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Algebraic Expressions and Equations: Applications II: Solving Problems

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve algebraic problems. By the end of the module students should be more familiar with the five-step method for solving applied problems and be able to use the five-step method to solve number problems and geometry problems.

Section Overview

  • The Five-Step Method
  • Number Problems
  • Geometry Problems

The Five Step Method

We are now in a position to solve some applied problems using algebraic methods. The problems we shall solve are intended as logic developers. Although they may not seem to reflect real situations, they do serve as a basis for solving more complex, real situation, applied problems. To solve problems algebraically, we will use the five-step method.

Strategy for Reading Word Problems

When solving mathematical word problems, you may wish to apply the following "reading strategy." Read the problem quickly to get a feel for the situation. Do not pay close attention to details. At the first reading, too much attention to details may be overwhelming and lead to confusion and discouragement. After the first, brief reading, read the problem carefully in phrases. Reading phrases introduces information more slowly and allows us to absorb and put together important information. We can look for the unknown quantity by reading one phrase at a time.

Five-Step Method for Solving Word Problems

  1. Let xx size 12{x} {} (or some other letter) represent the unknown quantity.
  2. Translate the words to mathematical symbols and form an equation. Draw a picture if possible.
  3. Solve the equation.
  4. Check the solution by substituting the result into the original statement, not equation, of the problem.
  5. Write a conclusion.

If it has been your experience that word problems are difficult, then follow the five-step method carefully. Most people have trouble with word problems for two reasons:

  1. They are not able to translate the words to mathematical symbols. (See (Reference).)
  2. They neglect step 1. After working through the problem phrase by phrase, to become familiar with the situation,

INTRODUCE A VARIABLE

Number Problems

Sample Set A

Example 1

What number decreased by six is five?

  1. Step 1: Let nn size 12{n} {} represent the unknown number.
  2. Step 2: Translate the words to mathematical symbols and construct an equation. Read phrases.

    What number: n decreased by: − six: 6 is: = five: 5 } n − 6 = 5 What number: n decreased by: − six: 6 is: = five: 5 } n − 6 = 5 size 12{ left none matrix { "What number:" {} # n {} ## "decreased by:" {} # - {} {} ## "six:" {} # 6 {} ## "is:" {} # ={} {} ## "five:" {} # 5{} } right rbrace n - 6=5} {}

  3. Step 3: Solve this equation.

    n−6=5n−6=5 size 12{n - 6=5} {} Add 6 to both sides.
    n − 6 + 6 = 5 + 6 n − 6 + 6 = 5 + 6 size 12{n - 6+6=5+6} {}
    n = 11 n = 11 size 12{n="11"} {}

  4. Step 4: Check the result.

    When 11 is decreased by 6, the result is 11−611−6 size 12{"11" - 6} {}, which is equal to 5. The solution checks.

  5. Step 5: The number is 11.

Example 2

When three times a number is increased by four, the result is eight more than five times the number.

  1. Step 1: Let x=x= size 12{x={}} {}the unknown number.
  2. Step 2: Translate the phrases to mathematical symbols and construct an equation.

    When three times a number: 3 x is increased by: + four: 4 the result is: = eight: 8 more than: + five times the number: 5 x } 3 x + 4 = 5 x + 8 When three times a number: 3 x is increased by: + four: 4 the result is: = eight: 8 more than: + five times the number: 5 x } 3 x + 4 = 5 x + 8 size 12{ left none matrix { "When three times a number:" {} # 3x {} ## "is increased by:" {} # +{} {} ## "four:" {} # 4 {} ## "the result is:" {} # ={} {} ## "eight:" {} # 8 {} ## "more than:" {} # +{} {} ## "five times the number:" {} # 5x{} } right rbrace 3x+4=5x+8} {}

  3. Step 3:
    3 x+4=5 x+8. Subtract 3 x from both  sides. 3 x + 4 − 3 x = 5 x + 8 − 3 x 4=2x+8 Subtract 8 from both  sides. 4 − 8 = 2x + 8 − 8 −4=2x Divide both  sides by 2. − 2 = x 3 x+4=5 x+8 size 12{3x+4=5x+8} {}. Subtract 3 x from both  sides. 3 x + 4 − 3 x = 5 x + 8 − 3 x size 12{3x+4 - 3x=5x+8 - 3x} {} 4=2x+8 size 12{4=2x+8} {} Subtract 8 from both  sides. 4 − 8 = 2x + 8 − 8 size 12{4 - 8=2x+8 - 8} {} −4=2x size 12{ - 4=2x} {} Divide both  sides by 2. − 2 = x size 12{ - 2=x} {}
  4. Step 4: Check this result.
     Three times - 2 - 2 is - 6 - 6 . Increasing - 6 - 6 by 4 results in −6+4=−2−6+4=−2 size 12{ - 6+4= - 2} {}. Now, five times - 2 - 2 is - 10 - 10 .
    Increasing - 10 - 10 by 88 size 12{g} {} results in −10+8=−2−10+8=−2 size 12{ - "10"+8= - 2} {}. The results agree, and the solution checks.
  5. Step 5: The number is - 2 - 2

Example 3

Consecutive integers have the property that if

n = the smallest integer, then n + 1 = the next integer, and n + 2 = the next integer, and so on. n = the smallest integer, then n + 1 = the next integer, and n + 2 = the next integer, and so on.

