Finding the Part

What number is 35% of 80?

Translate the words: "What number" is $x$, "is" is $=$, "of" is $\times$ = $x = 35\% \times 80$

Write the rate as a decimal: $35\% = 35 \div 100 = 0.35$ = $x = 0.35 \times 80$

Multiply: $0.35 \times 80 = 28$ = $x = 28$

Answer: 28 is 35% of 80.

Finding the Whole

30 is 25% of what number?

Translate the words: "30 is" gives $30 =$, "25% of what number" gives $0.25 \times x$ = $30 = 0.25x$

Divide both sides by the rate: $x = 30 \div 0.25$ = $x = 120$

Check: $0.25 \times 120 = 30$ = Correct

Answer: 30 is 25% of 120.

Finding the Percent

18 is what percent of 45?

Translate, using $r$ for the rate: "18 is" gives $18 =$, "what percent of 45" gives $r \times 45$ = $18 = 45r$

Divide both sides by 45: $r = 18 \div 45$ = $r = 0.4$

Change the decimal into a percent: $0.4 \times 100 = 40$ = $40\%$

Answer: 18 is 40% of 45.

A Percent Above 100%

42 is 150% of what number?

Write the rate as a decimal: $150\% = 150 \div 100 = 1.5$ = $1.5$

Write the equation: "42 is 150% of what number" = $42 = 1.5x$

Divide by the rate: $x = 42 \div 1.5$ = $x = 28$

Check that it makes sense: More than 100% means the part is bigger than the whole: $42 > 28$ = Makes sense

Answer: 42 is 150% of 28.

A Percent Below 1%, Compared With a Proportion

0.4% of what number is 2?

Write the rate as a decimal: $0.4\% = 0.4 \div 100 = 0.004$ (not $0.4$ and not $0.04$) = $0.004$

Write and solve the equation: $0.004x = 2$, so $x = 2 \div 0.004$ = $x = 500$

Solve again with the proportion method: $\frac{2}{x} = \frac{0.4}{100}$, so $0.4x = 200$ = $x = 500$

Compare the two methods: Both give the same whole. The equation uses one multiplication, the proportion uses cross products. = Same answer

Answer: 0.4% of 500 is 2.

Mistake: Using the percent number instead of the decimal: writing $30 = 25x$ and getting $x = 1.2$

Why: The rate in the equation must be a decimal. 25% means 25 per hundred, which is $0.25$, not $25$.

Correct: Divide the percent by 100 first: $30 = 0.25x$, so $x = 120$.

Mistake: Mixing up the part and the whole: for "30 is 25% of what number?" calculating $0.25 \times 30 = 7.5$

Why: The number after "of" is the whole. Here the whole is the unknown, and 30 is the part, so 30 goes on the left of the equal sign.

Correct: Translate word by word: $30 = 0.25 \times x$, so $x = 120$.

Mistake: Forgetting to multiply by 100 when finding the rate: answering "0.4%" for "18 is what percent of 45?"

Why: Dividing part by whole gives the rate as a decimal ($0.4$). A percent is that decimal times 100.

Correct: $18 \div 45 = 0.4$, and $0.4 \times 100 = 40$, so the answer is $40\%$.

Mistake: Writing a small percent with the wrong decimal: $0.4\% = 0.04$ or $0.4\% = 0.4$

Why: Every percent is divided by 100, also small ones. Move the decimal point two places to the left: $0.4 \to 0.004$.

Correct: $0.4\% = 0.004$, so 0.4% of 500 is $0.004 \times 500 = 2$.

Paying a Deposit

Stores often ask for a deposit that is a percent of the full price. The deposit is the part, and the full price is the whole you may need to find.

A 60 dollar deposit that is 20% of the price means $60 = 0.2x$, so the price is $60 \div 0.2 = 300$ dollars.

Test Scores and Free Throws

Teachers and coaches turn "how many out of how many" into a percent. Here the rate is the unknown.

A basketball player makes 34 of 40 free throws: $34 = r \times 40$, $r = 0.85$, so she makes $85\%$ of her shots.

Savings Interest Below 1%

Savings accounts often pay less than 1% interest per year. The interest is the part, the money in the account is the whole.

At 0.5% interest, 2400 dollars earns $0.005 \times 2400 = 12$ dollars in one year.

Q: Should I use the percent equation or the percent proportion?

A: Both always give the same answer. The equation $\text{part} = \text{rate} \times \text{whole}$ is shorter when you can turn the percent into a decimal easily. The proportion $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ (lesson "Percent Proportion") keeps the percent as a whole number and can feel safer with tricky decimals. Pick the one you find clearer.

Q: Can the part be bigger than the whole?

A: Yes, when the percent is above 100%. For example, 150% of 28 is 42, which is bigger than 28. In growth, prices and populations this happens often.

Q: How do I know which number is the whole?

A: Look for the word "of": the whole usually comes right after it. In "18 is what percent of 45", the whole is 45. The number next to "is" is usually the part.

Q: Why do I multiply by 100 at the end when finding a percent?

A: Part divided by whole gives the rate as a decimal, for example $0.4$. A percent means "per hundred", so $0.4 = \frac{40}{100} = 40\%$.