SAT Percentages: Multipliers, Reversals, and Stacked Discounts

The core idea

Here is a result that trips up almost every test-taker: a price rises 25%, then falls 20%. What's the net change?

Zero. The price is exactly back where it started. That sounds like a coincidence — it isn't, and once you see the mechanism, you'll own every SAT percentage problem on the test.

A percent is not a fixed dollar amount. It's a fraction of whatever it's applied to. That "whatever" — the number you're taking the percent of — is called the base. Almost every wrong answer on SAT percentage questions comes from using the wrong base. The right answer always comes from asking two questions first:

What is the base? Am I going forward (base → result) or backward (result → base)?

Here's the reframe that makes all of this mechanical. When a price increases by $p\%$, you're not adding a fixed amount — you're scaling:

So a 25% increase is the same as multiplying by $1.25$. A 20% decrease is the same as multiplying by $0.80$. These are your multipliers, and they're the engine of this whole topic.

• Increase by $p\%$ → multiply by $\left(1 + \dfrac{p}{100}\right)$
• Decrease by $p\%$ → multiply by $\left(1 - \dfrac{p}{100}\right)$

Now that opening puzzle makes sense: $+25\%$ then $-20\%$ is $1.25 \times 0.80 = 1.00$. Exactly back to the start. Not a coincidence — the multipliers cancelled.

The key sentence of this lesson — the one worth writing on your hand before the test:

Going forward, multiply by the multiplier. Going backward, divide by it. Never subtract the percent from the result.

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Worked examples

Example 1 — Forward: a straightforward discount

A jacket originally costs $80. It goes on sale at 25% off. What is the sale price?

The base is $80 — that's the original price, the number the 25% is measured against.

A 25% reduction means you keep $100\% - 25\% = 75\%$ of the original:

The sale price is $60.

You could instead compute the discount separately — $0.25 \times 80 = \dollar 20$ — then subtract: $80 - 20 = 60$. Same answer. But the two-step route has a trap built in: if you stop after finding $20, you've found the discount, not the price. The SAT puts $20 in the answer choices for exactly this reason. The multiplier route gets you to the final answer in one shot.

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Example 2 — Percent change: which number goes on the bottom?

A plant grows from 20 inches to 25 inches over one month. By what percent did its height increase?

This type has one classic wrong answer baked in: 20%. Here's where it comes from — and why it's wrong.

The formula is:

The original always goes in the denominator. You're measuring how far you've traveled relative to where you started, not where you ended up.

The plant's height increased by 25%.

The wrong answer of 20% comes from dividing by 25 — the new height — instead of 20. It's tempting because 25 is right there in the problem. But ask yourself: 5 inches is what fraction of where you started? Twenty inches. The original is always the anchor.

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Example 3 — Backward: recovering the original price

A gym membership costs $48 after a 20% increase. What was the original price?

This is where most points are lost. The tempting wrong move: subtract 20% from $48.

Why is that wrong? Because the 20% was never taken off $48 — it was added to the original price. You'd only be justified in removing 20% from $48 if the 20% had been measured against $48, and it wasn't. That's the wrong base.

The right move: write one equation, then divide.

Let $x$ be the original price. A 20% increase means the new price is 120% of $x$:

The original price was $40.

Check: $40 + 0.20(40) = 40 + 8 = 48$. ✓

The same logic handles discounts. A different jacket's sale price is $48 after a 20% markdown. What was the original price?

A 20% markdown means the sale price is 80% of the original:

Check: $20\%$ of $60$ is $12$, and $60 - 12 = 48$. ✓

Notice the trap the SAT sets here: multiplying $48 by 1.2 gives $57.60 — this is the most common wrong answer. It would only be correct if the 20% had come off the sale price. It didn't. The 20% came off the original. One equation, then divide.

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Example 4 — Successive changes: multiply the multipliers

A desk originally costs $60. Its price increases 25%, then decreases 20%. What is the final price?

Convert each change to a multiplier, then multiply:

The final price is $60 — exactly the original.

Or compose first: $1.25 \times 0.80 = 1.00$, so the net multiplier is exactly 1. The two changes cancel perfectly.

This is why $+25\%$ then $-20\%$ nets to zero: the 20% comes off the already-inflated $75, removing $15, while the 25% only added $15 on top of $60. Equal dollar amounts, different percent labels — because the bases are different.

The rule: when you see back-to-back percent changes, multiply the multipliers. Never add the percents.

$+10\%$ then $-10\%$ is not $0\%$. It's $1.10 \times 0.90 = 0.99$ — a 1% net loss. The interactive below makes this visible:

For a nonzero pair of equal and opposite percentage changes within these sliders, the final amount is below the positive starting amount: $(1+r)(1-r)=1-r^2<1$. If both changes are zero, it is unchanged. The base mismatch is built into the geometry.

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Example 5 — Two-level percent: finding a count within a count

A school has 400 students. 30% belong to at least one club. Of those club members, 25% are in the debate club. How many students are in the debate club?

The trap here is reaching for 25% of 400. The problem says 25% of club members, not 25% of all students. Each percent has its own base.

Work in two steps, left to right:

Step 1: How many students are club members?

Step 2: How many of those are in debate?

The answer is 30.

If you want one calculation: compose the multipliers first, then apply to the total.

Two traps to name, because they're both in the answer choices:

• $0.25 \times 400 = 100$ — wrong base (25% of all students, not of club members)
• $(0.30 + 0.25) \times 400 = 220$ — adding percents instead of tracking bases

The discipline is simple: before you multiply any percent, write down what it's of.

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Before harder practice

A percentage can change one factor in a product. If revenue equals price times quantity, a 15% price reduction and 20% increase in sales multiply revenue by $0.85(1.20)=1.02$: a 2% increase. Compare the final factor with 1 before reporting the percentage change.

For an unknown percentage, write its multiplier as an expression. A 10% increase followed by a decrease of $100r\%$ that leaves 99% of the original gives $1.10(1-r)=0.99$. Thus $1-r=0.90$ and $r=0.10$: the decrease is 10%. For a reversed chain or subgroup count, divide the known final amount by the product of the relevant nonzero factors.

Mixed groups require a weighted total. Suppose 40% of people belong to one group, 25% of that group choose an option, and 34% of all people choose it. If fraction $q$ of the remaining people choose it, $0.40(0.25)+0.60q=0.34$. Therefore $q=0.40$, or 40% of the remaining group. The group share and the percentage within that group have different bases.