3.1.3: Applications of Proportions

Last updated
Save as PDF

Page ID: 116772

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Learning Objectives

solve proportion problems using the five-step method

The Five-Step Method

In [link] 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:

By careful reading, determine what the unknown quantity is and represent it with some letter. There will be only one unknown in a problem.
Identify the three specified numbers.
Determine which comparisons are to be made and set up the proportion.
Solve the proportion (using the methods of [link]).
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

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

Solution

Step 1: The unknown quantity is miles.
Let \(x =\) 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 \(\to \dfrac{\text{2 inches}}{\text{25 miles}}\)
8 inches to \(x\) miles \(\to \dfrac{\text{8 inches}}{\text{x miles}}\)
Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations. \(\dfrac{2}{25} = \dfrac{8}{x}\)

Step 4: \(\dfrac{2}{25} = \dfrac{8}{x}\) Perform the cross multiplication.
\(\begin{array} {rcl} {2 \cdot x} & = & {8 \cdot 25} \\ {2 \cdot x} & = & {200 \ \ \ \ \ \ \ \text{Divide 200 by 2.}} \\ {x} & = & {\dfrac{200}{2}} \\ {x} & = & {100} \end{array}\)
In step 1, we let \(x\) represent the number of miles. So, \(x\) represents 100 miles.

Step 5: If 2 inches represents 25 miles, then 8 inches represents 100 miles.

Sample Set A

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?

Solution

Step 1: The unknown quantity is the number of parts of water.
Let \(x =\) 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 \(\to \dfrac{7}{2}\)
\(n\) parts water to 20 parts acid \(\to \dfrac{n}{20}\)
\(\dfrac{7}{2} = \dfrac{n}{20}\)

Step 4: \(\dfrac{7}{2} = \dfrac{n}{20}\) Perform the cross multiplication.
\(\begin{array} {rcl} {7 \cdot 20} & = & {2 \cdot n} \\ {140} & = & {2 \cdot n \ \ \ \text{Divide 140 by 2.}} \\ {\dfrac{140}{2}} & = & {n} \\ {70} & = & {n} \end{array}\)
In step 1, we let \(n\) represent the number of parts of water. So, \(n\) represents 70 parts of water.

Step 5: 7 parts water to 2 parts acid indicates 70 parts water to 20 parts acid.

Sample Set A

A 5-foot girl casts a \(3 \dfrac{1}{3}\) 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?

Solution

Step 1: The unknown quantity is the height of the person.
Let \(h = \text{ height of the preson.}\)

Step 2: The three specified numbers are
5 feet ( height of girl)
3\(\dfrac{1}{3}\) feet (length of shadow)
3 feet (length of shadow)

Step 3: The comparisons are
5-foot girl is to \(3 \dfrac{1}{3}\) foot shadow \(\to \dfrac{5}{3 \dfrac{1}{3}}\)
\(h\)-foot person is to 3-foot shadow \(\to \dfrac{h}{3}\)
\(\dfrac{5}{3\dfrac{1}{3}} = \dfrac{h}{3}\)

Step 4:

\(\begin{array} {rcl} {\dfrac{5}{3\dfrac{1}{3}}} & = & {\dfrac{h}{3}} \\ {5 \cdot 3} & = & {3 \dfrac{1}{3} \cdot h} \\ {15} & = & {\dfrac{10}{3} \cdot h \ \ \ \ \text{Divide 15 by } \dfrac{10}{3}} \\ {\dfrac{15}{\dfrac{10}{3}}} & = & {h} \\ {\dfrac{\begin{array} {c} {^3} \\ {\cancel{15}} \end{array}}{1} \cdot \dfrac{3}{\begin{array} {c} {\cancel{10}} \\ {^2} \end{array}}} & = & {h} \\ {\dfrac{9}{2}} & = & {h} \\ {h} & = & {4 \dfrac{1}{2}} \end{array}\)

Step 5: A person who casts a 3-foot shadow at this particular time of the day is \(4 \dfrac{1}{2}\) feet tall.

Sample Set A

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?

Solution

Step 1: The unknown quantity is the number of women in town.
Let \(x =\) 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 \(\to \dfrac{3}{5}\)
19,200 men to \(x\) women \(\to \dfrac{19,200}{x}\)
\(\dfrac{3}{5} = \dfrac{19,200}{x}\)

Step 4:

\(\begin{array} {rcl} {\dfrac{3}{5}} & = & {\dfrac{19,200}{x}} \\ {3 \cdot x} & = & {19,200 \cdot 5} \\ {3 \cdot x} & = & {96,000} \\ {x} & = & {\dfrac{96,000}{3}} \\ {x} & = & {32,000} \end{array}\)

Step 5: There are 32,000 women in town.

