. 15 is what percent of 50?

21. What number is 120% of 45? 22. 578 is what percent of 1345?

23. 18 is 36% of what number? 24. 74% of what number is 370?

38

Solve Verbal Model 6

Solve the following using the verbal model format. Show all work.

1. A motorist traveled 190 miles on the turnpike in 2.5 hours. How long will it take her at the same rate to

travel 380 miles?

2. Alex caught four fireflies in six minutes. How long will it take him to catch ten fireflies at the same rate?

3. The club washed eight cars in the first half-hour. At this rate, how many cars will it wash during its five-hour

car wash fundraiser?

4. Stan is planning a long-range trip. He knows he can travel 432 miles in 9 hours. At this rate, how long will it

take him to travel 3500 miles?

5. Helen earned $67 in 4 days. At these rates, how many days will it take Helen to earn $536?

6. Jennifer sold two houses in the last three weeks. If this trend continues, how many houses will she sell in

eight weeks?

7. If 3 gallons of paint cover a surface containing 658 square feet, how many gallons at the same rate will be

needed to paint a surface containing 1974 square feet?

8. A picture 2.5 inches wide and 3.25 inches high is to be enlarged. If the height of the enlargement will be

9.75 inches, how wide will it be?

9. Mr. Jones traveled 458 miles on 11 gallons of gas. At this rate, how far can he travel on 18 gallons of gas?

10. A recipe calls for 4 cups of flour and 6 tablespoons of shortening. How many tablespoons of shortening

are needed when 6 cups of flour are used?

∆ABC and ∆DEF are similar. For each set of measures given, find the measures of the remaining sides.

11. c = 11, f = 6, d = 5, e = 4

12. a = 5, d = 7, f = 6, e = 5

13. a = 17, b = 15, c = 10, f = 6

14. a = 16, e = 7, b = 13, c = 12

15. d = 2.1, b = 4.5, f = 3.2, e = 3.4

16. f = 12, d = 18, c = 18, e = 16

17. c = 5, a = 12.6, e = 8.1, f = 2.5

Cost or Money or Distance or Miles is almost always in the numerator.

Time is almost always in the denominator.

Glencoe Algebra 1, p. 204

E

F D

B

C A

a c

b

d f

e

39

Model 7 Notes -- Predicted Value = Original value + rate per unit • number of units

Example 1: A repair shop charges a $65 service charge plus a fee for labor of $32.50 per hour. If a

serviceman works for 1.5 hours, what is the final cost to the customer?

Example 2: An Apple I-pad mini 3 is on sale for $299. What is the final cost of the I-pad mini if there is a 6%

sales tax?

Example 3: You go to Friendly’s for lunch. The final cost, including a 15% tip, was $18.80. What was the

actual amount of tip that the wait-person received?

Example 4: What was the original cost of a pair of designer jeans if you paid $84.80 during a 40% off sale?

40

Solve the following using the verbal model format. Show all work. Verbal Model 7

1. Eleanor purchases come to $85.33. However, there is a 6% sales tax. What is her total purchase price?

2. A company charges a 7% delivery charge on an order of $1732.43. What was the total cost to the

customer?

3. The restaurant bill came to $18.50. If you leave the waitress a 15% tip, what was the real final cost of the

dinner?

4. What is the total cost to Warren if the item costs $16.80 and the sales tax is 6%?

5. A Nike windbreaker is marked $45 at Modell’s. There is a 40% off sign above the rack. What is the final

sale price of the windbreaker?

6. What is the final cost of a bookcase marked $250, if the sales tax is 7%?

7. How much did the entire dinner cost including an 18% tip, if the cost of the food alone was $98.52?

8. Sharon’s share of the bill came to $12. If she leaves a 15% tip for the waitperson, how much money must

she leave to cover her share of the bill including tip?

9. Adidas shorts are marked $22 in a store with a 25% Off Everything in the Store sale. The discount will be

taken at the cash register. What is the price of the shorts if you purchase them during the sale?

