SAT Math: Predator-Prey Percentage Increases

A hard Digital SAT grid-in question testing whether you translate percentage increases and percent-less-than language precisely.

Question

A researcher investigated two species of mites: a predator and its prey. At the start of a week, there was an equal number of the two species. At the end of the week, the number of prey had increased by $2900\%$ of the number of prey at the start of the week, and the number of predators had increased by $320\%$ of the number of predators at the start of the week. The number of predators at the end of the week was $p\%$ less than the number of prey at the end of the week. What is the value of $p$ ?

Step-by-Step Solution

Translate the population changes before comparing them.

1Choose a starting amount.

The two populations start equal. Let the starting number of prey be x and the starting number of predators also be x.

\text{Initial prey}=x \qquad \text{Initial predators}=x

2Convert each percent increase to a final amount.

An increase by 2900% means add 29x to the original x. So the final prey population is:

x+29x=30x

An increase by 320% means add 3.2x to the original x. So the final predator population is:

x+3.2x=4.2x

3Compare predators to prey at the end.

The ending predator population is 4.2x, and the ending prey population is 30x. The fraction of the prey population represented by predators is:

\frac{4.2x}{30x}=0.14

So the predator population is 14% of the prey population.

4Turn that into percent less than.

If predators are 14% of prey, then predators are 86% less than prey.

p=86