Can the part be bigger than the whole?

Yes, when the percent is more than 100. For example, 60 is $150\%$ of 40: $\frac{60}{40} = \frac{p}{100}$ gives $p = 150$.

Why does cross multiplication work?

Two equal fractions stay equal when you multiply both sides by both denominators. For $\frac{a}{b} = \frac{c}{d}$ this gives $a \times d = b \times c$.

What is the difference between the percent proportion and the percent equation?

They are two ways to write the same fact. The proportion compares two fractions; the percent equation (next lesson) writes part = rate × whole with the percent as a decimal. Both give the same answer.

Teacher Resources

Percent Proportion

Use the proportion part/whole = percent/100 to find a missing part, whole or percent with cross multiplication.

Intermediate25 minLesson

Printable worksheet

Definition

The percent proportion says that a part compared to its whole is the same as the percent compared to 100:

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

The whole is the total amount (it is usually the number after the word of).
The part is a piece of that total (it is usually the number next to the word is).
$p$ is the percent, the number in front of the % sign.

In every percent problem, two of these three numbers are known and one is missing. Put the known numbers in the proportion, write a letter for the missing one, then cross multiply and divide.

Example: What percent of 40 is 12?

\frac{12}{40} = \frac{p}{100} \quad\Rightarrow\quad 40p = 12 \times 100 = 1200 \quad\Rightarrow\quad p = 30

So 12 is

30\%

of 40.

Try it now

Which proportion matches the question "What percent of 50 is 20?"

Worked Examples

What percent of 40 is 12?

Identify the part and the whole

The whole comes after "of": 40. The part comes with "is": 12. The percent is missing. → part = 12, whole = 40

Write the proportion

$\frac{12}{40} = \frac{p}{100}$

Cross multiply

$40 \times p = 12 \times 100$ → $40p = 1200$

Divide by 40

$p = 1200 \div 40$ → $p = 30$

Answer: 12 is $30\%$ of 40.

Common Mistakes

Putting the whole on top: writing $\frac{40}{12} = \frac{p}{100}$ for "What percent of 40 is 12?"

Why it's wrong: The fraction on the left must be part over whole. Flipping it gives $p \approx 333$ , which is impossible here because 12 is smaller than 40.

Correct: Part on top, whole on the bottom: $\frac{12}{40} = \frac{p}{100}$ , so $p = 30$ .

Writing the percent in the part spot: $\frac{30}{x} = \frac{15}{100}$ for "15 is 30% of what number?"

Why it's wrong: The number with the % sign always goes over 100. The other known number is the part or the whole.

Correct: $\frac{15}{x} = \frac{30}{100}$ , so $30x = 1500$ and $x = 50$ .

Multiplying straight across instead of crosswise: $12 \times p = 40 \times 100$

Why it's wrong: Cross multiplication pairs each numerator with the OTHER fraction's denominator.

Correct: Multiply on the diagonals: $12 \times 100 = 40 \times p$ .

Answering 0.3 when the question asks for the percent

Why it's wrong: In the proportion $p$ is already the number of percent (out of 100), not a decimal.

Correct: If $p = 30$ , the answer is $30\%$ . Write the % sign.

Interactive Visual

12 is 30% of 40

Part: 30%

= 12

Rest: 70%

= 28

Whole: 100% = 40

= 10 (40)

Part: 30% value:12

Rest: 70% value:28

Equation Solver

Solve for $x$ :

\frac{12}{40} = \frac{x}{100}

\frac{12}{40} = \frac{x}{100}

Cross-multiply:

a \cdot d = b \cdot c

Numerator of each side times the denominator of the other side. It is the same as multiplying both sides by both denominators.

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.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which proportion matches the question "What percent of 50 is 20?"

Why It Matters

You already know how to find a percent of a number. Real problems are often the other way around:

Test scores: You got 34 out of 40 points. What percent is that?
Surveys: 45 students chose basketball, which is 18% of everyone asked. How many students were asked?
Shopping: You saved 12 dollars, which was 15% off. What was the original price?
Sports: A player made 27 of 30 free throws. What is the success rate?

The percent proportion handles all three kinds of questions with one setup, so you do not need a separate rule for each. In the next lesson you will meet the other method, the percent equation (part = rate × whole), which solves the same problems in one line.

Real World Applications

Test Scores

Teachers turn points into percents so that tests with different numbers of questions can be compared.

Example:

18 points out of 24 is $\frac{18}{24} = \frac{p}{100}$ , so $p = 75$ , a score of $75\%$ .

1Try It Yourself

Liam answered 27 of the 30 questions on a quiz correctly.

What percent did Liam get right?

Step 1: Write the mathematical expression

Solve $\frac{27}{30} = \frac{p}{100}$ : calculate $27 \times 100 \div 30$

Surveys and Polls

Polls report results as percents. If you know the percent and how many people gave an answer, you can find how many people were asked.

Example:

If 30 people are $25\%$ of a group, then $\frac{30}{x} = \frac{25}{100}$ and the group has 120 people.

2Try It Yourself

In a town poll, 120 people said yes to a new park. They were 60% of all the people asked.

How many people were asked in the poll?

Step 1: Write the mathematical expression

Solve $\frac{120}{x} = \frac{60}{100}$ : calculate $120 \times 100 \div 60$

Shopping and Discounts

A receipt often shows how much you saved but not the original price. The percent proportion finds the original price.

Example:

Saving 6 dollars at $20\%$ off means $\frac{6}{x} = \frac{20}{100}$ , so the original price was 30 dollars.

3Try It Yourself

A jacket is 15% off and you save 12 dollars.

What was the original price of the jacket in dollars?

Step 1: Write the mathematical expression

Solve $\frac{12}{x} = \frac{15}{100}$ : calculate $12 \times 100 \div 15$

Key Takeaways

1The percent proportion is $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$
2The whole usually follows the word "of", the part usually goes with "is"
3The number with the % sign always goes over 100
4Write a letter for the missing number, cross multiply, then divide
5The same setup finds the part, the whole or the percent
6Check your answer: the part should be that percent of the whole

Frequently Asked Questions

Ask: which number is the total? That is the whole (often after "of"). The part is a piece of it (often next to "is"). If the percent is under 100, the part is smaller than the whole.

Glossary

Percent proportion: The equation $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ that links a part, a whole and a percent.
Part: The amount that is a portion of the whole, for example the correct answers on a test.
Whole: The total amount, which counts as $100\%$ .
Percent: A number out of 100, written with the % sign.
Cross multiplication: In $\frac{a}{b} = \frac{c}{d}$ , multiplying on the diagonals to get $a \times d = b \times c$ .

Formula Card

Percent proportion

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

Part over whole equals percent over 100.

Missing percent

p = \frac{\text{part} \times 100}{\text{whole}}

What percent of 40 is 12? $p = 1200 \div 40 = 30$

Missing whole

\text{whole} = \frac{\text{part} \times 100}{p}

15 is 30% of what? $1500 \div 30 = 50$

Missing part

\text{part} = \frac{p \times \text{whole}}{100}

35% of 60: $2100 \div 100 = 21$

Percent Proportion

Definition

Worked Examples

Common Mistakes

Interactive Visual

12 is 30% of 40

Equation Solver

Tips:

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Test Scores

Surveys and Polls

Shopping and Discounts

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Percent proportion

Missing percent

Missing whole

Missing part

More in This Topic

Introduction to Percentages

Finding Percent of a Number

Percent Increase and Decrease

Percent Equation