10. To repair an electronic device, a repair shop charges a $50 service charge plus $45 per hour for labor. If

the serviceman works for 1.4 hours on an item, how much will the customer be charged?

11. A man agrees to put $2500 down and to pay $175 per month for three years to purchase a used car. What

is the final cost of the car after the three years?

12. What is the final cost to repair a car that the mechanic worked on for 0.75 hours, if the parts cost

$83.29 and labor cost $68 per hour?

13. John paid a total of $37.94 for a set of sheets and two extra pillowcases. If the extra pillowcase cost $4.97

each, what was the cost of the sheets?

14. A dress originally marked $69 sold for $47.90. What is the rate of discount during the sale?

15. Your credit card was charged $68.54 at the restaurant including an 18% tip. What is the original cost of

meal without the tip?

16. What is the original cost of a microwave if the final cost after paying a 6% sales tax was $31.78?

17. What is the original cost of a pair of Diesel Jeans during a 33% off sale if the final cost was $106?

18. Sue gave the sales clerk a discount coupon worth 25% off for the purchase of a dress. If the cost of the

dress after the discount was $101.93, what was the original cost of the dress?

19. Parts for the washer cost $163.29 while the labor cost $79 per hour. If the final cost for repairing the

washer was $262.04, how many hours did the repairman work on the washer?

41

1. Video Game Membership: A local store charges $8 to rent a video game for three days. You must be a

member to rent from the store, but the membership is free. A video game club in town charges only $3 to

rent a game for three days, but the membership in the club is $50 a year. Which membership is more

economical? 2. Bike Safety: Suppose you live near a park that has a bike trail you like to ride. The Parks Department

rents a bike with safety equipment for $5 a day. If you provide your own safety equipment, the bike rental

is $3 a day. You could buy the equipment at a sports store for $28. How many times must you use the

trail to justify buying your own safety equipment? 3. Catching Up with Sis: Kate is always reminding her younger brother Tony that she is taller than he is.

Kate is 63” tall and is growing at a rate of 13 inch a year. Tony is 60” tall and is growing at a rate of 2 1

3

inches per year. How long will it take Tony to catch up with Kate?

4. Summer Swimming: A new swimming pool is opening for 15 weeks during the summer. You can swim in

the afternoon for $3 or buy a membership for $80 and pay only $1 for the afternoon session. You must

decide whether to buy a membership or to buy daily passes. 5. Remote Control Cars: The Fast Track Company manufactures toy remote-control race cars, which it sells

for $18 each. The production cost for the company is $2000 per day plus $13 per race car. How many

cars must the company sell in a day to break even? 6. The Gazelle and the Cheetah: A gazelle can run 73 feet per second for several minutes. A cheetah can

run faster (88 feet per second), but it can only sustain its top speed for about 20 seconds. Gazelles seem

to have an instinct for this difference because they will not run from the prowling cheetah until it enters its

“safety zone.” This is the distance the cheetah would need to run to overtake the gazelle in 20 seconds if

both are running at top speed. How close should the gazelle let the cheetah come before it runs? 7. High School Enrollments: Cleveland High is in the city and West Lake High is in one of the suburbs.

Cleveland High’s enrollment has been decreasing at an average rate of 75 students per year, whereas

West Lake High’s enrollment has been increasing at an average rate of 60 students per year. Cleveland

High has 3150 students, and West Lake High has 2475. If enrollments continue to change at the same

rates, when will the two schools have the same enrollment?

8. Running a Race: Two runners are running on a 21 kilometer course. The first runs at 10 kilometers per

hour, and the second runs at 15 kilometers per hour. If the first runner is 7 kilometers past the starting line

before the second runner starts, how far does each run before they are side by side? 9. Left Behind: An elephant herd is migrating to greener plains. The herd is moving at about 10 miles per

hour. One elephant strays from the herd, stops, and is left behind. Then it senses danger and begins

running at about 25 miles per hour to reach the others. It takes the stray elephant 5 minutes to catch up

with the herd. How far had the herd traveled when the stray elephant became frightened?

