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.
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.
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.
A jacket originally priced at $80 is marked down by 25%. What is the sale price, in dollars?
After a 20% increase, a membership now costs $72. What was the original price, in dollars?
A town's population was 8,500 and is now 9,180. What was the percent increase?
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.
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.
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.
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.
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.