Learning goals
- Explain a percent as parts per hundred and convert between %, decimals, and fractions
- Find an amount, a percent, or a whole when you know the other two values
- Solve discount, tip, and “what percent of” problems with clear steps
What is a percentage?
A percentage expresses a ratio as parts per hundred. Saying 25% is the same as saying 25/100, or 0.25 of a whole.
The “whole” is whatever counts as 100% in your problem — a bill total, a test’s max score, last year’s sales, or a recipe batch. Naming the whole first prevents most percent errors.
You will usually solve one of three questions: What is X% of Y? What percent is A of B? If X% of something is A, what is the something?
Core percent formula
amount = (percent ÷ 100) × whole
Convert the percent to a decimal (divide by 100), then multiply by the whole. Rearrange to find percent = (amount ÷ whole) × 100, or whole = amount ÷ (percent ÷ 100).
Percent change
percent change = ((new − old) ÷ old) × 100
Use the original value as the whole. A positive result is an increase; a negative result is a decrease. “Increased by 20%” means multiply by 1.20, not add 20.
Example 1 — Find X% of a number
Given
What is 15% of 240?
Steps
- Identify the whole: 240.
- Convert 15% to a decimal: 15 ÷ 100 = 0.15.
- Multiply: 0.15 × 240 = 36.
Answer
15% of 240 is 36.
Tip: Mental check: 10% of 240 is 24, so 5% is 12, and 15% is 24 + 12 = 36.
Example 2 — Find what percent one number is of another
Given
42 is what percent of 70?
Steps
- Identify the part (amount) = 42 and the whole = 70.
- Divide: 42 ÷ 70 = 0.6.
- Convert to a percent: 0.6 × 100 = 60%.
Answer
42 is 60% of 70.
Example 3 — Sale price after a discount
Given
A jacket costs $80.00. It is 25% off. What is the sale price?
Steps
- Discount amount = 0.25 × 80 = 20.
- Sale price = original − discount = 80 − 20 = 60.
- Shortcut: pay 75% of the price → 0.75 × 80 = 60.
Answer
The sale price is $60.00.
Tip: “X% off” means you pay (100 − X)% of the original.
Mixing up part and whole
Common mistake
Treating the smaller number as the whole just because it appears first in the sentence.
Better approach
Ask “100% of what?” The whole is the reference total — often the original price, max score, or starting value.
Adding the percent instead of multiplying
Common mistake
“Increase 50 by 20%” → 50 + 20 = 70.
Better approach
Increase by 20% → 50 × 1.20 = 60. You add 20% of 50 (which is 10), not the number 20.
Check your understanding
1.What is 12% of 450?
Answer: 0.12 × 450 = 54.
2.18 is what percent of 60?
Answer: (18 ÷ 60) × 100 = 30%.
3.A $200.00 item is 10% off. What do you pay?
Answer: 0.90 × 200 = $180.00.
4.Sales grew from 500 to 575. What is the percent increase?
Answer: ((575 − 500) ÷ 500) × 100 = 15%.
Key takeaways
- A percent is a rate out of 100: amount = (percent ÷ 100) × whole.
- To find what percent one number is of another: percent = (part ÷ whole) × 100.
- Always identify which value is the whole (100%) before rearranging.
- “Increase by X%” multiplies by (1 + X/100); “X% off” multiplies by (1 − X/100).