Sample Set A

The rate of wins to losses of a particular baseball team is \(\dfrac{9}{2}\). How many games did this team lose if they won 63 games?

Solution

Step 1: The unknown quantity is the number of games lost.
Let \(n =\) number of games lost.

Step 2: Since \(\dfrac{9}{2} \to\) 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 \(\to \dfrac{9}{2}\)
63 wins to \(n\) losses \(\to \dfrac{63}{n}\)
\(\dfrac{9}{2} = \dfrac{63}{n}\)

Step 4:

\(\begin{array} {rcl} {\dfrac{9}{2}} & = & {\dfrac{63}{n}} \\ {9 \cdot n} & = & {2 \cdot 63} \\ {9 \cdot n} & = & {126} \\ {n} & = & {\dfrac{126}{9}} \\ {n} & = & {14} \end{array}\)

Step 5: This team had 14 losses.

Practice Set A

Solve each problem.

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:

Answer: 500 miles

Practice Set A

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:

Answer: 24 parts of alcohol

Practice Set A

A \(5 \dfrac{1}{2}\)-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:

Answer: \(3 \dfrac{9}{11}\) feet

Practice Set A

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:

Answer: 864 outside plants

Practice Set A

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:

Answer: The event occurs 99 times.

Practice Set A

The rate of passing grades to failing grades in a particular chemistry class is 7272 . If there are 21 passing grades, how many failing grades are there?

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Answer: 6 failing grades

Exercises

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

Exercise \(\PageIndex{1}\)

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

Answer: 24

Exercise \(\PageIndex{2}\)

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

Exercise \(\PageIndex{3}\)

A model is built to \(\dfrac{2}{15}\) scale. If a particular part of the model measures 6 inches, how long is the actual structure?

Answer: 45 inches

Exercise \(\PageIndex{4}\)

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 \(\PageIndex{5}\)

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?

Answer: 33 parts

Exercise \(\PageIndex{6}\)

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 \(\PageIndex{7}\)

The ratio of sulfur to air in a container is \(\dfrac{4}{45}\). How many ml of air are there in a container that contains 207 ml of sulfur?

Answer: 2328.75

Exercise \(\PageIndex{8}\)

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 \(\PageIndex{9}\)

A \(5\dfrac{1}{2}\)-foot woman casts a \(1\dfrac{1}{2}\)-foot shadow at a particular time of the day. How long a shadow does her \(3\dfrac{1}{2}\)-foot niece cast at the same time of the day?

Answer: \(\dfrac{21}{22}\) feet

Exercise \(\PageIndex{10}\)

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 \(\PageIndex{11}\)

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?

Answer: 12,250

Exercise \(\PageIndex{12}\)

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 \(\PageIndex{13}\)

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?

Answer: 12

Exercise \(\PageIndex{14}\)

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 \(\PageIndex{15}\)

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?

Answer: \(1\dfrac{5}{6}\)

Exercise \(\PageIndex{16}\)

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 \(\PageIndex{17}\)

A pond loses \(7\dfrac{1}{2}\) gallons of water every 2 days due to evaporation. How many gallons of water are lost, due to evaporation, in \(\dfrac{1}{2}\) day?

Answer: \(1\dfrac{7}{8}\)

Exercise \(\PageIndex{18}\)

A photograph that measures 3 inches wide and \(4\dfrac{1}{2}\) inches high is to be enlarged so that it is 5 inches wide. How high will it be?

Exercise \(\PageIndex{19}\)

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

Answer: \(31 \dfrac{1}{4}\)

Exercise \(\PageIndex{20}\)

Every \(\dfrac{1}{2}\) 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 \(\PageIndex{21}\)

Find the product, \(818 \cdot 0\).

Answer: 0

Exercise \(\PageIndex{22}\)

Determine the missing numerator: \(\dfrac{8}{15} = \dfrac{N}{90}\).

Exercise \(\PageIndex{23}\)

Find the value of \(\dfrac{\dfrac{3}{10} + \dfrac{4}{12}}{\dfrac{19}{20}}\).

Answer: \(\dfrac{2}{3}\)

Exercise \(\PageIndex{24}\)

Subtract 0.249 from the sum of 0.344 and 0.612.

Exercise \(\PageIndex{25}\)

Solve the proportion: \(\dfrac{6}{x} = \dfrac{36}{30}\).

Answer: 5