Consecutive odd or even integers have the property that if

n = the smallest integer, then n + 2 = the next odd or even integer (since odd or even numbers differ by 2), and n + 4 = the next odd or even integer, and so on. n = the smallest integer, then n + 2 = the next odd or even integer (since odd or even numbers differ by 2), and n + 4 = the next odd or even integer, and so on.

The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

  1. Step 1. Let n = the first odd integer. Then, n + 2 = the second odd integer, and n + 4 = the third odd integer. Let n = the first odd integer. Then, n + 2 = the second odd integer, and n + 4 = the third odd integer.
  2. Step 2. Translate the words to mathematical symbols and construct an equation. Read phrases.

    The sum of: add some numbers three consecutive odd integers: n , n + 2, n + 4 is equal to: = one less than: subtract 1 from twice the first odd integer: 2 n } n + ( n + 2 ) + ( n + 4 ) = 2 n − 1 The sum of: add some numbers three consecutive odd integers: n , n + 2, n + 4 is equal to: = one less than: subtract 1 from twice the first odd integer: 2 n } n + ( n + 2 ) + ( n + 4 ) = 2 n − 1 size 12{ left none matrix { "The sum of:" {} # "add some numbers" {} ## "three consecutive odd integers:" {} # n,n+2,n+4 {} ## "is equal to:" {} # ={} {} ## "one less than:" {} # "subtract 1 from" {} ## "twice the first odd integer:" {} # 2n{} } right rbrace n+ \( n+2 \) + \( n+4 \) =2n - 1} {}

  3. Step 3.

    n+n+2+n+4=2 n−1 3 n+6=2 n−1 Subtract 2 n from both  sides. 3 n + 6 − 2 n = 2 n − 1 − 2 n n+6=−1 Subtract 6 from both  sides. n + 6 − 6 = − 1 − 6 n=−7 The first integer is -7. n+2=−7+2=−5 The second integer is -5. n+4=−7+4=−3 The third integer is -3. n+n+2+n+4=2 n−1 size 12{n+n+2+n+4=2n - 1} {} 3 n+6=2 n−1 size 12{3n+6=2n - 1} {} Subtract 2 n size 12{2n} {} from both  sides. 3 n + 6 − 2 n = 2 n − 1 − 2 n size 12{3n+6 - 2n=2n - 1 - 2n} {} n+6=−1 size 12{n+6= - 1} {} Subtract 6 from both  sides. n + 6 − 6 = − 1 − 6 size 12{n+6 - 6= - 1 - 6} {} n=−7 size 12{n= - 7} {} The first integer is -7. n+2=−7+2=−5 size 12{n+2= - 7+2= - 5} {} The second integer is -5. n+4=−7+4=−3 size 12{n+4= - 7+4= - 3} {} The third integer is -3.

  4. Step 4. Check this result.
     The sum of the three integers is

    − 7 + ( − 5 ) + ( − 3 ) = − 12 + ( − 3 ) = − 15 − 7 + ( − 5 ) + ( − 3 ) = − 12 + ( − 3 ) = − 15

    One less than twice the first integer is 2(−7)−1=−14−1=−152(−7)−1=−14−1=−15 size 12{2 \( - 7 \) - 1= - "14" - 1= - "15"} {}. Since these two results are equal, the solution checks.

  5. Step 5. The three odd integers are -7, -5, -3.

Practice Set A

Exercise 1

When three times a number is decreased by 5, the result is -23. Find the number.

  1. Step 1: Let x = x =
  2. Step 2:
  3. Step 3:
  4. Step 4: Check:
  5. Step 5: The number is
    .

Solution

-6

Exercise 2

When five times a number is increased by 7, the result is five less than seven times the number. Find the number.

  1. Step 1: Let n = n =
  2. Step 2:
  3. Step 3:
  4. Step 4: Check:
  5. Step 5: The number is
    .

Solution

6

Exercise 3

Two consecutive numbers add to 35. Find the numbers.

  1. Step 1:
  2. Step 2:
  3. Step 3:
  4. Step 4: Check:
  5. Step 5: The numbers are
    and
    .

Solution

17 and 18

Exercise 4

The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.

