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Ratios and Rates: Applications of Proportions

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 applications of proportions. By the end of the module students should be able to solve proportion problems using the five-step method.

Section Overview

  • The Five-Step Method
  • Problem Solving

The Five-Step Method

In (Reference) we noted that many practical problems can be solved by writing the given information as proportions. Such proportions will be composed of three specified numbers and one unknown number represented by a letter.

The first and most important part of solving a proportion problem is to deter­mine, by careful reading, what the unknown quantity is and to represent it with some letter.

The Five-Step Method

The five-step method for solving proportion problems:

  1. By careful reading, determine what the unknown quantity is and represent it with some letter. There will be only one unknown in a problem.
  2. Identify the three specified numbers.
  3. Determine which comparisons are to be made and set up the proportion.
  4. Solve the proportion (using the methods of (Reference)).
  5. Interpret and write a conclusion in a sentence with the appropriate units of measure.

Step 1 is extremely important. Many problems go unsolved because time is not taken to establish what quantity is to be found.

When solving an applied problem, always begin by determining the unknown quantity and representing it with a letter.

Problem Solving

Sample Set A

Example 1

On a map, 2 inches represents 25 miles. How many miles are represented by 8 inches?

  • Step 1: The unknown quantity is miles.
    Let x=x= number of miles represented by 8 inches
  • Step 2: The three specified numbers are
    2 inches
    25 miles
    8 inches
  • Step 3: The comparisons are
    2 inches to 25 miles → 2 inches25 miles2 inches25 miles size 12{ { {"2 inches"} over {"25 miles"} } } {}
    8 inches to x miles → 8 inchesx miles8 inchesx miles size 12{ { {"8 inches"} over {"x miles"} } } {}
    Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations.
     2 25 = 8 x 2 25 = 8 x size 12{ { {2} over {"25"} } = { {8} over {x} } } {}
  • Step 4: 2 25 = 8 x Perform the cross multiplication. 2 25 = 8 x Perform the cross multiplication.
    2 ⋅ x = 8 ⋅ 25 2 ⋅ x = 200 Divide 200 by 2. x = 200 2 x = 100 2 ⋅ x = 8 ⋅ 25 2 ⋅ x = 200 Divide 200 by 2. x = 200 2 x = 100
    In step 1, we let xx size 12{x} {} represent the number of miles. So, xx size 12{x} {} represents 100 miles.
  • Step 5: If 2 inches represents 25 miles, then 8 inches represents 100 miles.
     Try Exercise 1 in Section 5.

Example 2

An acid solution is composed of 7 parts water to 2 parts acid. How many parts of water are there in a solution composed of 20 parts acid?

  • Step 1: The unknown quantity is the number of parts of water.
     Let n=n= number of parts of water.
  • Step 2: The three specified numbers are
     7 parts water
     2 parts acid
     20 parts acid
  • Step 3: The comparisons are
     7 parts water to 2 parts acid → 7272 size 12{ { {7} over {2} } } {}
     nn size 12{n} {} parts water to 20 parts acid → n20n20 size 12{ { {n} over {"20"} } } {}
     7 2 = n 20 7 2 = n 20 size 12{ { {7} over {2} } = { {n} over {"20"} } } {}
  • Step 4: 7 2 = n 20 Perform the cross multiplication. 7 2 = n 20 Perform the cross multiplication.
    7 ⋅ 20 = 2 ⋅ n 140 = 2 ⋅ n Divide 140 by 2. 140 2 = n 70 = n 7 ⋅ 20 = 2 ⋅ n 140 = 2 ⋅ n Divide 140 by 2. 140 2 = n 70 = n
    In step 1 we let nn size 12{n} {} represent the number of parts of water. So, nn size 12{n} {} represents 70 parts of water.
  • Step 5: 7 parts water to 2 parts acid indicates 70 parts water to 20 parts acid.
    Try Exercise 2 in Section 5.

Example 3

A 5-foot girl casts a 313313 size 12{ { {1} over {3} } } {}-foot shadow at a particular time of the day. How tall is a person who casts a 3-foot shadow at the same time of the day?