10. Population Growth: From 1987 to 1988 in the United States, the population of the western region

increased by 982,000 and that of the Midwest region increased by 222,000. In 1988, the population of the

western region was 50,679,000 and that of the Midwest region was 58,878,000. If the populations continue

to change at the same rates, when will the populations of the western region and Midwest region be the

same? 11. Temperature Change: Suppose you live in Greenville, South Carolina, where the temperature is 69

0F and

going up at a rate of 20F an hour. You are talking on the phone to a friend, who lives in Waterloo, Iowa,

where the temperature is 840F and going down at a rate of 3

0F an hour. If the temperatures continue to

change at the same rates, how long would you and your friend have to talk before they would be the

same?

Heath Algebra 1

Verbal Model 8

42

Use the verbal model format for the following. Verbal Models 1 to 4 Review

1. Seven more than three times a number is eighteen. Find the number.

2. Jason has fourteen more than three times Evelyn’s amount. Together they have 130 marbles.

How many marbles does each have?

3. Seven times the difference between eleven and twelve times a number is four. Find the number.

4. April is building a dollhouse that is 25 inches wide, 20 inches deep, and 30 inches tall. What is the volume

of the house?

5. A car traveled 1453 miles over a 27-hour time period. What was its average speed?

6. The quotient of nine times a number and fourteen is seven. Find the number.

7. One number is 28. This is nine more than three times a second number. Find the second number.

8. George has twelve books more than seven times Jonathan’s amount. Together, they have 44 books. How

many books does each have?

9. Julian has $65, which is $11 more than his friend Dylan. How many does Dylan have?

10. Find the radius of a sphere if its volume is 457.33 cubic feet.

11. Arthur has 23 books, which is twelve books less than Jonathan. How many books does Jonathan have?

12. How fast does a car have to go to travel 215 miles in 3.6 hours without stopping?

13. Find the edge of a cube when the volume is 56 yards3.

14. One number is sixteen more than five times a second number. Their sum is 58. Find the numbers.

15. Eleven is twice the sum of eight times a number and twelve. Find the number.

16. Greg has earned 4 more than twice the number of Matt’s points. Together, they have earned 19 points.

How many points has each earned?

17. Find the radius of a sphere with the volume of 13.65 feet3.

18. John has 18 more papers to deliver than Sam. If John delivers 103 papers, how many does Sam deliver?

19. The volume of a rectangular cylinder is 408 yards3 and its height is 17 yards. Find the radius of the base.

20. Nineteen less than twelve times a number is 91. Find the number.

21. Find the missing radius of a cone if its height is 8.3 mm and its volume is 454.49 mm3.

22. The first of three numbers is six times the second. The third is nine less than the first. Their sum is 184.

Find the numbers.

23. A sofa costs $11 more than twice the chair. The table costs half as much as the chair. The total of the

sofa, table and chair is $1852. Find the cost of each.

24. Find the number such that eleven times the difference between nine times a number and twenty is 462.

25. Lillian has 9 more dolls than Suzanne. If Lillian has 25 dolls, how many does Suzanne have?

26. Lillian has 9 more dolls than Suzanne. Together, they have 41dolls. How many does each have?

43

Solve the following using the verbal model format. Show all work. Verbal Models 5, 6, 7 Review

1. Marley purchased a ski jacket for $140 using a coupon for 15% off. How much will she pay for the jacket if

she did not use the coupon?

2. Alexandra had dinner at Friendly’s. The bill came to $27.35 including a 15% tip. How much did the food

alone cost?

3. Jan tries to convince her parents to buy her a pair of $126 rollerblades. How much tax will they have to

pay if the sales tax rate is 6%?

