5.3 Ratios and proportions: application of proportions

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

• Step 1 The unknown quantity is miles.
  Let $x=$ 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 → $\frac{\text{2 inches}}{\text{25 miles}}$
  8 inches to x miles → $\frac{\text{8 inches}}{\text{x miles}}$
  Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations.
  $\frac{2}{\text{25}}=\frac{8}{x}$
• Step 4 $\begin{array}{cccc}\hfill \frac{2}{25}& =& \frac{8}{x}\hfill & \text{Perform the cross multiplication.}\hfill \end{array}$
  $\begin{array}{cccc}\hfill 2\cdot x& =& 8\cdot \mathrm{25}\hfill & \\ \hfill 2\cdot x& =& 200\hfill & \text{Divide 200 by 2.}\hfill \\ \hfill x& =& \frac{200}{2}\hfill & \\ \hfill x& =& 100\hfill & \end{array}$
  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.
  Try [link] in [link] .

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 $n=$ 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 → $\frac{7}{2}$
  $n$ parts water to 20 parts acid → $\frac{n}{\text{20}}$
  $\frac{7}{2}=\frac{n}{\text{20}}$
• Step 4 $\begin{array}{cccc}\hfill \frac{7}{2}& =& \frac{n}{\mathrm{20}}\hfill & \text{Perform the cross multiplication.}\hfill \end{array}$
  $\begin{array}{cccc}\hfill 7\cdot \mathrm{20}& =& 2\cdot n\hfill & \\ \hfill 140& =& 2\cdot n\hfill & \text{Divide 140 by 2.}\hfill \\ \hfill \frac{140}{2}& =& n\hfill & \\ \hfill 70& =& n\hfill & \end{array}$
  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.
  Try [link] in [link] .

A 5-foot girl casts a $3\frac{1}{3}$ -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?

• Step 1 The unknown quantity is the height of the person.
  Let $h=\text{height of the person}$ .
• Step 2 The three specified numbers are
  5 feet ( height of girl)
  $3\frac{1}{3}$ feet (length of shadow)
  3 feet (length of shadow)
• Step 3 The comparisons are
  5-foot girl is to $3\frac{1}{3}$ foot shadow → $\frac{5}{3\frac{1}{3}}$
  h -foot person is to 3-foot shadow → $\frac{h}{3}$
  $\frac{5}{3\frac{1}{3}}=\frac{h}{3}$
• Step 4 $\begin{array}{cccc}\hfill \frac{5}{3\frac{1}{3}}& =& \frac{h}{3}\hfill & \end{array}$
  $\begin{array}{cccc}\hfill 5\cdot 3& =& 3\frac{1}{3}\cdot h\hfill & \\ \hfill 15& =& \frac{10}{3}\cdot h\hfill & \text{Divide}15\text{by}\frac{10}{3}\hfill \\ \hfill \frac{15}{\frac{10}{3}}& =& h& \\ \hfill \frac{\stackrel{3}{\cancel{15}}}{1}\cdot \frac{3}{\underset{2}{\cancel{10}}}& =& h\hfill & \\ \hfill \frac{9}{2}& =& h\hfill & \\ \hfill h& =& 4\frac{1}{2}\hfill & \end{array}$
• Step 5 A person who casts a 3-foot shadow at this particular time of the day is $4\frac{1}{2}$ feet tall.
  Try [link] in [link] .