Percents | SAT Problem Solving and Data | SAT Math

Introduction

Percents questions deal in one way or another with the relationship between a part and a whole. As the origin of the word “percent” implies (think century or centennial), the number $100$ is at the heart of the percent calculation. A percent is a way of expressing the relationship between a part and a whole if the whole were $100$ . And because that relationship involves dividing the part by the whole, percents are closely related to fractions and ratios. In sum, a percent calculation relates a number to $100$ but then lists the number without the $100$ in the final result. On the SAT, the embedded calculator will be of great use to you, but you must know how to convert a percent into a decimal. We will remind you of this many times through this lesson!

Approach Question

A biologist is monitoring cell division by observing cells in a petri dish. On Day 3 of her experiment, she notes that the number of cells has grown by $20%$ since the beginning of the experiment. If the number of cells on Day 3 is $132$ , how many cells were in the petri dish at the beginning of the experiment?

A. 110
B. 112
C. 152
D. 158

Explanation

The trickiest kinds of percent questions are those that provide a value at the end of a process rather than at the beginning. If you have an initial number and a percentage by which that number grows or shrinks, the calculation is straightforward. For example, a score that begins at $80$ and then grows by $15%$ is equal to $(80) (1.15) = 92$ . Note the $1$ in front of the decimal place; to deal with a quantity growing or decreasing, it helps to think of the starting point as the equivalent of $1$ , which is another way of saying the tautology that something starts at $100%$ of itself.

Because this question describes a process that ends with the value of $132$ , we cannot follow the sequence described above. In fact, the trap answer of $158$ comes from the mistake of multiplying $132$ by $1.2$ , which comes to $158.4$ . If you use the UnCLES method well, reading carefully and considering the answers, you might have figured out that answers greater than $132$ are impossible; after all, $132$ is the number of cells after the culture grows. The initial amount must be smaller.

The best approach when the initial amount is unknown is to set up an equation. If we denote the initial amount $x$ , we can multiply by the $1.2$ that represents a growth of $20%$ , setting the result equal to the final quantity of $132$ . If $1.2 x = 132$ , we simply need to divide each side by $1.2$ ; the calculator helps us identify the right answer as 110.

One final note: you may have learned to solve percent problems using a proportion with $100$ . If you are comfortable with this approach and realize that a $20%$ increase is equal to $120%$ , you could proceed as follows, using cross-multiplication:

$\frac{100}{120} 120 x x = \frac{x}{132} = 13200 = 110$

Definitions

Part: We note two formulas for you to memorize in the flashcard portion of this lesson. One of them references the words “part” and “whole.” The part in a percent expression is the number that is meant to be divided by the sum total or “whole.” Typically, the “part” is less than the whole, which makes sense when we think of the normal meaning of these words. However, this is not always the case, as we can see a population that grows from $1, 000$ to $1, 500$ . The new population is actually $150%$ of the old population because the “part” has grown larger than the original “whole”.
Whole: The “whole” is the entire quantity to which a part is compared, naturally following the words “out of” (as in “What percent is $45$ out of $60$ ?”). As shown above, a whole is usually, but not always, greater than the part.
Is: Why is this word here? Don’t we know what the word “is” means? We include it here because of translation from English to algebra. “Is” is the same as equals, so it should be replaced by the equals sign when doing this sort of translation. See the Difficulty 3 problem below to see how this works.
Of: This word is also included for “translation” purposes. When you encounter “of” in a percent problem, it should be replaced by the multiplication symbol.

Topics for Cross-Reference

Variations

Sometimes, the SAT will ask about percent change rather than simply percentage. You can tell this is happening when the question pairs the word “percent” or “percentage” with the words “more/greater” or “less” rather than “of” or “out of.” (The Approach question bears some similarity to this type of question but is a little different in that it doesn’t ask for the percent change, but rather the initial amount).

Strategy Insights

What number do you think might be a helpful starting point for a percent question? Probably $100$ , right? If you are not given a quantity and need a place to begin, 100 is a great place to start. This is especially helpful when the question involves increasing and then decreasing by a certain percentage (or decreasing and then increasing). We mention this here because it can sometimes be helpful to plug in a number when the initial amount is unknown rather than starting with a variable to represent that unknown amount.
On a related note, it might make more sense in the context of some questions to simply use $10$ instead of $100$ , to keep the values smaller. Our entire mathematical system is based on place values with 10, so take advantage of that fact!

Flashcard Fodder

There are two formulas you must know to be fully equipped for percent questions:

-percent = (part/whole) x 100
-percent change = (new - old)/old

(the “old” is the starting point in a situation before it moves to the “new” value)

Sample Questions

Difficulty 1

A dress is originally priced at $ $80$ but is on sale for $25%$ off. What is the sale price of the dress, in dollars? (Note: this is a free-response question.)

(spoiler)

The answer is 60. There are two ways to begin this question; both begin by recognizing that $25%$ must be converted to the decimal $0.25$ to be used in the calculator. (Remember that “percent” means out of 100; this means that $25%$ is the same as $25/100$ , which is the same as $0.25$ .)