  1. Step 1: Let x = x = smallest integer.
    = next integer.
    = largest integer.
  2. Step 2:
  3. Step 3:
  4. Step 4: Check:
  5. Step 5: The integers are
    ,
    , and
    .

Solution

-8, -6, and -4

Geometry Problems

Sample Set B

Example 4

The perimeter (length around) of a rectangle is 20 meters. If the length is 4 meters longer than the width, find the length and width of the rectangle.

  1. Step 1: Let x=x= size 12{x={}} {}the width of the rectangle. Then,
    x+4=x+4= size 12{x+4={}} {}the length of the rectangle.
  2. Step 2: We can draw a picture.

    A rectangle with base length the quantity x + 4 and height x.

    The length around the rectangle is
     x ︸ width + x + 4 ︸ length + x ︸ width + x + 4 ︸ length = 20 x ︸ width + x + 4 ︸ length + x ︸ width + x + 4 ︸ length = 20

  3. Step 3:

    x+x+4+x+x+4=20 4 x+8=20 Subtract 8 from both  sides. 4 x=12 Divide  both  sides by 4. x=3 Then, x + 4 = 3 + 4 = 7 x+x+4+x+x+4=20 size 12{x+x+4+x+x+4="20"} {} 4 x+8=20 size 12{4x+8="20"} {} Subtract 8 from both  sides. 4 x=12 size 12{4x="12"} {} Divide  both  sides by 4. x=3 size 12{x=3} {} Then, x + 4 = 3 + 4 = 7 size 12{x+4=3+4=7} {}

  4. Step 4: Check:

    A rectangle with base length 7 and height 3.
    graphics3.png

  5. Step 5: The length of the rectangle is 7 meters.
    The width of the rectangle is 3 meters.

Practice Set B

Exercise 5

The perimeter of a triangle is 16 inches. The second leg is 2 inches longer than the first leg, and the third leg is 5 inches longer than the first leg. Find the length of each leg.

  1. Step 1: Let x=x= size 12{x={}} {} length of the first leg.
    = length of the second leg.
    = length of the third leg.
  2. Step 2: We can draw a picture.
  3. Step 3:
  4. Step 4: Check:
  5. Step 5: The lengths of the legs are
    ,
    , and
    .
Solution

3 inches, 5 inches, and 8 inches

Exercises

For the following 17 problems, find each solution using the five-step method.

Exercise 6

What number decreased by nine is fifteen?

  1. Step 1. Let n=n= size 12{n={}} {}the number.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The number is
    .

Solution

24

Exercise 7

What number increased by twelve is twenty?

  1. Step 1. Let n=n= size 12{n={}} {}the number.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The number is
    .

Exercise 8

If five more than three times a number is thirty-two, what is the number?

  1. Step 1. Let x=x= size 12{x={}} {}the number.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The number is
    .

Solution

9

Exercise 9

If four times a number is increased by fifteen, the result is five. What is the number?

  1. Step 1. Let x=x= size 12{x={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The number is
    .

Exercise 10

When three times a quantity is decreased by five times the quantity, the result is negative twenty. What is the quantity?

  1. Step 1. Let x=x= size 12{x={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The quantity is
    .

Solution

10

Exercise 11

If four times a quantity is decreased by nine times the quantity, the result is ten. What is the quantity?

  1. Step 1. Let y=y= size 12{y={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The quantity is
    .

Exercise 12

When five is added to three times some number, the result is equal to five times the number decreased by seven. What is the number?

  1. Step 1. Let n=n= size 12{n={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The number is
    .

Solution

66 size 12{6} {}

Exercise 13

When six times a quantity is decreased by two, the result is six more than seven times the quantity. What is the quantity?

  1. Step 1. Let x=x= size 12{x={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The quantity is
    .

Exercise 14

When four is decreased by three times some number, the result is equal to one less than twice the number. What is the number?

  1. Step 1.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5.

Solution

1

Exercise 15

When twice a number is subtracted from one, the result is equal to twenty-one more than the number. What is the number?

  1. Step 1.
  2. Step 2.
  3. Step 3.
  4. Step 4.
  5. Step 5.

Exercise 16

The perimeter of a rectangle is 36 inches. If the length of the rectangle is 6 inches more than the width, find the length and width of the rectangle.

  1. Step 1. Let w=w= size 12{w={}} {}the width.
    = the length.
  2. Step 2. We can draw a picture.
    A rectangle.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The length of the rectangle is
    inches, and the width is
    inches.

Solution

Length=12 inches, Width=6 inches

Exercise 17

The perimeter of a rectangle is 48 feet. Find the length and the width of the rectangle if the length is 8 feet more than the width.

  1. Step 1. Let w=w= size 12{w={}} {}the width.
    = the length.
  2. Step 2. We can draw a picture.
    A rectangle.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The length of the rectangle is
    feet, and the width is
    feet.

