Lesson 64: Using a Unit Multiplier to Convert a Rate

New Concept

We convert rates using a unit multiplier. This is a fraction equal to one that cancels unwanted units and introduces the ones you need.

What’s next

Next, we'll walk through examples converting running speeds and driving rates. You’ll see exactly how to set up and solve these conversions.

Property

A rate, like $\frac{60 \text{ miles}}{1 \text{ hour}}$, is converted by multiplying it by a unit multiplier, which is a fraction of equivalent values like $\frac{1 \text{ hour}}{60 \text{ minutes}}$.

Examples

• Convert miles per hour to miles per minute: $\frac{72 \text{ mi}}{1 \text{ hr}} \times \frac{1 \text{ hr}}{60 \text{ min}} = \frac{1.2 \text{ mi}}{1 \text{ min}}$
• Convert tons per day to pounds per day: $\frac{0.5 \text{ tons}}{1 \text{ day}} \times \frac{2000 \text{ lbs}}{1 \text{ ton}} = \frac{1000 \text{ lbs}}{1 \text{ day}}$

Explanation

Think of unit multipliers as magic cancelers! To convert a rate, multiply it by a special fraction that has the unwanted unit on the opposite side. Poof! The old unit is gone, and the new one takes its place. It is all about strategic cancellation.

Property

To convert a rate like miles per minute to miles per hour, multiply by a unit multiplier that cancels the original time unit: $\frac{1 \text{ mile}}{5 \text{ min}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} = \frac{12 \text{ mi}}{1 \text{ hr}}$.

Examples

• A snail crawls 2 feet per minute. Find its speed in feet per hour: $\frac{2 \text{ ft}}{1 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} = \frac{120 \text{ ft}}{1 \text{ hr}}$
• Convert 440 yards per minute to yards per second: $\frac{440 \text{ yd}}{1 \text{ min}} \times \frac{1 \text{ min}}{60 \text{ sec}} = \frac{7.33 \text{ yd}}{1 \text{ sec}}$

Explanation

So, you ran a mile in 5 minutes and want to brag about your speed in miles per hour? No problem! Just multiply your rate by the unit multiplier $\frac{60 \text{ min}}{1 \text{ hr}}$. The 'minutes' cancel out, leaving you with your awesome new speed to share.

Property

First, convert the rate to new units, then use it in a formula like $d = rt$. For example, convert mph to kph: $\frac{50 \text{ mi}}{1 \text{ hr}} \cdot \frac{1.6 \text{ km}}{1 \text{ mi}} = \frac{80 \text{ km}}{1 \text{ hr}}$. Then find distance.

Examples

• Driving 50 mph, how many kilometers do you go in 2 hours? (1 mi ≈ 1.6 km). First: $\frac{50 \text{ mi}}{1 \text{ hr}} \times \frac{1.6 \text{ km}}{1 \text{ mi}} = \frac{80 \text{ km}}{1 \text{ hr}}$. Then: $d = \frac{80 \text{ km}}{1 \text{ hr}} \times 2 \text{ hr} = 160 \text{ km}$.
• A pump moves 3 gallons per minute. How many quarts does it move in 4 minutes? (1 gal = 4 qt). First: $\frac{3 \text{ gal}}{1 \text{ min}} \times \frac{4 \text{ qt}}{1 \text{ gal}} = \frac{12 \text{ qt}}{1 \text{ min}}$. Then: Volume = $\frac{12 \text{ qt}}{1 \text{ min}} \times 4 \text{ min} = 48 \text{ qt}$.

Explanation

This is a two-step dance! First, you convert the rate to the units you actually need for the problem, like changing miles into kilometers. Once you have your new rate, you can plug it into a formula to find your final answer, like calculating the total distance traveled.