Percent Change, Explained: The Math Behind Increases, Decreases, and the 10% Trap
Percentages are everywhere — sale tags, tip screens, pay raises, stock charts. And yet the single most useful percentage skill is one almost nobody practices deliberately: percent change. Once you understand it, you can see through misleading sales, decode a raise offer in seconds, and never get tripped up by the famous "10% trap" again.
The One Formula You Need
Percent change = (new value − old value) ÷ old value × 100.
The sign tells you the direction: positive means increase, negative means decrease.
- A $30 book now costs $39 → (39 − 30) ÷ 30 × 100 = 30% increase.
- A $50 sweater marked down to $38 → (38 − 50) ÷ 50 × 100 = −24%, i.e., a 24% decrease.
That's it. Every example below is just this formula wearing a different outfit.
Worked Example 1: The Sale-Price Test
A jacket is $80, and the store offers 25% off. What's the final price?
There are two ways to do this, and both are worth knowing:
Method A — subtract the discount: 25% of $80 = $20 → $80 − $20 = $60.
Method B — multiply by what's left: 100% − 25% = 75% → $80 × 0.75 = $60.
Method B is the superpower. Whenever a problem says "decrease by X%," multiply by (1 − X). For "increase by X%," multiply by (1 + X). A 30% raise on $3,000: $3,000 × 1.10 = $3,300. A 4% pay bump on $50,000: $50,000 × 1.04 = $52,000.
Worked Example 2: The 10% Trap
Here's the classic. A $100 pair of shoes goes up 10%, then a month later goes down 10%. What does it cost now?
Most people's gut says $100. The gut is wrong.
- After the 10% increase: $100 × 1.10 = $110.
- After the 10% decrease: $110 × 0.90 = $99.
Why? Because the 10% down applies to the new, larger base ($110), so the dollar drop ($11) is bigger than the dollar gain ($10). Successive percentages don't cancel — they compound. This single idea is why "was $X, now back down" sales aren't always the deal they look like, and why investors get burned when they assume a 50% loss needs only a 50% gain to recover (it needs a 100% gain).
Worked Example 3: The Sneaky Stack — When Two Discounts Hit
An $200 item gets a 25% increase, then a 25% decrease. Final price?
If you multiply naively: 200 × 1.25 × 0.75 = $187.50. Not $200 — the trap again, costing you $12.50.
But here's a fun twist worth memorizing: an increase of 25% followed by a decrease of 20% brings you exactly back to start (1.25 × 0.80 = 1.0). Order doesn't even matter — 0.80 × 1.25 = 1.0 too. Whenever you see complementary percent pairs like this, the multiplication shortcut beats doing it step by step.
Percentage Points vs. Percent — Stop Mixing These Up
This is the one that trips up adults most. A savings account rate goes from 4% to 6%. Two things are true:
- It rose by 2 percentage points (6 − 4 = 2).
- It rose by 50 percent ((6 − 4) ÷ 4 = 50%).
Both are correct; they answer different questions. If a bank advertises "50% higher interest," that sounds dramatic — but it's just 4% → 6%. Whenever someone quotes a percentage change of a percentage, ask which one they mean. Headlines love this ambiguity; you don't have to fall for it.
Worked Example 4: Sanity-Check a "Deal"
A store lists a book at $63, marked "40% off the list price." What was the list price?
Work backward: 100% − 40% = 60% → list price × 0.60 = 63 → list = 63 ÷ 0.60 = $105. And check: (63 − 105) ÷ 105 = −40%. ✓
This backward version is where most shopping mistakes happen — people try to "add 40% to $63" ($88.20, wrong) instead of dividing by 0.60. When you want a raise, a bonus, or a discount reversed, the divisor is always (1 ± percent), not a simple addition. It's the kind of check a free percentage increase calculator does for you in seconds, step by step.
The 30-Second Summary
- Formula: (new − old) ÷ old × 100. Sign = direction.
- "X% off" → multiply by (1 − X). "X% up" → multiply by (1 + X).
- Successive percentages multiply, never cancel: 10% up then 10% down lands at $99, not $100.
- Reversing a discount? Divide by (1 − X), never add the percent back on.
- Percentage points ≠ percent change.
The only way to make this automatic is repetition. Try a free percentage increase calculator to check your work on real problems — like whether that "70% off + extra 10%" sale is really the steal it claims (it's 73% off total, since 0.30 × 0.90 = 0.27). And if you're comparing two sales side by side, run both through a percentage decrease calculator to see the true discount on each before you decide. A calculator is just training wheels; the math behind it is yours to keep.