Teacher Guide: Percent Equation

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

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Percentages. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Translate percent sentences into equations, with "is" as = and "of" as ×
Use part = rate × whole with the rate written as a decimal
Solve for the whole and for the rate, and write the rate as a percent
Work with percents greater than 100% and smaller than 1%
Compare the percent equation with the percent proportion method

Prerequisites

• Introduction to Percentages: changing percents into decimals
• Finding Percent of a Number: multiplying a whole by a percent
• Solving one-step equations such as $0.25x = 30$
• Percent Proportion: $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ (helpful for the comparison)

Discussion Starters

1. In "30 is 25% of what number?", which number is the part and which is missing? How do you know?
2. Can 150% of a number be smaller than the number? Why or why not?
3. Why is 0.4% not the same as 0.4? What would go wrong in a calculation?
4. You solved the same problem with an equation and with a proportion. Which felt easier, and why?
5. Where might you know the part and the percent but not the whole in real life?

Common Misconceptions

The bigger number is always the whole

Remediation: Show "42 is 150% of what number?": the part 42 is bigger than the whole 28. Teach students to find the whole by the word "of", not by size.

The rate in the equation can stay as the percent number (25 instead of 0.25)

Remediation: Solve $30 = 25x$ together and get $x = 1.2$ . Ask whether 30 can be 25% of 1.2. Then solve $30 = 0.25x$ and check with $0.25 \times 120 = 30$ .

A percent below 1% is the same as a decimal below 1

Remediation: Write $40\%$ , $4\%$ , $0.4\%$ under each other and divide each by 100: $0.4$ , $0.04$ , $0.004$ . Link to money: 0.4% of 1000 dollars is only 4 dollars.

Differentiation Ideas

For Struggling Students:

• Give a template "___ is ___% of ___" and let students fill in two numbers and mark the unknown
• Start with friendly rates (10%, 25%, 50%) before rates like 35% or 0.4%
• Let students check every answer by plugging it back into part = rate × whole

For On-Level Students:

• Mix problems that ask for the part, the whole or the percent in one set
• Include word problems about deposits, scores and interest
• Solve two problems with both the equation and the proportion and compare

For Advanced Students:

• Work with rates such as 0.25% and 260% and explain what they mean
• Write their own percent word problem for each missing quantity
• Find the whole when the given part is the price after an increase, for example 72 is 120% of last year's price

Standards Alignment

7.RP.A.3 (CCSS.MATH.CONTENT.7.RP.A.3)
Use proportional relationships to solve multistep ratio and percent problems
6.RP.A.3c (CCSS.MATH.CONTENT.6.RP.A.3.C)
Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent
7.EE.B.4a (CCSS.MATH.CONTENT.7.EE.B.4.A)
Solve word problems leading to equations of the form px + q = r and p(x + q) = r

Lesson Resources

visualStep-by-Step Equation Solver
Students solve percent equations such as $0.25x = 30$ one step at a time
visualPercent Tape Diagram
A tape split into 25% and 75% shows how a part of 30 belongs to a whole of 120
activityTranslate the Sentence
Cards with percent sentences; students circle "is" and "of" and write the matching equation
worksheetFind the Missing Piece
A table of part, rate and whole with one value missing in each row, including rates above 100% and below 1%

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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

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 $0.004 \times 500 = 2$ .

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

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.

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\%

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.

Teacher Guide: Percent Equation

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

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