Percentages
A percentage represents a part of 100, expressing fractions and ratios in a standardized form. The word "percent" comes from the Latin "per centum," meaning "by the hundred." Converting between percentages, decimals, and fractions forms the foundation for solving proportion problems across mathematics.
Why it matters
Percentages appear throughout daily life in sales tax calculations, discount pricing, test scores, and statistical reporting. A 20% tip on a $45 meal equals $9, while a 15% discount on a $120 jacket saves $18. In finance, compound interest rates like 3.5% annually determine loan payments and investment returns. Medical statistics often report success rates as percentages, such as a 95% effectiveness rate for vaccines. Grade 6 students encounter percentages in CCSS.6.RP standards when finding percent of a quantity, while Grade 7 extends to multi-step problems under CCSS.7.RP. Advanced mathematics builds on percentage concepts in probability theory, where events are expressed as percentages of favorable outcomes. Economics relies heavily on percentage changes to track inflation, unemployment rates, and market performance indicators.
How to solve percentages
Percentages — how to
- Convert the percent to a decimal by dividing by 100.
- Multiply the decimal by the base number.
- For discounts: subtract the discount from the original.
Example: 20% of 80 → 0.20 × 80 = 16.
Worked examples
10% of 20 = _______. Is it closer to 0 or 20?
Answer: 2
- Convert percent to fraction → 10% = 110 — 10% is a common fraction — memorise these.
- Apply to the base → 20 × 10100 = 2 — Take a tenth of 20.
- Verify → 2 × 100 ÷ 20 = 10% ✓ — Check backwards.
What is 30% of 60?
Answer: 18
- Convert percent to decimal → 30% = 0.3 — 30% means 30 per hundred, so divide by 100.
- Multiply by the base → 0.3 × 60 = 18 — Multiplying the decimal by the base gives the percentage amount.
- Verify → 18 ÷ 60 × 100 = 30% ✓ — Working backwards confirms the percent.
40% of what number is 80?
Answer: 200
- Write as equation → 0.4 × x = 80 — Translate to equation.
- Divide both sides by 0.4 → x = 80 ÷ 0.4 = 200 — Solve for x by dividing.
- Verify → 40% × 200 = 80 ✓ — Check the answer.
Common mistakes
- Confusing percent with decimal form leads to errors like calculating 25% of 80 as 25 × 80 = 2000 instead of 0.25 × 80 = 20.
- Adding percentages incorrectly produces results like claiming 30% + 40% = 70% of a number equals the sum of individual calculations, ignoring that percentages of different bases cannot be directly combined.
- Reversing the base and percentage in word problems creates errors such as finding 20% of 15 when the problem asks for 15% of 20, yielding 3 instead of the correct answer of 3.