Ratios Rates Proportional Relationships and Units Pattern - Rate Application

Rate Application

A rate tells you how much of something happens per unit of time (or per unit of something else). This pattern gives you a rate and asks you to find a total, a time, or the rate itself.

The Three Rate Questions

$\text{Total} = \text{Rate} \times \text{Time}$ (or $\text{Rate} \times \text{Quantity}$)

Rearranging: $\text{Time} = \dfrac{\text{Total}}{\text{Rate}}$ and $\text{Rate} = \dfrac{\text{Total}}{\text{Time}}$

Worked Examples

Example 1. A machine produces $120$ bottles per hour. How many bottles does it produce in $8$ hours?

Total $= 120 \times 8 = 960$ bottles.

Example 2. A printer prints $15$ pages per minute. How long does it take to print $225$ pages?

Time $= \dfrac{225}{15} = 15$ minutes.
Gotcha: Don't multiply $225 \times 15 = 3{,}375$. When finding time, you divide the total by the rate.

Example 3. A car uses $12$ gallons to drive $384$ miles. What is the fuel efficiency in miles per gallon?

Rate $= \dfrac{384}{12} = 32$ miles per gallon.
Gotcha: Option traps may show $\dfrac{12}{384}$ (gallons per mile). Read what unit is asked: "miles per gallon" means miles in the numerator.

Example 4. A download speed is $5$ megabytes per second. A file is $800$ megabytes. How many minutes does the download take?

Time in seconds: $\dfrac{800}{5} = 160$ seconds.
Convert to minutes: $\dfrac{160}{60} \approx 2.67$ minutes.
Gotcha: The rate is in seconds, but the answer is in minutes. Don't forget the unit conversion.

What to Do on Test Day

• Total = Rate $\times$ Time. Memorize this relationship and rearrange as needed.
• Unit rate = divide total by the number of units. "Miles per gallon" = $\dfrac{\text{miles}}{\text{gallons}}$.
• Watch the units. If the rate is per second but the answer asks for minutes, you need an extra conversion step.
• Numerator vs. denominator: "Per" tells you what goes in the denominator. "Miles per gallon" = miles $\div$ gallons.