Converting Rates
Converting rates uses unit multipliers to change a rate expressed in one pair of units to an equivalent rate in different units. To convert 9 yards per second to feet per second, multiply by the unit multiplier 3 ft/1 yd: (9 yd/1 sec) times (3 ft/1 yd) = 27 ft/sec. This Grade 7 math skill from Saxon Math, Course 2 extends unit conversion to compound units (two-unit rates), and is directly applicable in physics (converting m/s to km/h), sports analytics (converting statistics), and engineering.
Key Concepts
Property To convert a rate, write the rate as a ratio and multiply it by a unit multiplier to convert to the desired unit. For example, to change yards per second to feet per second: $$\frac{9 \text{ yd}}{1 \text{ sec}} \cdot \frac{3 \text{ ft}}{1 \text{ yd}}$$.
Examples Convert 20 miles per gallon to miles per quart ($1 \text{ gal} = 4 \text{ qt}$): $$\frac{20 \text{ mi}}{1 \text{ gal}} \cdot \frac{1 \text{ gal}}{4 \text{ qt}} = \frac{5 \text{ mi}}{1 \text{ qt}}$$ Convert 60 beats per minute to beats per hour: $$\frac{60 \text{ beats}}{1 \text{ min}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} = \frac{3600 \text{ beats}}{1 \text{ hr}}$$.
Explanation Is your speed in miles per hour, but you need feet per second? No sweat! Treat the rate as a fraction and use a unit multiplier to swap the unit you do not want for the one you do. You can convert distance, time, or both! It's like unit conversion on beast mode.
Common Questions
How do I convert a rate using unit multipliers?
Write the rate as a ratio, then multiply by a unit multiplier fraction that cancels the unwanted unit and introduces the desired unit. For 9 yd/sec times 3 ft/1 yd = 27 ft/sec.
What is a unit multiplier?
A unit multiplier is a fraction equal to 1 where numerator and denominator are equivalent measurements. Multiplying by it does not change the value — only the units.
How do I convert miles per hour to feet per second?
Multiply by the chain of unit multipliers: (miles per hour) times (5280 ft/1 mile) times (1 hour/3600 seconds). The miles and hours cancel, leaving feet per second.
Why is it important to set up unit multipliers with the right orientation?
The unit you want to cancel must appear in the denominator of the multiplier (to cancel with the unit in the numerator of the rate). If the units are flipped, they will not cancel.
When do students learn to convert rates?
Rate conversion is a Grade 7 skill. Saxon Math, Course 2 covers it in Chapter 9 as an application of unit multipliers.
What are common mistakes when converting rates?
Students sometimes flip the unit multiplier, causing units to not cancel. Always confirm that the unit you want to eliminate appears in the denominator of the multiplier.
How does rate conversion connect to dimensional analysis?
Rate conversion IS dimensional analysis. Tracking units through multiplication to ensure the final result has the desired units is the core of dimensional analysis in all sciences.