Math — Percentages

Understanding Percentages

Master one of the most practical math skills: calculating discounts, test scores, tips, tax, and more — step by step.

Learning Objectives

  • Define what a percentage means and why it is based on 100
  • Convert freely between fractions, decimals, and percentages
  • Calculate a percentage of any given number
  • Solve percentage increase and decrease problems
  • Apply percentage thinking to sales tax, discounts, and data

Prerequisites

You should be comfortable multiplying and dividing whole numbers, and have a basic idea of what fractions and decimals look like. If those feel shaky, visit Khan Academy Arithmetic first.

The Lesson

Step 1 — What does "percent" mean?

The word percent comes from the Latin per centum, meaning per hundred. So 40% simply means 40 out of every 100. Think of it as slicing any whole into exactly 100 equal pieces — the percentage tells you how many pieces you have.

Visualisation: Imagine a grid of 100 squares. If 35 are shaded, that is 35%. The whole grid always represents 100%.

Step 2 — Converting between fractions, decimals, and percentages

These three notations describe the same number in different forms. Fluency with conversions is essential.

Fraction → Percent: Divide numerator by denominator, multiply by 100.
3/4 → 3 ÷ 4 = 0.75 → 0.75 × 100 = 75%

Decimal → Percent: Multiply by 100 (shift decimal two places right).
0.06 → 0.06 × 100 = 6%

Percent → Decimal: Divide by 100 (shift decimal two places left).
45% → 45 ÷ 100 = 0.45

Step 3 — Finding a percentage of a number

The most common percentage task: "What is 20% of 150?" Convert the percentage to a decimal, then multiply.

Example A: 20% of 150
20% = 0.20  →  0.20 × 150 = 30

Example B: 8.5% of 400 (sales tax)
8.5% = 0.085  →  0.085 × 400 = 34 — so total price = $400 + $34 = $434

Step 4 — Finding what percentage one number is of another

Formula: (part ÷ whole) × 100

Example: You scored 46 out of 58 on a test. What percentage?
46 ÷ 58 = 0.7931...  →  × 100 = 79.3%

Step 5 — Percentage increase and decrease

Used for price changes, population growth, salary raises, etc.

Percentage increase formula:
((new − old) ÷ old) × 100

Example: A jacket was $60, now $75. Increase?
(75 − 60) ÷ 60 = 15 ÷ 60 = 0.25 → 0.25 × 100 = 25% increase

Percentage decrease example: Item dropped from $80 to $68.
(80 − 68) ÷ 80 = 12 ÷ 80 = 0.15 → 15% decrease

Step 6 — Finding the original value (reverse percentage)

If you know the final value and the percentage change, work backwards.

Example: After a 20% discount, a TV costs $320. What was the original price?
$320 = 80% of original (because 100% − 20% = 80%)
Original = 320 ÷ 0.80 = $400

Practice Problems

  1. Q: What is 35% of 260?
    Solution: 0.35 × 260 = 91
  2. Q: A class of 30 students — 18 passed. What percentage passed?
    Solution: 18 ÷ 30 = 0.60 → 60%
  3. Q: A shirt costs $45 and is on sale for 15% off. What is the sale price?
    Solution: 0.15 × 45 = $6.75 discount → $45 − $6.75 = $38.25
  4. Q: A town grew from 12,000 to 15,600 people. What is the percentage increase?
    Solution: (15600 − 12000) ÷ 12000 = 3600 ÷ 12000 = 0.30 → 30%
  5. Q: After a 12% raise, an employee earns $3,360/month. What was the original salary?
    Solution: 3360 ÷ 1.12 = $3,000/month

Common Mistakes

Mistake 1 — Forgetting to convert to decimal first. Writing 30% × 90 as 30 × 90 = 2700 instead of 0.30 × 90 = 27. Always divide by 100 before multiplying.
Mistake 2 — Applying % increase incorrectly. A price increases 10% then decreases 10% — students assume you are back to the start. Actually: $100 × 1.10 × 0.90 = $99. Percentage changes are multiplicative, not additive.
Mistake 3 — Confusing "percent of" with "percent more than." "30% more than 80" = 80 + 0.30×80 = 104, not 0.30×80 = 24.
Mistake 4 — Mixing up part and whole. In (part ÷ whole) × 100, always put the larger "total" as the denominator, not the smaller piece.
Mistake 5 — Rounding too early. Keep at least 4 decimal places during the calculation; round only the final answer.

Further Practice Resources

Khan Academy — Solving Percent Problems (video + exercises) Math Is Fun — Percentages with interactive examples Britannica — Percentage definition and history Wikipedia — Percentage (deeper mathematical background) MIT OCW — Calculus foundation (for advanced applications)

Frequently Asked Questions

What does percent mean?

Percent means "per hundred." 40% means 40 out of every 100 parts of a whole.

How do I convert a decimal to a percent?

Multiply the decimal by 100 and add the % sign. Example: 0.75 × 100 = 75%.

How do I find a percentage of a number?

Convert the percent to a decimal, then multiply. 30% of 80: 0.30 × 80 = 24.

What is percentage increase?

Percentage increase = ((new value − old value) ÷ old value) × 100. It measures how much something grew relative to its starting value.

Why are percentages useful in real life?

They create a standard scale out of 100, making it easy to compare quantities of very different sizes — test scores, interest rates, discounts, and statistics all rely on them.

Can a percentage be over 100?

Yes! 150% means one-and-a-half times the whole. This arises frequently in growth comparisons — a value doubling is a 100% increase; tripling is 200%.