  • Step 1: The unknown quantity is the height of the person.
     Let h= height of the personh= height of the person size 12{h=" height of the person"} {}.
  • Step 2: The three specified numbers are
     5 feet ( height of girl)
     313313 size 12{3 { {1} over {3} } } {} feet (length of shadow)
     3 feet (length of shadow)
  • Step 3: The comparisons are
    5-foot girl is to 313313 size 12{3 { {1} over {3} } } {}foot shadow → 53135313 size 12{ { {5} over {3 { {1} over {3} } } } } {}
     h-foot person is to 3-foot shadow → h3h3 size 12{ { {h} over {3} } } {}
     5 3 1 3 = h 3 5 3 1 3 = h 3 size 12{ { {5} over {3 { {1} over {3} } } } = { {h} over {3} } } {}
  • Step 4: 5 313 = h 3 5 313 = h 3
    5 ⋅ 3 = 3 1 3 ⋅ h 15 = 10 3 ⋅ h Divide15by103 15 103 = h 15 3 1 ⋅ 3 10 2 = h 9 2 = h h = 4 1 2 5 ⋅ 3 = 3 1 3 ⋅ h 15 = 10 3 ⋅ h Divide15by103 15 103 = h 15 3 1 ⋅ 3 10 2 = h 9 2 = h h = 4 1 2
  • Step 5: A person who casts a 3-foot shadow at this particular time of the day is 412412 size 12{4 { {1} over {2} } } {} feet tall.
    Try Exercise 3 in Section 5.

Example 4

The ratio of men to women in a particular town is 3 to 5. How many women are there in the town if there are 19,200 men in town?

  • Step 1: The unknown quantity is the number of women in town.
     Let x=x= number of women in town.
  • Step 2: The three specified numbers are
    3
     5
     19,200
  • Step 3: The comparisons are 3 men to 5 women → 3535 size 12{ { {3} over {5} } } {}
     19,200 men to xx size 12{x} {} women → 19,200x19,200x size 12{ { {"19","200"} over {x} } } {}
    3 5 = 19 , 200 x 3 5 = 19 , 200 x size 12{ { {3} over {5} } = { {"19","200"} over {x} } } {}
  • Step 4: 3 5 = 19,200 x 3 5 = 19,200 x
    3 ⋅ x = 19,200 ⋅ 5 3 ⋅ x = 96,000 x = 96,000 3 x = 32,000 3 ⋅ x = 19,200 ⋅ 5 3 ⋅ x = 96,000 x = 96,000 3 x = 32,000
  • Step 5: There are 32,000 women in town.

Example 5

The rate of wins to losses of a particular baseball team is 9292 size 12{ { {9} over {2} } } {} . How many games did this team lose if they won 63 games?

  • Step 1: The unknown quantity is the number of games lost.
    Let n=n= number of games lost.
  • Step 2: Since 9292 size 12{ { {9} over {2} } } {}→ means 9 wins to 2 losses, the three specified numbers are
    9 (wins)
     2 (losses)
     63 (wins)
  • Step 3: The comparisons are
    9 wins to 2 losses→ 9292 size 12{ { {9} over {2} } } {}
    63 wins to nn size 12{n} {} losses → 63n63n size 12{ { {"63"} over {n} } } {}
    9 2 = 63 n 9 2 = 63 n size 12{ { {9} over {2} } = { {"63"} over {n} } } {}
  • Step 4: 9 2 = 63 n 9 2 = 63 n
    9 ⋅ n = 2 ⋅ 63 9 ⋅ n = 126 n = 126 9 n = 14 9 ⋅ n = 2 ⋅ 63 9 ⋅ n = 126 n = 126 9 n = 14
  • Step 5: This team had 14 losses.
    Try Exercise 4 in Section 5.

Practice Set A

Solve each problem.

Exercise 1

On a map, 3 inches represents 100 miles. How many miles are represented by 15 inches?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

500 miles

Exercise 2

An alcohol solution is composed of 14 parts water to 3 parts alcohol. How many parts of alcohol are in a solution that is composed of 112 parts water?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

24 parts of alcohol

Exercise 3

A 512512 size 12{5 { {1} over {2} } } {} -foot woman casts a 7-foot shadow at a particular time of the day. How long of a shadow does a 3-foot boy cast at that same time of day?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

39113911 size 12{3 { {9} over {"11"} } } {} feet

Exercise 4

The rate of houseplants to outside plants at a nursery is 4 to 9. If there are 384 houseplants in the nursery, how many outside plants are there?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

864 outside plants

Exercise 5

The odds for a particular event occurring are 11 to 2. (For every 11 times the event does occur, it will not occur 2 times.) How many times does the event occur if it does not occur 18 times?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

The event occurs 99 times.

Exercise 6

The rate of passing grades to failing grades in a particular chemistry class is 7272 size 12{ { {7} over {2} } } {} . If there are 21 passing grades, how many failing grades are there?

  • Step 1:



  • Step 2:



  • Step 3:



  • Step 4:



  • Step 5:



Solution

6 failing grades

Exercises

For the following 20 problems, use the five-step method to solve each problem.