4. Keith traveled 1453 miles in one month for his company and was reimbursed money for each mile he

drove. What was the reimbursement rate if he was reimbursed $813.68?

5. Cedric types 63 words a minute. How many words can he type in 120 minutes?

6. Abe makes 17% commission on all the clothing he sells. One customer spent $687.92 with Abe’s help.

How much money did Abe earn from the sale?

7. Erica collected $23.75 for every goal she makes in the Lacrosse Shot-a-thon. If Erica makes 84 goals,

how much money will Erica make for her team?

8. Jiffy-Lube promises to take care of each car in 25 minutes. How many hours will it take the company to

service 17 cars if only one mechanic works on all 17 cars?

9. A man purchased six tickets to an amusement park and was charge $285 on his credit card. Find the cost

per ticket to the park.

10. A Philadelphia store charges a 8% sales tax on bathing suits. How much tax will you have to pay for a

bathing suit marked $84.99?

11. Benton drove the limousine for 2.5 hours and covered over 162.5 miles. What was the average speed of

limo in miles per hour?

12. Will has a 25% off coupon for Kohl’s. He wants to buy an $85 pair of designer jeans. How much will he

save with the coupon?

13. Jim left $51.48 to cover the 20% tip and the cost of the food. How much was the meal alone? How much

was the 20% tip?

14. How much will you save on an item that costs $12.54 during a “15% Off Everything Sale”?

15. Claire likes a carpet that sells for $14.99 per yard2. If she needs 9.7 yards

2, how much is her bill?

16. How much sales tax will you have to pay on computer equipment selling for $1899 if the sales tax is 8%?

17. How far will a train travel going 86.9 mph for 9.6 hours?

18. How much is the 17% tip on a meal that cost $87.59?

19. The final cost of a $135 outfit during a sale was $36.99. What was the discount rate?

20. Monica purchased matching gifts for $37.99 each. If her final cost was $151.96, how many items did she

buy?

44

Solve the following using the verbal model format. Show all work. Verbal Models 1 to 7 Review

1. Scarlet was sent to the store to purchase 5 loaves of bread that cost $1.59 each. What was her total bill?

2. John has 16 more soda can tabs than twice Alex. If they have 643 tabs altogether, how many does each

have?

3. Eighteen is to 48 as what number is to eight?

4. Find the number if six times the sum of twelve and twice that number is 492.

5. What is the original cost of a washing machine if the customer has to pay a 7% sales tax and the final cost

was $469?

6. Find the rectangular prism’s length when its volume is 1116 cm3 and its dimensions are 12 cm, 14cm?

7. John can purchase 12 articles for $108. How much will he pay for 15 articles at the same rate?

8. The measure of the first angle of a triangle is 104°. The measure of the second angle is 23°.

Find the measure of the third angle.

9. Eighty-four is what percent of 142?

10. Joe paid $75 for shoes during a 25% off sale. What was their original cost?

11. Lady Godiva rode through the village on a horse for a total of eight miles. If she rode at an average rate of

three miles per hour, how long was her horse ride?

12. Mrs. Sanchez is giving beach towels to seven cousins. Each towel costs the same price. What is the cost

of the towels before tax if the total cost was $102.13?

13. A dozen donuts cost $7.80. At this same rate, how much will eight donuts cost?

14. How fast does a car have to go to travel 315 miles in 4.5 hours without stopping?

15. Elizabeth bought six novels at Target that cost $7.99 each. What was her cost if she purchased them in

Delaware where there is no sales tax?

16. There are 24 sport cars in a car lot. Each car is either red or white. There are four fewer red cars than

white cars. How many cars of each color are there in the lot?

17. Twenty-five is to forty-four as 150 is to what number?

18. Phil has 28 apps for his cell phone. This is one more than three times the number of Dylan’s apps. How

many apps does Dylan have for his computer?

