2-9A Solving Equations from Word Problems

Basic Amount/Sum ProblemsRectangle

Problems

Algebra 1 Glencoe McGraw-Hill Linda Stamper

let f = first basket

There are 54 kilograms of apples in two baskets. The second basket of apples weighs 12 kilograms more than the first. How many kilograms are in each basket?

first basket + second basket = total

flet f + 12 = second basket

+

(f + 12)

=

54

Assign Labels.

Verbal Model

Algebraic Model. (Equation)

Remove parentheses

when there is an addition sign

beside it.

Solve.

5412ff

Sentence.

There are 21 kg of apples in the first basket and 33 kg in the second basket.

21f2242f212125412f2

12f

3312)21(

Does 21 and 33 equal 54?

Does the second basket weigh 12 kg more than the first basket?

Why is it f+12 and not

12+f?

8w2

The length of a rectangle is 8cm longer than twice the width. If the perimeter is 34 find the dimensions of the rectangle.

Perimeter =

34

Solve.

Labels. Let w = width Let 2w + 8 = length

V.M.

A.M. =

(2w + 8)

+ 2

length

length

widthwidthw

two lengths+two widths 2

Sentence.

The width is 3cm and the length is 14cm.

w216w434 16w634 1616

w618 6 6

w3

832 86

14

Remember when asked for the dimensions of a rectangle, you are being asked for the measurement of the width and the length.

Check

34 = 2(14) + 2(3)

Perimeter = 2 lengths + 2 widths

34 = 28 + 634 = 34

Is the length twice the width plus 8?

The length of a rectangle is 8cm longer than twice the width. If the perimeter is 34 find the dimensions of the rectangle.

Perimeter =

34

Labels. Let w = width Let 2w + 8 = length

V.M.

A.M. = (2w + 8)+ 2

length

length

widthwidth

w

two lengths+two widths

2

Complete the steps highlighted above for the seven Complete the steps highlighted above for the seven class work problems. When you have finished those class work problems. When you have finished those steps, go back and solve the equations.steps, go back and solve the equations.

Example 1 The sum of the ages of two sisters is 25. The second sister’s age is 5 more than three times the first sister’s age. Find the two ages.

Assign Labels. Let f = first sister’s age

Let 3f + 5 = second sister’s age

Verbal Model. first sister’s age + second sister’s age =total

Algebraic Model.

f

Solve. 255f4

5 520f4

4 4 5f

5f3

553 515

20

Sentence. The first sister is 5 and the second sister is 20.

5f3

25

+ =255f3f

short board + long board = total

2s3

Example 2 A carpenter cut a board that was 10 feet long into two pieces. The longer piece is two feet longer than three times the length of the shorter piece. What is the length of each piece? Assign Labels. Let s = short board

Let 3s + 2 = long board

Verbal Model.

Algebraic Model.

s

Solve. 102s4

2 28s4

4 4 2s

2s3

223 26

8Sentence. The short board is 2 feet

and the long board is 8 feet long.

10

+ =102s3s

Example 3 The length of a rectangle is 1 meter less than twice its width. If the perimeter is 112 meters, find the dimensions.

Perimeter =

112

Solve.

Labels. Let w = width Let 2w - 1 = length

V.M.

A.M. =

(2w - 1)

+ 2

length

length

widthwidthw

two lengths+ two widths 2

Sentence.

The width is 19 meters and the length is 37 meters.

w22w4112 2w6112 22

w6114 6 6

w19

1w2

1192 138

37

Example 4 The length of a rectangle is twice its width. If the perimeter is 60 meters, find the dimensions (length and width) of the rectangle.

Perimeter =

60

Solve.

Labels. Let w = width Let 2w = length

V.M.

A.M. =

(2w)

+ 2

length

length

widthwidthw

two lengths+ two widths 2

Sentence.

The width is 10 meters and the length is 20 meters.

w2w460 w660 6 6

w10

w2

10220

Example 5 There are three numbers. The first is twice as big as the second, and the second is twice as big as the third. The total of the numbers is 224. What are the numbers?

Assign Labels. Let t = third number Let 2t = second number

Verbal Model. first # + second # + # number = total

Algebraic Model.Solve.

224t7 7 7

32t 32264

3222

Sentence. The number are 32, 64, and 128.

224tt2t4

Let 2(2t) = first number

224tt2t22

642128

Check 2241286432

Example 6 Mary and Betty have saved $43. Betty has saved $3 more than three times the amount Mary has saved. How much money has each girl saved?

Assign Labels. Let m = Mary’s savings

Let 3m + 3 = Betty’s savings

Verbal Model. Mary’s savings + Betty’s savings = total Algebraic Model.

Solve. 433m4 3 3

40m4 4 4

10m

3103 330

33

Sentence. Mary saved $10 and Betty saved $33.

433m3m

Example 7 Marge worked three times as many problems as Sue. They worked a total of 32 problems. How many problems did Sue work? Assign Labels. Let s = Sue’s problems

Let 3s = Marge’s problems

Verbal Model. Sue’s problems + Marge’s problems = total Algebraic Model.

Solve. 32s4 4 4

8s

8324

Sentence. Sue worked 8 problems.

32s3s

Check 32824

2-A13 Handout A13