Example

As of March, $2016$ the world's population was estimated at $7.4$ billion. [footnote] "Current World Population." World Population Clock: $7.4$ Billion People $(2016)$. Accessed June $21, 2016$. http://www.worldometers.info/world-population/. [/footnote].  According to water.org, $1$ out of every $3$ people on the planet lives without access to a toilet.  Find the number of people on the planet that do not have access to a toilet.

Answer: We can use a proportion to find the unknown number of people who live without a toilet since we are given that $1$ in $3$ do not have access, and we are given the population of the planet. We know that $1$ out of every $3$ people do not have access, so we can write that as a ratio (fraction).

$\frac{1}{3}$

Let the number of people without access to a toilet be x. The ratio of people with and without toilets is then

$\frac{x}{7.4\text{ billion }}$

Equate the two ratios since they are representing the same fractional amount of the population.

$\frac{1}{3}=\frac{x}{7.4\text{ billion }}$

Solve:

$\begin{array}{l}\frac{1}{3}=\frac{x}{7.4}\\\text{}\\7.4\cdot\frac{1}{3}=\frac{x}{7.4}\cdot{7.4}\\\text{}\\2.46=x\end{array}$

The original units were billions of people, so our answer is $2.46$ billion people do not have access to a toilet. Wow, that is a lot of people.

Example

It has been shown that a person's height is proportional to the length of their femur [footnote]Obialor, Ambrose, Churchill Ihentuge, and Frank Akapuaka. "Determination of Height Using Femur Length in Adult Population of Oguta Local Government Area of Imo State Nigeria." Federation of American Societies for Experimental Biology, April $2015$. Accessed June $22, 2016$. http://www.fasebj.org/content/29/1_Supplement/LB19.short.[/footnote]. Given that a person who is $71$ inches tall has a femur length of $17.75$ inches, how tall is someone with a femur length of $16$ inches?

Answer: Height and femur length are proportional for everyone, so we can define a ratio with the given height and femur length. We can then use this to write a proportion to find the unknown height. Let x be the unknown height. Define the ratio of femur length and height for both people using the given measurements.

Person $1$:  $\frac{\text{femur length}}{\text{height}}=\frac{17.75\text{ }\text{inches}}{71\text{ }\text{inches}}$

Person $2$:  $\frac{\text{femur length}}{\text{height}}=\frac{16\text{ }\text{inches}}{x\text{ }\text{inches}}$

Equate the ratios since we are assuming height and femur length are proportional for everyone.

$\frac{17.75\text{ }\text{inches}}{71\text{ }\text{inches}}=\frac{16\text{ }\text{inches}}{x\text{ }\text{inches}}$

 Solve by using the common denominator to clear fractions. The common denominator is $71x$.

$\begin{array}{c}\frac{17.75}{71}=\frac{16}{x}\\\\71x\cdot\frac{17.75}{71}=\frac{16}{x}\cdot{71x}\\\\17.75\cdot{x}=16\cdot{71}\\\\17.75\cdot{x}=1136\\\\x=\frac{1136}{17.75}=64\end{array}$

The unknown height of person $2$ is $64$ inches. In general, we can simplify the fraction $\frac{17.75}{71}=0.25=\frac{1}{4}$ to find a general rule for everyone. This would translate to saying a person's height is 4 times the length of their femur.

Example

Given a scale factor of $1:557$ on a map of the US, if the distance from Seattle, WA to San Jose, CA is $1.5$ inches on the map,  define a proportion to find the actual distance between them.

Answer: We need to define a proportion to solve for the unknown distance between Seattle and San Jose.  The scale factor is $1:557$, and we will call the unknown distance x. The ratio of inches to miles is $\frac{1}{557}$. We know the inches between the two cities, but we do not know miles, so the ratio that describes the distance between them is $\frac{1.5}{x}$. The proportion that will help us solve this problem is $\frac{1}{557}=\frac{1.5}{x}$. Solve using the common denominator $557x$ to clear fractions. $$\begin{array}{ccc}\frac{1}{557}=\frac{1.5}{x}\\557x\cdot\frac{1}{557}=\frac{1.5}{x}\cdot{557x}\\x=1.5\cdot{557}=835.5\end{array}$$ We used the scale factor $1:557$ to find an unknown distance between Seattle and San Jose. We also checked our answer of $835.5$ miles with Google maps and found that the distance is $839.9$ miles, so we did pretty well!

Example

Two cities are $2.5$ inches apart on a map.  Their actual distance from each other is $325$ miles.  Write a proportion to represent and solve for the scale factor for one inch of the map.

Answer: We know that for each $2.5$ inches on the map, it represents $325$ actual miles. We are looking for the scale factor for one inch of the map. The ratio we want is $\frac{1}{x}$ where x is the actual distance represented by one inch on the map.  We know that for every $2.5$ inches, there are $325$ actual miles, so we can define that relationship as $\frac{2.5}{325}$ We can use a proportion to equate the two ratios and solve for the unknown distance.

$\begin{array}{ccc}\frac{1}{x}=\frac{2.5}{325}\\325x\cdot\frac{1}{x}=\frac{2.5}{325}\cdot{325x}\\325=2.5x\\x=130\end{array}$

The scale factor for one inch on the map is $1:130$ or for every inch of map there are $130$ actual miles.