19. Monica bought a bathing suit for $35.98 after paying a 6% sales tax. What was the original cost of the

bathing suit?

20. The dinner came to $64 and Mike volunteered to leave a 15% tip. How much money did the waitperson

receive?

21. The perimeter of a rectangle is 78 square feet. If the length is six feet more than the width, find the width.

45

Solve the following using the verbal model format. Show all work. Verbal Model 1 to 7 Review

22. Jeans were on sale for 20% off. Mary paid $63.92. How much were the jeans before the sale?

23. Jill worked 45 hours, which was 16 hours more than Samantha. How many hours did Samantha work?

24. If 5 items cost $75, how much will 8 items cost at the same rate?

25. The bill for dinner came to $256 and Mr. Smith left a 20% tip. What was the final cost of the dinner?

26. Amanda worked 9 hours more than twice Jesse. If Amanda worked 23 hours, how long did Jesse work?

27. Find the length of a edge of a cube if its volume is 2197 feet3.

28. What number is 35% of 157?

29. One number is three more than seven times the second number. Find both numbers if their sum is 95.

30. Forty-one is 65% of what number?

31. A chair, originally marked $14.99, is on sale. If the sale is 25% off everything in the store, how much is

the sale price of the chair? DO NOT include sales tax.

32. Two men traveled to Dodge City. The first man took twice as long as the second man to reach Dodge City.

If they rode a total of 45 hours, how long was each on the road traveling?

33. What is the actual 6% sales tax on a Bluetooth speaker that is marked as $144.99?

34. Alexis rode her bike for eight miles longer than three times Monica’s ride. If Alexis rode for 44 miles, how

far did Monica ride her bike?

35. How much is a 15% tip on a dinner that cost $37.54?

36. The measure of one angle of a triangle is 88° more than the second angle and the measure of the third

angle is 3° less than the second. Find the measures of all three angles.

37. Mary purchased a $24 pair of pants, a $16 shirt, a $49 pair of shoes, and a new $20 purse. It was all 20%

off. What was her final cost for everything after the discount? There was no sales tax.

38. Alan played the piano seven minutes less than four times Colleen. Together they played 55 minutes. How

long did each play?

39. Sam purchased some tools that cost $13.99, $29.98, $56.99 and $2.49. He had to pay an 8% sales tax.

What was the total amount he had to pay?

40. Nine times the difference between twelve and twice a number is 848. Find the number.

41. Sue walked for 15 miles longer than Christine during the walk-a-thon. If Sue walked for 22 miles, how far

did Christine walk?

42. George has six more trophies than twice David. Together, they have 18 trophies. How many does each

have?

43. Find a number such that eleven times the difference between nine times that number and twenty is 462.

46

Pythago

rean T

heorem

: c2 = a

2 + b2

Pythago

rean T

riples:

47

Which of the following are right triangles?

1. 3, 4, 5 2. 6, 4, 5 3. 13, 12, 5 4. 11, 6, 9

5. 7, 24, 25 6. 13, 10, 8 7. 6, 11, 157 8. 9, 14, 115

9. 9, 7, 32 10. 12, 20, 24 11. 9, 40, 41 12. 2, 2.5, 1.5

13. 16, 356 , 10 14. 150 , 4, 13 15. 15 , 7 , 8 16. 1.7, 1.5, 0.8

Solve the following using the Pythagorean Theorem. Draw a picture.

1. A fire truck parks 15 ft away from a building. The fire truck extends its ladder 39 ft. How far up the building

from the truck's roof does the extension ladder reach?

2. Carson found an old tent in the attic of his house and decided to set it

up in the back yard. However, the support sticks for the tent are

missing. If the tent is 60 inches across on the bottom and 34 inches on

each side, how tall of a stick does he need to set up the tent?

3. A television has a rectangular screen with a diagonal measurement of 30 inches. If the screen has a height

of 18 inches, what is the width of the screen?