Exercise 18

The sum of three consecutive integers is 48. What are they?

  1. Step 1. Let n=n= size 12{n={}} {}the smallest integer.
    = the next integer.
    = the next integer.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The three integers are
    ,
    , and
    .

Solution

15, 16, 17

Exercise 19

The sum of three consecutive integers is -27. What are they?

  1. Step 1. Let n=n= size 12{n={}} {} the smallest integer.
    = the next integer.
    = the next integer.
  2. Step 2.
  3. Step 3.
  4. Step 4. Check:
  5. Step 5. The three integers are
    ,
    , and
    .

Exercise 20

The sum of five consecutive integers is zero. What are they?

  1. Step 1. Let n=n= size 12{n={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4.
  5. Step 5. The five integers are
    ,
    ,
    ,
    , and
    .

Solution

-2, -1, 0, 1, 2

Exercise 21

The sum of five consecutive integers is -5. What are they?

  1. Step 1. Let n=n= size 12{n={}} {}
  2. Step 2.
  3. Step 3.
  4. Step 4.
  5. Step 5. The five integers are
    ,
    ,
    ,
    , and
    .

Continue using the five-step procedure to find the solutions.

Exercise 22

The perimeter of a rectangle is 18 meters. Find the length and width of the rectangle if the length is 1 meter more than three times the width.

Solution

Length is 7, width is 2

Exercise 23

The perimeter of a rectangle is 80 centimeters. Find the length and width of the rectangle if the length is 2 meters less than five times the width.

Exercise 24

Find the length and width of a rectangle with perimeter 74 inches, if the width of the rectangle is 8 inches less than twice the length.

Solution

Length is 15, width is 22

Exercise 25

Find the length and width of a rectangle with perimeter 18 feet, if the width of the rectangle is 7 feet less than three times the length.

Exercise 26

A person makes a mistake when copying information regarding a particular rectangle. The copied information is as follows: The length of a rectangle is 5 inches less than two times the width. The perimeter of the rectangle is 2 inches. What is the mistake?

Solution

The perimeter is 20 inches. Other answers are possible. For example, perimeters such as 26, 32 are possible.

Exercise 27

A person makes a mistake when copying information regarding a particular triangle. The copied information is as follows: Two sides of a triangle are the same length. The third side is 10 feet less than three times the length of one of the other sides. The perimeter of the triangle is 5 feet. What is the mistake?

Exercise 28

The perimeter of a triangle is 75 meters. If each of two legs is exactly twice the length of the shortest leg, how long is the shortest leg?

Solution

15 meters

Exercise 29

If five is subtracted from four times some number the result is negative twenty-nine. What is the number?

Exercise 30

If two is subtracted from ten times some number, the result is negative two. What is the number?

Solution

n=0n=0 size 12{n=0} {}

Exercise 31

If three less than six times a number is equal to five times the number minus three, what is the number?

Exercise 32

If one is added to negative four times a number the result is equal to eight less than five times the number. What is the number?

Solution

n=1n=1 size 12{n=1} {}

Exercise 33

Find three consecutive integers that add to -57.

Exercise 34

Find four consecutive integers that add to negative two.

Solution

-2, -1, 0, 1

Exercise 35

Find three consecutive even integers that add to -24.

Exercise 36

Find three consecutive odd integers that add to -99.

Solution

-35, -33, -31

Exercise 37

Suppose someone wants to find three consecutive odd integers that add to 120. Why will that person not be able to do it?

Exercise 38

Suppose someone wants to find two consecutive even integers that add to 139. Why will that person not be able to do it?

Solution

…because the sum of any even number (in this case, 2) o even integers (consecutive or not) is even and, therefore, cannot be odd (in this case, 139)

Exercise 39

Three numbers add to 35. The second number is five less than twice the smallest. The third number is exactly twice the smallest. Find the numbers.

Exercise 40

Three numbers add to 37. The second number is one less than eight times the smallest. The third number is two less than eleven times the smallest. Find the numbers.

Solution

2, 15, 20

Exercises for Review

Exercise 41

((Reference)) Find the decimal representation of 0.34992÷4.320.34992÷4.32 size 12{0 "." "34992" div 4 "." "32"} {}.

Exercise 42

((Reference)) A 5-foot woman casts a 9-foot shadow at a particular time of the day. How tall is a person that casts a 10.8-foot shadow at the same time of the day?

Solution

6 feet tall

Exercise 43

((Reference)) Use the method of rounding to estimate the sum: 4512+151254512+15125 size 12{4 { {5} over {"12"} } +"15" { {1} over {"25"} } } {}.

Exercise 44

((Reference)) Convert 463 mg to cg.

Solution

46.3 cg

Exercise 45

((Reference)) Twice a number is added to 5. The result is 2 less than three times the number. What is the number?

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