Percentages: Pick 100 and Watch the Problem Fall Apart

By EMPOWERgmat on July 28, 2020

Here is a scene I watch play out every single week. A Test Taker reads a percent question. There is not one real number in it. Just "a certain price" and "some number of employees." So they do what they learned in school. They call the price P. They write 1.2P. They write 0.75(1.2P). Four lines later they are staring at 0.9P and wondering if they dropped a decimal somewhere. Two and a half minutes are gone. Maybe they got it right. Maybe not.

THAT is the long way. On the GMAT, the long way costs you points.

Here is the short way. When a percent question gives you no real numbers, YOU pick the number. And the number you pick is 100.

Why 100 makes the problem fall apart

"Percent" means "out of 100." So when you start at 100, every percent turns into a plain number. 20% of 100 is 20. 35% of 100 is 35. No decimals. No variables. No algebra.

This tactic is called TESTing VALUES. You swap the unknown for a real number you choose. Then you do simple arithmetic and read the answer off your notes. If the question wants a percent, the answer will be the same percent no matter what number you start with. So start with the friendliest number there is.

Stop reading for a second. If a price goes UP 20% and then DOWN 20%, is it back where it started? Decide before you read the next line.

It is not. 100 up 20% is 120. Now 20% of 120 is 24, so 120 down 20% is 96. You ended up 4% lower than you started. Why? Because the second percent is taken from a BIGGER number. I'll bet a lot of you said "yes, it's back to 100." That is exactly the instinct the GMAT is built to punish, and starting at 100 shows you the truth in about five seconds.

A tall stack of identical blocks grows taller as blocks are added, then a larger slice is lifted off the top, leaving the stack a little shorter than where it began. — Up a percent, then down the same percent, and you land below where you started.

Why this matters on the test you are actually taking

Right now the Quant section gives you 31 questions in 62 minutes, split between Problem Solving (regular multiple choice with five answers) and Data Sufficiency (where you decide whether the facts given are enough to answer the question). You can see the full layout on GMAC's page on the sections, timing and format.

That works out to an AVERAGE of 2 minutes per question. Average is not a speed limit. Some questions should take 40 seconds so others can take 3 minutes. Percent questions with no real numbers should be your 40 second questions, but only if you stop treating them like algebra homework.

There is a second reason. The test is adaptive question by question, and you cannot go back to change an answer. A decimal slip in line three of your algebra is permanent. A clean chain of whole numbers on your scratch work is much harder to mess up.

Try it

A store raises the price of a jacket by 20%. A month later, the store puts the jacket on sale at 25% off its new price. The sale price is what percent of the original price?

10%
90%
92%
95%
105%

Show the answer

Step 1: Pick 100. The original price is 100. Write it down.

Step 2: Run the first change. 20% of 100 is 20. So the new price is 120. Write 100 then 120.

Step 3: Run the second change on the NEW number. 25% of 120 is 30. So the sale price is 120 minus 30, which is 90. Your notes now read 100, 120, 90.

Step 4: Answer the question they asked. 90 out of an original 100 is 90%. The answer is (B).

Check it a second way: 1.2 times 0.75 is 0.9. Same answer.

The traps: (D) is for people who add the percents: up 20, down 25, so "down 5," so 95%. That ignores that the 25% comes off 120, not 100. (A) is for people who did the math right and then answered the wrong question. The price DROPPED 10%, but the question asked what percent OF the original the sale price is. Read the question one more time before you click.

Now look at the variety. Data Sufficiency questions love this idea too, and 100 works just as well there.

Try it

On Monday, the price of a stock rose by p percent. On Tuesday, the price fell by p percent. Is the price at the end of Tuesday lower than the price at the start of Monday?

(1) p = 20
(2) p is a positive integer less than 50

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Show the answer

Step 1: Pick 100 as the Monday morning price.

Statement (1): p = 20. 100 up 20% is 120. 20% of 120 is 24, so 120 down 20% is 96. 96 is lower than 100. The answer is a definite YES. Sufficient.

Statement (2): p could be many values, so TEST several. Try p = 1: 100 becomes 101, and 1% of 101 is 1.01, so Tuesday ends at 99.99. Lower. Try p = 49: 100 becomes 149, and 49% of 149 is 73.01, so Tuesday ends at 75.99. Lower. Try p = 10: 110, then 110 minus 11 is 99. Lower. Every value gives YES, because the drop is always taken from a bigger number than the rise was. Sufficient.

Each statement alone is sufficient. The answer is (D).

The trap: people who think "up p, down p, it cancels" decide the price ends where it started and never test anything. Others pick (A) because they believe they need ONE exact value of p. You do not. You only need the same answer to the yes or no question every time.

The mistakes I see every week

Picking a "realistic" number. Someone reads "a jacket" and starts with $80 because that feels like a jacket price. Now 15% of 80 is 12, which is fine, but 35% of 80 is 28 and 12.5% of 80 is 10, and the arithmetic gets slower with every step.
INSTEAD: The question does not care what the jacket costs. Start at 100.
Adding and subtracting the percents. Up 30%, down 10%, so "up 20%." No.
INSTEAD: Run each change, one at a time, on the number you have right now. 100, then 130, then 130 minus 13, which is 117. Up 17%, not 20%.
Doing the chain in your head. You are "smart enough" to hold 100, 130 and 117 in your head. Sure. Until question 24, when you are tired and you hold 130 and 113 instead.
INSTEAD: Write every number in the chain. It takes three seconds. Skipping it is pride, and pride is a terrible test strategy.
Forcing 100 when the problem also has fractions. If the question says "one third of the remaining stock," then 100 gives you 33.33 and you are back in decimal land.
INSTEAD: Pick a multiple of 100 that also plays nicely with the fraction. For thirds, use 300. For sevenths, 700. Same idea, still whole numbers.
Answering the question you expected instead of the one they asked. "Percent of the original" and "percent decrease" are different answers, and the GMAT puts BOTH in the answer choices.
INSTEAD: Before you click, look back at the last sentence of the question and match your answer to it word for word.
Using algebra because it feels more "real." Some strong math students feel like picking numbers is cheating. It is not. The test rewards the right answer, not the prettiest work.
INSTEAD: Solve one problem both ways and time each. Then decide what you want MORE: elegant algebra or a higher score.

Where 100 fits in the bigger picture

Starting at 100 is one tool in a bigger toolbox. If you want the full rundown on the percent question types you will face, read our guide to tackling percentage problems. And if you still think simple arithmetic is "too slow" to be a real strategy, look at how brute force can work really fast on an official question. The theme is the same every time: Quant is not a math test. It is a thinking test that uses math to keep score. The Test Takers who score high are the ones who pick the cheap route on purpose.

Your challenge for today

Go back through your practice work or the official practice materials and pull five percent questions that had no real numbers in them. Pick the ones you got wrong, or the ones you got right the long way. Redo all five starting at 100. Write every number in the chain. Time each one.

Then put your new times next to your old ones. I'll bet at least three of the five got FASTER, and I'll bet you made zero decimal slips. If that bet pays off, you have just found free minutes for the hard questions. Go collect them.