4. Two ships leave port at the same time. Ship X is heading due north and Ship Y is heading due east. Six

hours later they are 300 miles apart. If the Ship X had traveled 240 miles from the port, how many miles

had Ship Y traveled?

5. Jeremy goes to White Water Amusement Park. While there he decides to go down the park's huge

waterslide called Lightning. If the slide is 120 feet high and the base of the slide is 90 feet from the pool,

then what is the length of the slide?

In #6 to 10, find the length of each of these lines in a coordinate plane.

7

6

9

1 0

8

48

Solve the following using the PYTHAGOREAN THEOREM. c2 = a2 + b2 where c is the longest side.

1. What is the length of the diagonal of a square that measures 16.3 inches on a side?

2. The diagonal of an i-phone 4S is 12 cm and its width is 6 cm. Find its length.

3. A tent is 15 feet across the bottom and has no sidewalls. There is a 6-foot pole in the middle that holds up

the top of the tent. How long is the top of the tent from side to side?

4. A base of a pyramid is formed by a right triangle. The longest side of the triangle is 26 m while the shortest

side is 17.7 m. How long is the other side?

5. A roofer needs to inspect the roof of a house that is 35 feet tall. If he sets the ladder 22 feet away from the

base of the house, how tall must his ladder be to reach the roof of the house?

6. Can the following be measurements for the sides of a right triangle? 31.5 feet, 75.6 feet, 81.9 feet Explain why or why not.

7. A 18-foot flagpole casts a 13-foot shadow. If a string were connected from the tip of the flagpole to the

furthest point of the shadow away from the pole, how long would the string have to be?

8. A circle is drawn on the ground so that two of its radii form a right angle. The area of the circle is 803.84

feet2. How long is the chord that connects the two radii?

9. The perimeter of a triangle is 36 cm. Its second side is 3 cm longer than the first side while the third side is

6 cm longer than the first side. Is the triangle a right triangle? Explain.

10. A circle is drawn on the ground so that two of its radii form a right angle. The chord that connects the two

radii is 15.56 inches long. How long is the radius of the circle?

11. A football field measures 160 feet wide by 300 feet long. If a person walked diagonally across the field,

how far would they have to walk?

12. The diagonal of a flat screen television is 55 inches. If its height is 33 inches, what is its width?

13. Two co-workers leave their office at the same time. Mr. Porter heads due east while Mr. Adams heads

due south. After one hour, they were 15 miles apart. If Mr. Porter had traveled 9 miles, how many miles did

Mr. Adams travel?

14.

14. In the rectangular prism above, the length of MR is 8 inches; the length of RS is 9 inches, and the length of

ST is 12 inches. What is the length of a line segment drawn from point T to point M?

15. In the rectangular pyramid, L = 32 inches, w = 24 inches, and h = 30 inches. What is the length of s?

16. What is the distance between A and B in the coordinate plane above?

15. 16.

49

Solve completely the following word problems using the verbal model format: Verbal Models 1 to 9 Review

1. The volume of a rectangular prism is 43200 m3

. If its length is 36 m and its height is 48 m, find its width.

2. Alice earned $54, which is $36 less than three times Sue’s amount. How much does Sue earn?

3. John drove 450 miles in 7.4 hours. What was his average rate?

4. The final cost of a refrigerator (including an 8% tax) was $744. What was the original cost of the refrigerator?

5. The radius of a sphere is 6.2 inches. Find its volume.

6. Find the number such that six times a number less than nine is the same as four times the same number subtract one

7. The hypotenuse of a right triangle measures 36 feet and one leg measures 8 feet. Find the length of the other leg.

8. A tree cast a 25-foot shadow at the same time as a 6-foot pole casts a 2.5-foot shadow. How tall is the tree?

9. The waiter received a tip of $29.67, which represented 20% of the food check. How much was the check for the food?

10. The volume of a cone is 314.16 m3

. If its height is 12 m, what is the radius of the cone?

