Percent Proportion

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.

Finding the Percent

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

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.

Finding the Whole

15 is 30% of what number?

Identify what you know: The part is 15 (next to "is"), the percent is 30. The whole (after "of") is missing. = part = 15, $p = 30$

Write the proportion: $\frac{15}{x} = \frac{30}{100}$ = $\frac{15}{x} = \frac{30}{100}$

Cross multiply: $30 \times x = 15 \times 100$ = $30x = 1500$

Divide by 30: $x = 1500 \div 30$ = $x = 50$

Answer: 15 is $30\%$ of 50. Check: $30\%$ of 50 is $0.3 \times 50 = 15$.

Finding the Part

Use the percent proportion to find 35% of 60.

Identify what you know: The whole is 60, the percent is 35. The part is missing. = whole = 60, $p = 35$

Write the proportion: $\frac{x}{60} = \frac{35}{100}$ = $\frac{x}{60} = \frac{35}{100}$

Cross multiply: $100 \times x = 35 \times 60$ = $100x = 2100$

Divide by 100: $x = 2100 \div 100$ = $x = 21$

Answer: $35\%$ of 60 is 21. The same setup works for every unknown.

Test Score as a Percent

Maria answered 34 of the 40 questions on a test correctly. What percent did she get right?

Identify the part and the whole: All questions are the whole: 40. Correct answers are the part: 34. = part = 34, whole = 40

Write the proportion: $\frac{34}{40} = \frac{p}{100}$ = $\frac{34}{40} = \frac{p}{100}$

Cross multiply: $40p = 34 \times 100$ = $40p = 3400$

Divide by 40: $p = 3400 \div 40$ = $p = 85$

Answer: Maria got $85\%$ of the questions right.

Survey: How Many Were Asked?

In a school survey, 45 students chose basketball as their favorite sport. That is 18% of all students who answered. How many students answered the survey?

Identify what you know: 45 students are a part of everyone. The percent is 18. The whole (all students) is missing. = part = 45, $p = 18$

Write the proportion: $\frac{45}{x} = \frac{18}{100}$ = $\frac{45}{x} = \frac{18}{100}$

Cross multiply: $18x = 45 \times 100$ = $18x = 4500$

Divide by 18 and check: $x = 4500 \div 18 = 250$; check: $0.18 \times 250 = 45$ = $x = 250$

Answer: 250 students answered the survey.

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.

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

Why: 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 $p = 30$.

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

Why: 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 $x = 50$.

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

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

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

Mistake: Answering 0.3 when the question asks for the percent

Why: 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.

Test Scores

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

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

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.

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

Shopping and Discounts

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

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

The percent proportion is $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$

The whole usually follows the word "of", the part usually goes with "is"

The number with the % sign always goes over 100

Write a letter for the missing number, cross multiply, then divide

The same setup finds the part, the whole or the percent

Check your answer: the part should be that percent of the whole

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

A: 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.

Q: Can the part be bigger than the whole?

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

Q: Why does cross multiplication work?

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

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

A: 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.

Exit Presentation

Percent Proportion

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Decimals & Percents / Percentages

Percent Proportion

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

Intermediate25 min

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