How to Solve SAT Percentage Problems Without Mixing Up the Cases

The three cases my students mix up are all distinct. The first case is when you see the word of. This is the simplest version. If a question asks for a percent of a number, you convert the percent to a decimal and multiply by the base number.

For example, if the question says 10% of 200, you move the decimal two places left to get 0.10, then multiply 0.10 by 200. You get 20. That is the direct percent-of case.

The second case is when you hear wording like greater than, more than, increased by, or increased over. For these, I want you to think tax. If you have a $100 subtotal and there is a 10% tax, you do not pay $10 total. You pay the original $100 plus the $10 tax, so the total is $110.

That means 10% greater than 200 is not 0.10 times 200. It is 110% of 200, which is 1.10 times 200. That gives you 220.

The third case is the opposite: less than, decreased by, or discounted by. For this one, think discount. If you have a 20% off coupon, the 20% is the part you are not paying. What remains is 80%, so you multiply by 0.80.

If you are paying 10% less than 200, you are paying 90% of 200. That is 0.90 times 200, which gives 180. Same percent, completely different multiplier. That is why the wording matters so much.

There are a few word cues that save a lot of time. In math word problems, the word of often acts like multiplication. The word is often acts like an equal sign. Greater than should make you think tax. Less than should make you think discount.

That might sound too simple, but this is exactly how you avoid randomly grabbing numbers from the question. You are not hunting for numbers. You are translating a sentence into an equation.

When the SAT says a population is K times another population, I literally split the sentence at is. Everything before is goes on the left side. Everything after is goes on the right side. That one habit keeps the relationship clean.

Here is the first practice question. A store has 240 items in stock. If 35% are on sale, how many items are on sale?

This is the simple case because the question is asking for 35% of 240. You are not increasing the 240. You are not discounting the 240. You are finding the portion of the stock that is on sale.

So we convert 35% to 0.35 and multiply: 240 times 0.35. That gives 84. So 84 items are on sale.

Notice that there is no second step here. A lot of students make percentage questions harder because they assume every percent question needs some trick. This one does not. It is just the of case.

Now try this one. This year's enrollment is 20% greater than last year's enrollment of 350 students. How many students are enrolled this year?

Last year's enrollment is 350 students, and that original amount represents 100% of last year's class. This year is 20% greater than that, so we add 20% on top of the original 100%.

That gives 120% total. As a decimal, 120% is 1.20. So we calculate 350 times 1.20, which gives 420.

The common wrong answer is 70 because 20% of 350 is 70. But 70 is only the increase. The question asks for this year's enrollment, which includes last year's 350 plus the increase.

The third basic example is the discount case. The sales price is 40% less than the original price of $150. What is the sales price?

The original price is 100%. A 40% discount means we chop away 40% of that original price. What remains is 60%.

So the sales price is 60% of 150. We calculate 150 times 0.60, which gives 90. The new price is $90.

Again, the percent in the sentence is not always the percent you pay. In a less-than question, the percent in the sentence is usually the amount removed. You have to ask what is left.

Now let us look at a more SAT-like version. The population of Greenville increased by 7% from 2015 to 2016. If the 2016 population is K times the 2015 population, what is the value of K?

First, I see the word is. That is my equal sign. The left side is the 2016 population. The right side is K times the 2015 population.

The question says the population increased by 7%. That means the 2016 population is the original 2015 population plus 7% of that population. In other words, it is 107% of the 2015 population.

As a decimal, 107% is 1.07. Since the question says the 2016 population is K times the 2015 population, K must be 1.07.

This is the greater-than case again, just with a variable instead of a final number. The logic is exactly the same.

The question that trips up a lot of students is the chained one. The regular price of a shirt at a store is $11.70. The sales price is 80% less than the regular price, and the sales price is 30% greater than the store's cost for the shirt. What was the store's cost?

When I see a question like this, I do not try to combine the whole thing in my head. I translate each sentence into an expression or equation one by one.

The first sentence is simple: the regular price is $11.70. I can call the regular price R, so R equals 11.70.

The second sentence says the sales price is 80% less than the regular price. Let the sales price be S. If the sales price is 80% less than regular price, then the customer is paying the remaining 20%. So S equals 0.20R.

Since R is 11.70, the sales price is 0.20 times 11.70, which is 2.34. So S equals 2.34.

The third sentence says the sales price is 30% greater than the store's cost. Let the cost be C. Greater than means tax, so the sales price is 130% of the cost. That means S equals 1.30C.

We already know S is 2.34, so 2.34 equals 1.30C. Divide both sides by 1.30 and you get C equals 1.80. The store's cost was $1.80.

The reason this question feels hard is not the arithmetic. It is that the SAT stacks two percentage relationships on top of each other. If you translate one sentence at a time, it becomes manageable.

If you nail this topic, it is an easy place to pick up points because percentage language comes up so often. These questions are really testing how well you can translate words into expressions.

Do the translation first. Then do the arithmetic. If you remember of means multiply, greater than means tax, and less than means discount, you will stop mixing up questions that look almost the same on the surface.