11. A pair of jeans went on sale for $38.86. What was their original cost if the discount rate during the sale was 33%?

12. The first number is twelve more than five times the second number. Their sum is 432. Find each number.

13. A simple recipe calls for 4 cups of flour for every 1.5 cups of sugar. Amanda only has 3 cups of flour. How much

sugar must she use with 3 cups of sugar to keep the taste the same?

14. Julie saved $11.62 on an item that was marked $64.58. What was the rate of discount?

15. Find the missing sides in these sets A. B.

of similar triangles.

16. Sloan made 35 more cards than five times Monica’s amount. If he made 245 cards, how many did Monica make?

17. Seven times the difference between eight times a number and five is the same as four times the sum of eleven times

that number and twelve. Find the number.

18. Jeanne ran 15 miles more than twice Alison’s distance. Together they ran 72 miles. How far did each girl run?

19. Two boats left the same dock at the same time. One traveled due north at 11 knots per hour. The second traveled

due south at 14 knots per hour. After how many hours were the boats 150 knots apart? How far had each boat gone?

20. What number is 84% of 26?

21. The sum of nine times a number plus three is to eight as the difference between seven and six times the same

number is to six. Find the number.

22. A square has an area of 990 square feet. How long is one of its sides?

23. Fifty-five is to thirty-three as twelve is to what number?

24. A rancher rode 32 miles on horseback over 8 hours. At this rate, how far can he go in 10 hours?

25. Emily has $4536 and deposits $35 each week into that account. Joanne has $2936 and deposits $55 each week into

her account at the same bank. If this continues, when will their savings accounts contain the same amount?

19 ft 8 ft y ft 12 ft 14 ft k ft

26 m 10 m 32m w m 12 m x m

50

Solve completely the following word problems using the verbal model format: Verbal Models 1 to 9 Review

26. Two boats left the same dock at the same time. One traveled due north at 16 knots per hour. The second traveled

due east at 15 knots per hour. After nine hours, how far apart were the boats?

27. A prism has a square base with an area of 289 feet2

and a height of 7 yards. Find the volume of the prism.

28. The legs of an isosceles right triangle are each 11.5 inches long. How long is the triangle’s base?

29. A 25-feet high statue and casts a 6-feet shadow while a building casts a 14-foot shadow. How high is the building?

30. The perimeter of a rectangle is 86 feet. Its length is 25 feet more than five times its width. What is the length and

width of the rectangle?

31. John bought a $24.99 frying pan, a $48.89 set of dishes, and $19.56 of assorted cooking utensils. If he purchased all

of these items during a 15% off sale, what was the final cost of all the items?

32. The diagonal of a rectangle measures 38 feet while the length is 21 feet. What is the width of the rectangle?

33. The page of your school yearbook is 8 ½ inch by 11 in. The left and right margins are ¾ in. and 2 7

8 in, respectively.

The space between pictures is 3

16 inch. How wide should you make each picture to fit three across the page?

34. The volume of a prism is 2536 in3

. Its length is 15 inches and its height is 13.5 inches. Find its width.

35. The first number is twelve times a second number while a third is three more than fourteen times the second number.

The sum of all three numbers is 65.1. Find the values of all three numbers.

36. A cylinder has a volume of 1778.18 feet3

and a height of 20.1 feet. Find its radius.

37. Twelve times a number less than fifteen equals twenty-four times the same number plus one. Find the number.

38. Matt mows the 1200 ft2

lawn in 85 minutes. At the same rate, how long will it take him to mow a 1600 ft2

lawn?

39. Monroe traveled 125 miles on the first day, 479 miles of day 2, and 635 miles on day 3. If his average speed was 45

mph over the three-day period, how many hours was he actually driving during the trip?

