Solving Problems with Rate Conversion
Grade 8 math lesson on solving multi-step problems that require converting between units and applying rate conversions. Students learn to use dimensional analysis and unit fractions to convert rates like miles per hour to feet per second or dollars per hour to cents per minute.
Key Concepts
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.
Common Questions
What is a rate conversion?
A rate conversion changes a rate from one unit combination to another. For example, converting miles per hour to feet per second, or meters per minute to kilometers per hour. You multiply by conversion fractions that equal 1.
How do you use dimensional analysis to convert rates?
Write the original rate, then multiply by conversion fractions where the unit you want to eliminate is on the opposite side of the fraction from where it appears in the original rate. Cancel matching units.
What is an example of a rate conversion problem?
Convert 60 miles per hour to feet per second: 60 miles/hour x 5280 feet/mile x 1 hour/3600 seconds = 88 feet per second. The miles and hours cancel, leaving feet per second.
Why do we multiply by fractions equal to 1 in rate conversions?
Multiplying by a fraction equal to 1 (like 5280 feet / 1 mile) does not change the value, only the units. Since 5280 feet equals 1 mile, the fraction equals 1 and we can multiply by it freely.