Can the part be bigger than the whole?

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.

How do I know which number is the whole?

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.

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

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

Teacher Resources

Percent Equation

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

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.

Solve any percent problem with one equation: part = rate × whole. Find the part, the whole or the percent, even above 100% and below 1%.

Intermediate25 minLesson

Printable worksheet

Definition

The percent equation connects three quantities with one multiplication:

\text{part} = \text{rate} \times \text{whole}

The whole is the total you take a percent of (100%).
The rate is the percent, written as a decimal: $25\% = 0.25$ , $150\% = 1.5$ , $0.4\% = 0.004$ .
The part is the amount that the percent describes.

Translating words into the equation:

"is" becomes $=$
"of" becomes $\times$
"what number" or "what percent" becomes the unknown $x$

So "30 is 25% of what number?" becomes

30 = 0.25x

If you know two of the three quantities, you can always find the third:

part: multiply, $\text{part} = \text{rate} \times \text{whole}$
whole: divide, $\text{whole} = \text{part} \div \text{rate}$
rate: divide, $\text{rate} = \text{part} \div \text{whole}$ , then multiply by 100 to get the percent

Try it now

Which equation matches the question "What number is 40% of 65?"

Worked Examples

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.

Common Mistakes

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

Why it's wrong: 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 $x = 120$ .

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

Why it's wrong: 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 $x = 120$ .

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

Why it's wrong: 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 $40\%$ .

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

Why it's wrong: 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 $0.004 \times 500 = 2$ .

Interactive Visual

Equation Solver

Solve for $x$ :

0.25x = 30

0.25x = 30

Multiply both sides by the LCD,

4

Now there are no fractions left

Tips:

• Whatever you do to one side, do to the other
• Undo operations in reverse order (PEMDAS backwards)
• Check your answer by substituting back

Follow along as we solve the equation step by step.

Ratio Tape Diagram

25% (the part)

= 30

75% (the rest)

= 90

100% = the whole

= 4 (120)

If total is:

25% (the part) value:30

75% (the rest) value:90

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which equation matches the question "What number is 40% of 65?"

Why It Matters

In the lesson "Finding Percent of a Number" you always knew the whole and the percent. Real problems often hide a different unknown:

Shopping: you paid a 45 dollar deposit, which is 15% of the price. What is the full price? (the whole is missing)
Grades: you got 34 of 40 points. What percent is that? (the rate is missing)
Savings: an account pays 0.8% interest per year. How much must you save to earn 36 dollars? (a rate below 1%)
Growth: a city now has 260% of its old population. (a rate above 100%)

The percent equation handles all of these with the same short line,

\text{part} = \text{rate} \times \text{whole}

, and the equation solving you already know.

Real World Applications

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.

Example:

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

1Try It Yourself

Mia pays a deposit of 45 dollars for a bike. The deposit is 15% of the full price.

What is the full price of the bike in dollars?

Step 1: Write the mathematical expression

Divide the part by the rate:

Test Scores and Free Throws

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

Example:

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

2Try It Yourself

On a test worth 80 points, Leo scores 68 points.

What percent of the points did Leo get? Give the number without the % sign.

Step 1: Write the mathematical expression

Divide the part by the whole and multiply by 100:

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.

Example:

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

3Try It Yourself

A savings account pays 0.8% interest per year. After one year Sara earned 36 dollars of interest.

How many dollars were in Sara's account?

Step 1: Write the mathematical expression

Divide the interest by the rate as a decimal:

Key Takeaways

1The percent equation is $\text{part} = \text{rate} \times \text{whole}$ , with the rate written as a decimal
2Translate word by word: "is" becomes $=$ , "of" becomes $\times$ , "what" becomes the unknown
3To find the whole, divide: $\text{whole} = \text{part} \div \text{rate}$
4To find the percent, divide part by whole and multiply by 100
5Above 100% the rate is greater than 1 ( $150\% = 1.5$ ) and the part is bigger than the whole; below 1% the rate starts with $0.00$ ( $0.4\% = 0.004$ )
6The equation and the proportion $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ always give the same answer

Frequently Asked Questions

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.

Glossary

Percent equation: The equation $\text{part} = \text{rate} \times \text{whole}$ that links a percent with two amounts.
Whole: The total amount you take a percent of; it stands for 100%. Usually it comes after the word "of".
Part: The amount that a percent of the whole is equal to.
Rate: The percent written as a decimal, for example $35\% = 0.35$ .
Unknown: The missing quantity in the equation, written with a letter such as $x$ or $r$ .

Formula Card

Percent equation

\text{part} = \text{rate} \times \text{whole}

Use it to find the part: multiply the rate (as a decimal) by the whole.

Finding the whole

\text{whole} = \text{part} \div \text{rate}

Divide the part by the rate as a decimal, for example $30 \div 0.25 = 120$.

Finding the percent

\text{rate} = \text{part} \div \text{whole}

Divide the part by the whole, then multiply by 100 to get the percent.

Percent Equation

Definition

Worked Examples

Common Mistakes

Interactive Visual

Equation Solver

Tips:

Ratio Tape Diagram

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Paying a Deposit

Test Scores and Free Throws

Savings Interest Below 1%

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Percent equation

Finding the whole

Finding the percent

More in This Topic

Introduction to Percentages

Finding Percent of a Number

Percent Increase and Decrease

Percent Proportion