Exercise 7

On a map, 4 inches represents 50 miles. How many inches represent 300 miles?

Solution

24

Exercise 8

On a blueprint for a house, 2 inches represents 3 feet. How many inches represent 10 feet?

Exercise 9

A model is built to 215215 size 12{ { {2} over {"15"} } } {} scale. If a particular part of the model measures 6 inches, how long is the actual structure?

Solution

45 inches

Exercise 10

An acid solution is composed of 5 parts acid to 9 parts of water. How many parts of acid are there in a solution that contains 108 parts of water?

Exercise 11

An alloy contains 3 parts of nickel to 4 parts of silver. How much nickel is in an alloy that con­tains 44 parts of silver?

Solution

33 parts

Exercise 12

The ratio of water to salt in a test tube is 5 to 2. How much salt is in a test tube that contains 35 ml of water?

Exercise 13

The ratio of sulfur to air in a container is 445445 size 12{ { {4} over {"45"} } } {}. How many ml of air are there in a container that contains 207 ml of sulfur?

Solution

2328.75

Exercise 14

A 6-foot man casts a 4-foot shadow at a particu­lar time of the day. How tall is a person that casts a 3-foot shadow at that same time of the day?

Exercise 15

A 512512 size 12{5 { {1} over {2} } } {}-foot woman casts a 112112 size 12{1 { {1} over {2} } } {}-foot shadow at a particular time of the day. How long a shadow does her 312312 size 12{3 { {1} over {2} } } {}-foot niece cast at the same time of the day?

Solution

21222122 size 12{ { {"21"} over {"22"} } } {} feet

Exercise 16

A man, who is 6 feet tall, casts a 7-foot shadow at a particular time of the day. How tall is a tree that casts an 84-foot shadow at that same time of the day?

Exercise 17

The ratio of books to shelves in a bookstore is 350 to 3. How many books are there in a store that has 105 shelves?

Solution

12,250

Exercise 18

The ratio of algebra classes to geometry classes at a particular community college is 13 to 2. How many geometry classes does this college offer if it offers 13 algebra classes?

Exercise 19

The odds for a particular event to occur are 16 to 3. If this event occurs 64 times, how many times would you predict it does not occur?

Solution

12

Exercise 20

The odds against a particular event occurring are 8 to 3. If this event does occur 64 times, how many times would you predict it does not occur?

Exercise 21

The owner of a stationery store knows that a 1-inch stack of paper contains 300 sheets. The owner wishes to stack the paper in units of 550 sheets. How many inches tall should each stack be?

Solution

156156 size 12{1 { {5} over {6} } } {}

Exercise 22

A recipe that requires 6 cups of sugar for 15 serv­ings is to be used to make 45 servings. How much sugar will be needed?

Exercise 23

A pond loses 712712 size 12{7 { {1} over {2} } } {} gallons of water every 2 days due to evaporation. How many gallons of water are lost, due to evaporation, in 1212 size 12{ { {1} over {2} } } {} day?

Solution

178178 size 12{1 { {7} over {8} } } {}

Exercise 24

A photograph that measures 3 inches wide and 412412 size 12{4 { {1} over {2} } } {} inches high is to be enlarged so that it is 5 inches wide. How high will it be?

Exercise 25

If 25 pounds of fertilizer covers 400 square feet of grass, how many pounds will it take to cover 500 square feet of grass?

Solution

31143114 size 12{"31" { {1} over {4} } } {}

Exercise 26

Every 112112 size 12{1 { {1} over {2} } } {} teaspoons of a particular multiple vitamin, in granular form, contains 0.65 the mini­mum daily requirement of vitamin C. How many teaspoons of this vitamin are required to supply 1.25 the minimum daily requirement?

Exercises for Review

Exercise 27

((Reference)) Find the product, 818⋅0818⋅0 size 12{"818" cdot 0} {}.

Solution

0

Exercise 28

((Reference)) Determine the missing numerator: 815=N90815=N90 size 12{ { {8} over {"15"} } = { {N} over {"90"} } } {}.

Exercise 29

((Reference)) Find the value of 310+4121920310+4121920 size 12{ { { { {3} over {4} } + { {4} over {"12"} } } over { { {"19"} over {"20"} } } } } {}.

Solution

2323 size 12{ { {2} over {3} } } {}

Exercise 30

((Reference)) Subtract 0.249 from the sum of 0.344 and 0.612.

Exercise 31

((Reference)) Solve the proportion: 6x=36306x=3630 size 12{ { {6} over {x} } = { {"36"} over {"30"} } } {}.

Solution

5

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