ACT Math · Lesson 2 of 5
Percentages, rates, ratios, and averages
Choose the correct base for a percent or rate, and convert an average back into a total when a value is missing.
Rates and averages
A rate compares different units, such as miles per hour. If a car travels 150 miles in 2.5 hours at a constant speed, the speed is 150/2.5 = 60 miles per hour. Distance, rate, and time stay linked by distance = rate × time.
The mean is the total divided by the number of values. If five scores have a mean of 80, their total is 400. A missing score is the total minus the known scores, not the mean minus those scores.
Percent and ratio language
“20% of the class” uses the whole class as the base. “The ratio of boys to girls is 3:2” compares two parts. In a class of 30, those parts total 5 shares, so each share is 6 students: 18 boys and 12 girls.
A successive percent change multiplies factors. A 10% increase followed by a 10% decrease is ×1.10 × 0.90 = ×0.99, which is a 1% decrease overall, not a return to the start.
Worked example
Four test scores are 70, 80, 90, and 85. What fifth score gives a mean of 84?
- Five scores with a mean of 84 have total 5 × 84 = 420.
- The known scores sum to 70 + 80 + 90 + 85 = 325.
- The fifth score is 420 − 325 = 95.
Why this works. Turn the mean into a total before you subtract.
Check your understanding
- Include units in a rate and check that they cancel correctly.
- Recover a total from a mean before finding a missing value.
- Multiply percent-change factors instead of adding the percentages.