Percentage calculator

Question: An item costs $85. It's 20% off. What's the sale price?

Solution:

• Discount amount: 20% of $85 = 0.20 × $85 = $17
• Sale price: $85 − $17 = $68
• Or, direct: $85 × 0.80 = $68

Why it works: 20% off means you pay 80% of original price (100% − 20% = 80%).

Question: You bought stock at $50, sold at $65. What's your return percentage?

Solution:

• Gain: $65 − $50 = $15
• Return %: ($15 ÷ $50) × 100 = 30%

Common mistake: Dividing by $65 instead of $50. Always divide by the original (base) value.

Question: After a 20% price increase, an item costs $72. What was the original price?

Solution:

• After 20% increase, new price is 120% of original (100% + 20%)
• $72 = 1.20 × original
• Original = $72 ÷ 1.20 = $60

Check: $60 × 1.20 = $72 ✓

Question: A stock goes up 10%, then up another 10%. Is that a 20% total gain?

Solution:

• Starting value: $100
• After first +10%: $100 × 1.10 = $110
• After second +10%: $110 × 1.10 = $121
• Total gain: 21%, not 20%

Why: The second 10% applies to $110, not $100. Percentage changes compound.

Divide 72 by a growth rate to estimate doubling time. At 8% annual growth, money doubles in about 9 years (72 ÷ 8). At 6%, about 12 years. Works for any rate between 4-12% with reasonable accuracy.

A 200% increase means the value tripled (original + 200% of original = 3× original). A 1,000% return means you got 10× your money back. Percentages have no upper limit, though anything above a few hundred percent is usually exceptional.

50% of men vs. 50% of women means very different absolute numbers if there are more of one gender in the group. Always consider the base when comparing percentages across different groups or time periods.

[(New − Original) ÷ Original] × 100. If something went from $80 to $100: ($100 − $80) ÷ $80 × 100 = 25% increase.

[(Original − New) ÷ Original] × 100. If something went from $100 to $80: ($100 − $80) ÷ $100 × 100 = 20% decrease. Note: 25% increase and 20% decrease between the same values — the asymmetry is real.

Percent is relative; percentage point is absolute. Going from 10% to 15% is a 5 percentage point increase but a 50% increase (5 is 50% of 10). Use "percentage points" when discussing rates that are themselves percentages.

To find the original before a percentage change: if the final value is X after P% increase, the original was X ÷ (1 + P/100). After a 20% increase to $120, the original was $120 ÷ 1.20 = $100.

|Measured − True| ÷ True × 100. If you expected 50 and got 45, error is 10%. Always use the reference value in the denominator that your teacher or lab specifies.

Applying successive percent changes multiplies growth factors. +10% then +10% is 1.1×1.1 = 21% total, not 20%. Use sequential factors for sales, population growth, or investment streaks.

Decide your base: pre-tax vs post-tax changes the tip amount. Our tip calculator separates those explicitly; here, remember the base you chose before comparing to someone else's rule of thumb.

Sampling noise. A 3% margin on a 48% headline means the true population value likely sits in a band—not that the headline is wrong, but that repeated samples would wiggle. It is a different use of percent from retail math but the same arithmetic underneath.

Multiply the percentages as decimals. 20% of 50% = 0.20 × 0.50 = 0.10 = 10%. Example: If 50% of people are women and 20% of women are engineers, then 10% of people are female engineers. Convert to decimals, multiply, convert back.

One basis point (bp) = 0.01% = 1/100th of a percent. Used in finance to describe small changes in interest rates or yields. A rate going from 5.00% to 5.25% increased by 25 basis points. It avoids confusion: saying "up 0.25%" could mean 5.00% → 5.25% or 5.00% → 5.0125%.

Multiply by 100 and add the % symbol. 0.75 = 75%. 1.5 = 150%. 0.03 = 3%. Going backwards: divide by 100. This is why 'percent' literally means 'per hundred' — you're expressing the number as a fraction of 100.

'50% more' means 150% of the original (1.5×). '50% as much' means exactly 50% (0.5×). If you earned $100 and got 50% more, you now have $150. If you earned 50% as much, you have $50. This is a common source of miscommunication in contracts and comparisons.

Translate word problems into part ÷ whole × 100. Check both directions: part given whole vs whole given part and percent.

If monthly churn is 2%, retention is 98%—not 100% − 2% in compound multi-period models; use (1 − churn)^months for cohort survival over time.

Compute discounted subtotal first, then apply tax percent to the taxable amount your jurisdiction uses—some places tax fees differently.

Use percent change from current score to target to see how much a final exam must move the average—pair with our test-grade tools for fuller scenarios.

Cite engine version when publishing worked examples so readers can reproduce the rounding policy you used.