Percents

Percents express what the numerator of a fraction would be if the denominator was 100 (80% = 80/100). 100 is the standard base, and whenever a student is thinking about percentage questions, they must always want to think about that idea of the base.

It is possible to have percentages of numbers other than 100.

75% of 160 = 120. Put another way 75% = 120/160. In this case 160 is the base, and that determines the %.

The most common mistake ACT students make on percentage questions is to accidentally change the base of the percentage fraction, oftentimes without knowing it. ACT percentage questions will always contain wrong answers based off of this mistake.

For example, here is Problem #59 from the June 2020 ACT.

This problem includes two different price markdowns, and so there is a chance to make two different base errors here. Adding together the two discounts 10% + 30% leads to answer E. However, these percentages were discounts off of two different prices (and therefore also two different bases). Trying to add them together is like adding fractions with different denominators: it simply cannot be done.

Instead, a student must use the inverse percentage, so that everything stays in terms of the original price. Saying that something is 20% off of the original price is the same thing as saying it is 80% of the original price.

In the problem above, the sales price of a jacket is 10% off of the original price. This means that the sale price = 90% of the original price, or that the Sale Price = 0.9(Original Price).

The clearance price is 30% off of the sale price. This means that the clearance price = 70% of the sale price, or that Clearance Price = 0.7(Sale Price).

Combining these two equations together, shows that Clearance Price = 0.7(0.9)(Original Price) = 0.63(Original Price).

The clearance price is 63% of the original price, meaning that it is 37% off (100-63) of the original price.

The other wrong answers in this question come from getting caught up with the original percentages given. For example, it is easy to think that since the sales price is 10% off of the original price, then the original price is just 110% of the sales price.

However, that’s flipping the base of the percentage. The store took 10% of the original price, not 10% of the sales price. So adding 10% to the sales price is likely to underestimate the original change. This is why it’s good to simply keep things in terms of the original price Sale Price = 0.9(Original Price). 

You are much less likely to make an easy error if you keep things in the units/terms of the original numbers in the problem.