## Proportions and Rates

### Learning Outcomes

• Write a proportion to express a rate or ratio
• Solve a proportion for an unknown

If you wanted to power the city of Lincoln, Nebraska using wind power, how many wind turbines would you need to install? Questions like these can be answered using rates and proportions.

## RATES

A rate is the ratio (fraction) of two quantities.

A unit rate is a rate with a denominator of one.

### Recall Reducing Fractions

The Equivalent Fractions Property states that

If $a,b,c$ are numbers where $b\ne 0,c\ne 0$, then

${\dfrac{a\cdot c}{b\cdot c}}={\dfrac{a}{b}}$.

Ex. $\dfrac{500}{20}=\dfrac{25\cdot 20}{1\cdot 20}=\dfrac{25}{1}=25$

### Example

Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.

## Proportion Equation

A proportion equation is an equation showing the equivalence of two rates or ratios.

For an overview on rates and proportions, using the examples on this page, view the following video.

### Using Variables to represent unknowns

Recall that we can use letters we call variables to “stand in” for unknown quantities. Then we can use the properties of equality to isolate the variable on one side of the equation. Once we have accomplished that, we say that we have “solved the equation for the variable.”

In the example below, you are asked to solve the proportion (an equality given between two fractions) for the unknown value $x$.

Ex. Solve the proportion $\dfrac{7}{3}=\dfrac{x}{15}$

We see that the variable we wish to isolate is being divided by 15. We can reverse that by multiplying on both sides by 15.

$\dfrac{7}{3}=\dfrac{x}{15}$

$15\cdot \dfrac{7}{3}=x$, giving $x=35$.

### Example

Solve the proportion $\displaystyle\frac{5}{3}=\frac{x}{6}$ for the unknown value x.

### Example

A map scale indicates that ½ inch on the map corresponds with 3 real miles. How many miles apart are two cities that are $\displaystyle{2}\frac{1}{4}$ inches apart on the map?

### Example

Your car can drive 300 miles on a tank of 15 gallons. How far can it drive on 40 gallons?

A worked example of this last question can be found in the following video.

Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.

You have likely encountered distance, rate, and time problems in the past. This is likely because they are easy to visualize and most of us have experienced them first hand. In our next example, we will solve distance, rate and time problems that will require us to change the units that the distance or time is measured in.

### Example

A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?

View the following video to see this problem worked through.

### Example

Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?

Other quantities just don’t scale proportionally at all.

### Example

Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend$10,000?

Matters of scale in this example and the previous one are explained in more detail here.

Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.

### Examples

The 2010 U.S. military budget was $683.7 billion. To gain perspective on how much money this is, answer the following questions. 1. What would the salary of each of the 1.4 million Walmart employees in the US be if the military budget were distributed evenly amongst them? 2. If you distributed the military budget of 2010 evenly amongst the 300 million people who live in the US, how much money would you give to each person? 3. If you converted the US budget into$100 bills, how long would it take you to count it out – assume it takes one second to count one \$100 bill.

### Example

Compare the electricity consumption per capita in China to the rate in Japan.

Working with large numbers is examined in more detail in this video.