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Math

Percent Change

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Percent change shows up everywhere on the test — price markups, discounts, population growth, interest. Master the one multiplier trick and these go from scary to instant.

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Percent Change

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A percent change tells you how much something grew or shrank, measured relative to where it started. The core formula is:

percent change = (new − old) / old × 100

The number you divide by — the old (original) value — is the key. Percent change is always measured against the starting amount, not the new one.

The multiplier shortcut. Instead of finding the change and adding it back, just multiply once.

A percent increase of p% multiplies by (1 + p/100). A 15% increase → multiply by 1.15.
A percent decrease of p% multiplies by (1 − p/100). A 20% decrease → multiply by 0.80.

So if a $740 item rises 15%, the new price is 740 × 1.15 = 851. No need to compute the $111 increase separately.

Why the multiplier matters: it lets you chain changes and reverse them.

Reversing: "After a 25% increase, the price is $50. What was the original?" The original times 1.25 gives 50, so divide: 50 / 1.25 = 40. Don't subtract 25% of $50 — that's the classic trap.
Chaining: A 10% increase followed by a 10% decrease is × 1.10 × 0.90 = 0.99 — a net 1% decrease, not zero. Percent changes don't simply add.

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Percent change multipliers

Situation	Multiplier	Example (start = 100)
15% increase	× 1.15	100 → 115
20% decrease	× 0.80	100 → 80
+10% then −10%	× 1.10 × 0.90 = 0.99	100 → 99
Reverse a 25% increase	÷ 1.25	125 → 100

Turn every percent change into a single multiplier — increase, decrease, chain, or reverse.

Simple interest is a special percent application. The formula is:

Interest = P × r × t

where P is the principal (starting money), r is the rate as a decimal (4% = 0.04), and t is time in years. After finding interest, add it to P for the total balance. Example: $2000 at 4% for 4 years earns 2000 × 0.04 × 4 = 320 in interest.

Common conversions: to turn a percent into a decimal, divide by 100 (move the decimal two places left): 7% = 0.07, 150% = 1.50. To turn a decimal into a percent, multiply by 100.

The whole topic boils down to: identify the original value, decide increase vs. decrease, and use the right multiplier.

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Quick check

$2000 is invested at a simple interest rate of 4% per year. How much interest is earned after 4 years?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

A jacket originally priced at $80 is marked down by 25%. What is the sale price, in dollars?

Example 2

After a 20% increase, a membership now costs $72. What was the original price, in dollars?

Example 3

A town's population was 8,500 and is now 9,180. What was the percent increase?

Common pitfalls

Dividing by the new value instead of the original

Percent change uses the starting amount as the denominator. Dividing by the new value gives a close-but-wrong answer that's often a listed trap choice.

Reversing a change by subtracting

To undo a 20% increase, you divide by 1.20, not subtract 20% from the new price. Subtracting 20% of the new number overshoots because the new number is bigger than the original.

Assuming +x% then −x% cancels out

A 10% increase then a 10% decrease is × 1.10 × 0.90 = 0.99, a net 1% LOSS. Chained percent changes multiply; they don't add to zero.

Forgetting to convert percent to decimal in interest problems

In I = P × r × t, the rate r must be a decimal. Use 0.04 for 4%, not 4 — otherwise your interest is 100× too big.

Key takeaways

Percent change = (new − old) / old × 100 — always divide by the original.
Increase by p%: multiply by (1 + p/100). Decrease by p%: multiply by (1 − p/100).
Reverse a percent change by DIVIDING by the multiplier, never subtracting.
Chained percent changes multiply together; they don't simply add.
Simple interest: I = P × r × t, with r as a decimal.

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Where this lesson fits

Percent Change

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