Percentage: How to Calculate Shares, Change & Discounts

Fraction

Percent

Fraction

Percent

1/2

50%

1/8

12.5%

1/3

33.33%

1/10

10%

1/4

25%

1/12

8.33%

1/5

20%

1/16

6.25%

1/6

16.67%

1/20

Core answer: the three formulas — share = part ÷ whole; growth = (new − old) ÷ old; sale price = original × (1 − discount). Classic traps: after a 20% drop you need a 25% RISE to recover (not 20%); going from 3% to 5% is +2 percentage points but +66.7%.

The three basic formulas

Scenario	Formula	Example
Share	part ÷ whole × 100%	9 of 45 students nearsighted: 20%
Find the part	whole × percentage	15% of 300 = 45
Find the whole	part ÷ percentage	60 is 30% → whole = 200

Growth and decline

Growth rate = (new − old) ÷ old × 100%

Example 1: salary 8,000 → 9,200: (9200−8000)÷8000 = +15%

Example 2: home price ¥2M → ¥1.7M: (1.7−2)÷2 = −15%

Chained changes: up 10% then down 10% = 100×1.1×0.9 = 99 — a net 1% loss, not back to start.

Discounts and markups

"n-zhe" (Chinese discount notation) = ×n/10: 8 zhe = ×0.8, 7.5 zhe = ×0.75
x% off = ×(1−x%): 15% off = ×0.85
x% markup = ×(1+x%): +20% = ×1.2

Example: ¥399 at 8 zhe: 399×0.8 = ¥319.2.

Stacked discounts: 8 zhe then 9 zhe = 0.8×0.9 = 7.2 zhe, not 7.

Percentage points vs percent

Percentage points: the plain difference between two percentages
Percent change: the relative change rate

Example: a rate rises from 3% to 3.5%:

Up 0.5 percentage points
Change = (3.5−3)÷3 = +16.7%

"Rates rose 16.7%" and "rates rose 0.5 points" describe the SAME event with very different vibes — always check which one a headline uses.

Common mistakes

Down 20%, then up 20% = back to even: wrong. 100→80→96. Recovery needs 80→100 = +25%. Rule: after an x% drop you need x÷(100−x) to recover.
"Grew 300%" = became 3×: growing 300% means 4× (100+300); "became 3×" is only +200%.
Averaging growth rates arithmetically: +50% one year, −30% the next is NOT +10%/year — the geometric mean is √(1.5×0.7)−1 = +2.5%.
Comparing absolute gains across different bases: A: 100→150 (+50), B: 20→50 (+30) — B grew 150% vs A's 50%. Compare rates, not absolutes.

Frequently Asked Questions

How do I calculate X% of a number?

Multiply the number by X ÷ 100. 15% of 240 = 240 × 0.15 = 36. Mental shortcut: 10% first (24), then half of that for 5% (12), add them (36). Works for any percentage by decomposing into 10%s and 5%s.

How do I find what percent A is of B?

A ÷ B × 100. Example: 45 of 300 students passed → 45 ÷ 300 = 0.15 = 15%. Remember: the number after "of" is always the divisor (the base).

Why is +50% then −50% not back to the start?

Because the base changes. 100 + 50% = 150; then 150 − 50% = 75. You lost 25% of the original. The rule: equal up-and-down percentages always net a loss of (x%)². To undo +50% you need −33.3%, since 150 × (1 − 1/3) = 100.

What is the difference between percent and percentage points?

Percentage points (pp) measure absolute change in a rate; percent measures relative change. Unemployment moving 4% → 5% is +1pp but +25% relatively. Media often confuses the two — a "+2% mortgage rate increase" from 3% could mean 5% (pp) or 3.06% (relative). Always check which is meant.

How do I calculate a discount quickly?

20% off means pay 80%: multiply by 0.8. For stacked discounts, multiply the factors: 20% off then 10% off = 0.8 × 0.9 = 0.72 → 28% total off, NOT 30%. A "50% off then extra 50% off" deal = 75% off, not free.

How do I reverse a percentage to find the original?

Divide by the multiplier. A price rose 25% to 100 → original = 100 ÷ 1.25 = 80. After a 20% drop to 80 → original = 80 ÷ 0.8 = 100. The classic error is applying the same percentage in reverse (100 − 25% = 75 ≠ 80).

Key Takeaways