40. Can the following dimensions form a right triangle? Prove your conclusion. 6 inches, 7 inches, 13 inches.

41. Jen works 4 hours less than twice Maggie’s hours. If Jen works 42 hours this week, how long did Maggie work?

42. How far can a car travel on 14 gallons of gas if at the same rate, it can travel 1584 miles on 49.5 gallons of gas?

43. The volume of a sphere is 104.72 yd3

. Its height is 3 feet. Find its radius.

44. The length of a rectangle is 42 meters less than six times its width. If the perimeter measures 1134 meters, find the

length and width of the rectangle.

45. What is the original cost of a dress if you paid $110.49 during a 15% off sale?

46. The area of a triangle is 123.58 yard2

. If its base measures 14.8 yards, what is its height?

47. The population in Hargrove is 5,504 and is increasing at a rate of 15.2% each year. Across the state, Joshua has a

population of 6,195 but is decreasing at a yearly rate of 19.8%. When will the populations be the same?

48. Mr. Roman is going to paint his house using a 20-foot ladder. He sets the base of the ladder 15 feet away from the

house. How far up the side of the house will the ladder reach?

51

Solve the following using SYSTEMS OF EQUATIONS.

1. The perimeter of a rectangle is 154 inches. The length is seven less than twice the width. Find the

dimensions of the rectangle.

2. John has $19 more than three times Harry. Together they have $383. How much does each have?

3. George has $65 to purchase 20 plants, a combination of marigolds and geraniums for his garden.

Marigolds cost $2.80 per container and geraniums cost $3.50 per container. How many of each plant can he

purchase?

4. Five times Lulu’s amount added to six times Mike’s amount is 38, but Lulu has one more than Mike. How

much does each have?

5. The Smith family order four hamburgers and five sodas that cost $23.31. The Kuncio family order eight

hamburgers and ten sodas that cost $46.61. What is the cost of each hamburger and each soda?

6. The width of a rectangle is 13 yards less than four times its length. The perimeter is 114 yards. Find the

dimensions of the rectangle.

7. Mark wants to buy chocolate chip and peanut butter cookies for a party. Chocolate chip cookies cost $3.69

a pound while peanut butter cookies cost $2.79 a pound. He pays $14.30 and gets 4 pounds of cookies. How

many pounds of each kind of cookie did he purchase?

8. Mr. Jordan bought 0.25 pounds of American cheese and 1.33 roast beef for $12.93. Mrs. Armstrong bought

1.25 pounds of American cheese and 1.25 roast beef for $17.73. What was the cost per pound of the

American cheese and of the roast beef?

9. Ellie has $167 more than twice Tom. Together, they have $593. How much money does each have?

10. How many liters of a 22% solution of antifreeze must be combined with pure antifreeze to obtain 12 liters

of a 64% solution of antifreeze?

11. How many ounces of pure orange juice must be combined with a 35% orange juice concentrate to obtain

64 ounces of 60% concentrate?

12. The Logan Family spent $21.10 on two small blizzards and three banana splits. The Marcos Family spent

$14.95 on three small blizzards and one banana split. How much did each blizzard and each banana split

cost? What would be the cost of five small blizzards and two banana splits?

13. Twenty of item A and eleven of item B cost $701. Fifteen of item A and seventeen of item B cost $797.

What is the cost of each item? How much will seven of item A and nine of item B cost?

14. How many pounds of chocolate-covered peanuts at $8.50 a pound can be combined with chocolate

covered raisins at $6.45 a pound to create a mixture of 15 pounds at $7.75 a pound?

15. Joe won five more games than twice Kevin on the first night. On the second night, Joe won three less

games than four times Kevin. If they played the same amount of games each night, how many games did

each person win?

1. Identify two variables.

2. Write two equations from the given information.

3. Solve the equations using the substitution or elimination methods.

52

• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80

5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120

7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140

8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160

9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180

10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220

12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240

13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260

14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280

15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300

16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320

17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340

18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360

19 19 38 57 76 95 114 133 152 171 190 209 28 247 266 285 304 323 342 361 380

20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400