What this SAT skill is testing
This unit combines ratios, rates, proportional relationships, and unit conversion. The SAT often hides a short proportion inside a real-world context, so careful labeling matters more than complicated algebra.
Key relationships
- A ratio compares quantities in the same order every time.
- A rate compares quantities with different units, such as miles per hour.
- A unit rate has a denominator of 1.
- In a proportional relationship, (y=kx), where (k) is the constant of proportionality.
- Conversion factors are fractions equal to 1, chosen so unwanted units cancel.
Write units beside every number. If the units do not cancel to the unit requested, the setup is not finished.
Worked SAT-style example
A machine packages 84 items in 6 minutes. Its unit rate is ( rac{84}{6}=14) items per minute. At the same rate, the number packaged in 15 minutes is (14(15)=210).
The equation is (y=14x), where (x) is time in minutes and (y) is the number of items. Because the relationship is proportional, the graph passes through the origin.
Unit conversion by cancellation
To convert 72 miles per hour to miles per minute, use ( rac{1\text{ hour}}{60\text{ minutes}}):
(72\frac{\text{miles}}{\text{hour}}\cdot\frac{1\text{ hour}}{60\text{ minutes}}=1.2\frac{\text{miles}}{\text{minute}}).
The hour units cancel, leaving the requested unit.
Step-by-step method
- Define what each quantity represents and attach units.
- Decide whether the relationship is proportional or includes a starting value.
- Find the unit rate or constant of proportionality.
- Scale, substitute, or multiply by conversion factors.
- Check that the final unit and size make sense.
Common traps
- Reversing a ratio midway through the solution.
- Treating (y=kx+b) with (b\ne0) as a proportional relationship.
- Multiplying by a conversion factor in the wrong orientation.
- Mixing minutes and hours or inches and feet in one equation.
- Using a total rate when the question asks for a per-unit rate.
Practice strategy
Build the rate first, then apply it. This two-step habit makes word problems easier to audit and prevents you from multiplying or dividing by the wrong quantity.
