Read and Understand: We can use a proportion to find the unknown number of people who live without a toilet since we are given that [latex]1[/latex] in [latex]3[/latex] do not have access, and we are given the population of the planet.

Define and Translate: We know that [latex]1[/latex] out of every [latex]3[/latex] people do not have access, so we can write that as a ratio (fraction).

[latex]\frac{1}{3}[/latex]

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

[latex]\frac{x}{7.4\text{ billion }}[/latex]

Write and Solve: Equate the two ratios since they are representing the same fractional amount of the population.

[latex]\frac{1}{3}=\frac{x}{7.4\text{ billion}}[/latex]

Solve:

[latex]\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}[/latex]

Interpret: The original units were billions of people, so our answer is [latex]2.46[/latex] billion people do not have access to a toilet. Wow, that is a lot of people.

Read and Understand: 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.

Define and Translate: Let x be the unknown height. Define the ratio of femur length and height for both people using the given measurements.

Person [latex]1[/latex]:  [latex]\frac{\text{femur length}}{\text{height}}=\frac{17.75\text{ }\text{inches}}{71\text{ }\text{inches}}[/latex]

Person [latex]2[/latex]:  [latex]\frac{\text{femur length}}{\text{height}}=\frac{16\text{ }\text{inches}}{x\text{ }\text{inches}}[/latex]

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

[latex]\frac{17.75\text{ }\text{inches}}{71\text{ }\text{inches}}=\frac{16\text{ }\text{inches}}{x\text{ }\text{inches}}[/latex]

 Solve by using the common denominator to clear fractions. The common denominator is [latex]71x[/latex].

[latex]\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}=1,136\\\\x=\frac{1,136}{17.75}=64\end{array}[/latex]

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

Read and Understand: We need to define a proportion to solve for the unknown distance between Seattle and San Jose.

 Define and Translate: The scale factor is [latex]1:557[/latex], and we will call the unknown distance x. The ratio of inches to miles is [latex]\frac{1}{557}[/latex].

We know the inches between the two cities, but we do not know miles, so the ratio that describes the distance between them is [latex]\frac{1.5}{x}[/latex].

Write and Solve: The proportion that will help us solve this problem is [latex]\frac{1}{557}=\frac{1.5}{x}[/latex].

Solve using the common denominator [latex]557x[/latex] to clear fractions.

[latex]\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}[/latex]

Interpret: We used the scale factor [latex]1:557[/latex] to find an unknown distance between Seattle and San Jose. We also checked our answer of [latex]835.5[/latex] miles with Google maps and found that the distance is [latex]839.9[/latex] miles, so we did pretty well!

Read and Understand: We know that for each [latex]2.5[/latex] inches on the map, it represents [latex]325[/latex] actual miles. We are looking for the scale factor for one inch of the map.

Define and Translate: The ratio we want is [latex]\frac{1}{x}[/latex] where x is the actual distance represented by one inch on the map.  We know that for every [latex]2.5[/latex] inches, there are [latex]325[/latex] actual miles, so we can define that relationship as [latex]\frac{2.5}{325}[/latex]

Write and Solve: We can use a proportion to equate the two ratios and solve for the unknown distance.

[latex]\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}[/latex]

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