PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

On test day
Ratios, proportions, and percents
Data analysis

When a question features multiple percentages, you have to make a key strategic decision: can I do the arithmetic on the percentages themselves and get the answer right away or do I have to calculate each percentage individually and do the arithmetic on the actual values?

For example, suppose a car traveling 50 miles per hour increases its speed by 20 percent and then decreases its speed by 20 percent. Can you just say that its final speed is 50 miles per hour since +20% − 20% = 0? No, because after a 20% increase, the car’s speed becomes 120% of the original: 1.2(50) = 60. When the car “decreases its speed by 20 percent,” that 20 percent is calculated based on the new speed, 60, not the original speed, and 20 percent of 60 is greater than 20 percent of 50. Thus, the car’s final speed is lower than its starting speed: 50(1.2)(0.8) = 48 miles per hour.

By contrast, suppose you have to find how many more meat eaters than vegans live in a certain region where there are 13,450 residents, given that 62 percent of them eat meat and 8 percent of them are vegan. It may be tempting to find 62 percent of 13,450 (0.62 × 13,450 = 8,339), then find 8 percent of 13,450 (0.08 × 13,450 = 1,076), and finally subtract those two numbers to get the answer (8,339 − 1,076 = 7,263). This is a waste of time, though. Instead, you can quickly find the difference between the two percentages (62 − 8 = 54) and take 54 percent of the total to get the answer in one step: 13,450 × 0.54 = 7,263, the same answer.

If you can do arithmetic using the percentages but choose to do arithmetic on the raw numbers instead, you’ll waste time doing unnecessary work. But if you can’t do arithmetic on the percentages (as in the first example) but do anyway, then you’ll get an incorrect answer. So, being able to tell whether you can or can’t do the arithmetic on the percentages is a useful skill.

Luckily, the fundamental principle is simple: you can always do arithmetic on the percentages as long as the percentages are out of the same total. If the totals are different, then you must convert the percentages into actual values. Practice applying this principle on the following question.

21. There are 400 seniors and 420 juniors in a certain high school. Of the seniors, 65% are eligible for an advanced placement world history course. Among the juniors, 75% are not eligible to enroll in that course. How many more seniors than juniors could enroll in the course?

The answer and explanation can be found at the end of this chapter.