We could multiply $0.25$ by $80$ to arrive at the discount of $$20$ , then subtract that discount from the original amount to get $$60$ . However, the process is even simpler if you recognize that discounting something by $25%$ is the same thing as multiplying it by $75%$ . $(80) (.75) = 60$ .

Difficulty 2

Jay got $40$ questions correct on one test and then $46$ correct on the next one. Assuming the two tests have the same number of questions, Jay’s percent correct on the second test was what percent greater than his score on the first test?

A. 6
B. 12
C. 15
D. 20

(spoiler)

The answer is 15. A great change to use the percent change formula! The “new” amount is $46$ because it happened second; the “old” amount is $40$ . Plugging these into the formula, we arrive at $15$ , because $\frac{46 - 40}{40} \times 100 = 15$ . Watch out for the trap answer of $6$ ; that’s the absolute amount that Jay’s score grows, not the percent growth.

Difficulty 3

$18$ is $p$ percent of $80$ . What is the value of $p$ ? (Note: this is a free-response question.)

(spoiler)

The answer is 22.5. Problems like this often cause students trouble, but they can be dependably solved through a careful “translation” process. Let’s take the sentence “ $18$ is $p$ percent of $80$ ” and transfer each part into an algebraic equation, translating where necessary. $18$ can remain $18$ ; “is” becomes an equals sign; “ $p$ percent” becomes $p /100$ ; “of” becomes multiply; $80$ remains $80$ . The result looks like this:

$18181800 22.5 = \frac{p}{100} (80) = \frac{80 p}{100} = 80 p = p$

Had this been a multiple-choice question, plugging in the answers for p would have been an excellent strategy. Had an answer choice proved too large or too small, we would have been able to “triangulate” our way to the answer fairly quickly. But in a free-response situation, the textbook algebra approach is best.

Difficulty 4

A statistician records the number of people attending a four-game series between two baseball teams. After counting the attendance on the first day, the statistician notes that attendance falls by $20%$ from the first day to the second, grows by $10%$ from the second day to the third day, and grows by $25%$ from the third day to the last day. If the attendance on the first day is denoted $a$ , what is the attendance, in terms of $a$ , on the last day?

A. 1.01a
B. 1.05a
C. 1.1a
D. 1.15a

(spoiler)

The answer is 1.1a. We are helped tremendously on this question by remembering that a $1$ can always represent our initial amount before it changes. Just as the initial amount could be written here as 1a, we use this principle to observe that a decline of $20%$ means multiplying by $0.8$ (that is, $1 - 0.2$ ); a growth of $10%$ means multiplying by $1.1$ ; and a growth of $25%$ means multiplying by $1.25$ . We can now simply multiply all the values together: $(a) (0.8) (1.1) (1.25) = 1.1 a$ .

The main way to go wrong on this question is to think you can simply subtract and add all the percents. If you start with $100$ , then subtract $20$ , add $10$ , and add $25$ , you will end up at $115%$ . That could lead you to erroneously answer $1.15 a$ .

However, starting with $100$ is a great way to “plug and chug” here if done correctly! Beginning with $100$ shows readily that the next value must be $80$ (after a $20%$ decrease). From there, we must add $10%$ of $80$ , so $8$ , bringing us to $88$ . Then, $25%$ of $88$ is $22$ which, when added to $88$ , yields $110$ . Since $110%$ as a decimal is $1.1$ , the answer becomes apparent.

Difficulty 5

Three students are competing to see who can score the highest on a test. Student A scores the highest, achieving a score of $97.5$ correct. Student B’s score is $15%$ less than Student A’s score, and Student B’s score is $22%$ greater than Student C’s score. Which of the following is closest to Student C’s score?

(spoiler)

The answer is 68. The fact that both of the comparisons in this question are centered around Student B makes this question more challenging; in one case, we can multiply a percent by a known amount (Student A’s score) to get Student B’s score, but in the other case, we have to start with the unknown Student C to make sense of the comparison.

Let’s begin by recognizing, similar to what we’ve seen before, that “ $15%$ less” is the same thing as multiplying by $0.85$ ( $1 - 0.15$ ). We can take the original percent of $97.5$ and multiply it by $0.85$ , yielding $82.875$ .

For the other percent calculation, we need to begin with our unknown and make an equation. Since it’s Student C’s score, let’s call it $c$ ; we need to multiply that $c$ by $1.22$ because the score we’re moving toward (Student B’s score, which we already know) is $22%$ greater. So our equation is $1.22 c = 82.875$ . Using the calculator to divide by $1.22$ , we discover that $68$ is the closest whole number to our result.

This question also lends itself to backsolving (plugging in the answers). If you started with one of the middle answers, as recommended, you might begin with $66$ . If you multiply that score by $1.22$ , you’d come up with something less than $82.875$ , which is still Student B’s score, found the same way as before. That means $66$ is not quite high enough. Guess which answer, then, is correct!

For Reflection

How will you approach percent questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
Can you explain to yourself the difference between percent questions and percent change questions? Try to give yourself a couple of examples of each kind.