How to Calculate Percentages
What is a Percentage?
A percentage is a number expressed as a fraction of 100. The word comes from the Latin 'per centum', meaning 'by the hundred'. Percentages are used to express how large or small one quantity is relative to another quantity.
Enter plain signed decimals or scientific notation such as 1e-8, without commas or a % sign. Use at most 100 coefficient digits and an exponent from −100 to 100. Press Enter in either active field to calculate. Arithmetic uses exact decimal fractions; displayed results use up to 12 significant digits, with ≈ or “approximately” for rounded values. For money, round the final amount to the required currency unit.
Here P is the number before the percent sign: use P = 15 for 15%. Reference values must be nonzero. Percentage change requires a positive original value; a negative new value is allowed and can produce a decrease greater than 100%.
Common Percentage Formulas
Finding the percentage:
Example: 20 is what percent of 80?
(20 ÷ 80) × 100 = 25, so 25%
Finding the value:
Example: What is 15% of 80?
(15 ÷ 100) × 80 = 12
Finding the original value:
Example: 12 is 15% of what number?
12 ÷ (15 ÷ 100) = 80
Percentage increase/decrease:
Example: From 80 to 100:
((100 - 80) ÷ 80) × 100 = 25, so a 25% increase
Confession: I once tipped 42 % at a diner because I read the receipt wrong. The waiter loved me; my wallet cried. Five minutes with the right percentage shortcut would’ve saved lunch money for a week. Let’s make sure that never happens to you.
1-Minute Cheat Sheet 🚀
| Need it now? | Do this |
|---|---|
| Formula | P = (Part ÷ Whole) × 100 |
| Find the Part | (P × Whole) ÷ 100 |
| Find the Whole | Part ÷ (P ÷ 100) |
Sticky-note these three for a quick reference. In this table, P is the numeric percentage (for example, 20 for 20%); the whole must be nonzero when dividing by it, and P must be nonzero when finding the whole.
Quick Story: The Concert Ticket Mix-Up 🎸
Two friends bought tickets—one at $75, one with a 20 % student discount. My buddy asked, “How much did I save?” Brain freeze!
Part = ?
Whole = $75
% = 20 → (20 ÷ 100) × 75 = $15 savings.
Now he owes me popcorn for doing the math on the train.
Step-by-Step Walk-Through
Identify the Whole
– Original price, total points, or full population.Grab the Part or %
– Sale discount, test score, battery level—whatever chunk you’re eyeing.Plug & Chug
Part ÷ Whole → multiply by 100. Done. Calculator welcome, sanity preserved.
Example: Exam Score
You scored 42 out of 50.
P = (42 ÷ 50) × 100 = 84, so the score is 84%. The letter grade depends on your school’s grading scale.
Visual Hack 🎨 (Use in your notes)
Draw a pizza: whole pie = 100 %. Divide it into four equal slices and shade one: 1/4 = 25%. Suddenly “25 %” looks as tasty as mozzarella.
Everyday Scenarios & Mini-Calculations
| Scenario | Math in 10 sec | Result |
|---|---|---|
| Black-Friday 30 % off $120 | 0.30 × 120 | $36 off → Pay $84 |
| Battery drops from 100 % to 37 % | 100 – 37 | 63 percentage points lower; 63% of the original full charge used |
| Restaurant bill $48 + 18 % tip | 0.18 × 48 | $8.64 tip |
Pro tip: Round the bill to nearest $10 for mental math, then adjust. This gives an estimate; calculate from the actual bill and round the final tip to cents when needed.
Watch-Out Zone ⚠️
Percent of vs. Percent increase
– 20% of $50 is a $10 part. Increasing $50 by 20% adds that $10 and gives a new total of $60.Stacking Discounts
– 50 % + 20 % off isn’t 70 %—it’s 50% first, then 20% of the new subtotal: $100 → $50 → $40, a 60% overall discount.Over-100 % Results
– Scores can exceed 100 % with bonus points. Perfectly legal in math land.
Pull Quote
“Percentages are just fractions wearing fancy cologne.” —My stats professor, mid-lecture yawn
Mental-Math Hacks You’ll Actually Use
Move the Decimal – 10 % off? Shift it left one place. 20 %? Double the 10 %.
Build-Up Trick – Want 15 % tip? Find 10 % (easy), plus half of that. Boom.
Cross-Multiplication for Unknowns – a/b = x/100 → x = (a × 100)/b. Saves phone battery.
Nerd Corner (Optional but Geek-Approved)
Percent Error = |Actual − Expected| ÷ |Expected| × 100, with Expected ≠ 0. State which value is the reference.
Compound Percent Change = [(1 + p₁)(1 + p₂)… − 1] × 100, where each p is a decimal rate (20% = 0.20; a 20% decrease uses −0.20). For example, +20% then −20% gives (1.2 × 0.8 − 1) × 100 = −4, so the overall change is −4%.
Index Numbers (CPI, etc.) compare levels with a reference base; percentage changes between index levels use (new − old) ÷ old × 100. Some index series are chain-linked.
Feel-Good Challenge
Grab your latest grocery receipt. Pick any item, calculate how much you’d save with a 12 % coupon, and drop the number in the comments. Funniest product wins a virtual high-five emoji.
Remember: Next time you stand at a cash register, calculator panic = obsolete.
Percentage points versus percent change
A rate rising from 20% to 25% increases by 5 percentage points. Relative to the original 20%, it rises by (25 − 20) ÷ 20 × 100 = 25, so the relative increase is 25%. Always state the baseline.