Multiply
Use the percent and whole to find an amount contained within the whole.
Part = Percent × Whole
Free Math Guide
Percent Word Problems Explained
Learn how to identify the part, whole, and percent, solve percent increase and decrease problems, reverse a percent change, and explain what each answer means.
Most percent word problems are built from three quantities: a part, a whole, and a percent. The method depends on which quantity is unknown. Problems involving discounts, markups, tips, tax, growth, or decreases add one more decision: whether the question wants the amount of change, the final amount, or the rate of change.
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Part = Percent × Whole
The percent must be written as a decimal before it is used in the equation. For example, 25% becomes 0.25 and 7.5% becomes 0.075.
Main Percent Equation
Part = Percent as a Decimal × Whole
Divide by the Percent
Use a known part and its percent to recover the original total.
Whole = Part ÷ Percent
Compare Part to Whole
Divide the part by the whole, then convert the decimal result to a percent.
Percent = Part ÷ Whole
Translating “Is” and “Of”
In a statement such as “18 is 30% of 60,” the number before “is” is the part, the number before the percent sign is the percent, and the number after “of” is the whole: 18 = 0.30 × 60.
Identify Which Quantity Is Unknown
Before calculating, label the information as the part, whole, or percent. The unknown quantity determines which version of the relationship to use.
| Question Form | Unknown | Operation | Example |
|---|---|---|---|
| What is 30% of 80? | Part | Multiply percent × whole | 0.30 × 80 = 24 |
| 24 is 30% of what number? | Whole | Divide part ÷ percent | 24 ÷ 0.30 = 80 |
| 24 is what percent of 80? | Percent | Divide part ÷ whole | 24 ÷ 80 = 0.30 = 30% |
The Whole Is the Reference Amount
The whole is not automatically the larger number in every situation. It is the amount that represents 100% and serves as the reference for the comparison.
How to Find the Part
When the percent and whole are known, change the percent to a decimal and multiply. The result is the amount that represents that percent of the whole.
Part = Percent as a Decimal × Whole
Identify the whole
Find the original amount or total that represents 100%.
Convert the percent
Move the decimal point two places left: 18% becomes 0.18.
Multiply
Multiply the decimal percent by the whole and attach the requested unit.
Discount Amount Versus Sale Price
A discount amount is the part removed from the original price. The sale price is what remains after subtracting the discount.
Sale Price = Original Price − Discount Amount
How to Find the Whole
If a known amount represents a stated percent of a total, divide that known part by the percent written as a decimal.
Whole = Part ÷ Percent as a Decimal
Quick Reasonableness Check
If the stated percent is less than 100%, the whole should be greater than the part. For example, if $24 is 30% of a price, the original price must be greater than $24.
How to Find the Percent
Divide the part by the whole. The quotient is a decimal, so multiply by 100 or move the decimal point two places right to write the result as a percent.
Percent = Part ÷ Whole × 100%
The Division Order Matters
In “18 is what percent of 60?” divide 18 by 60. The part goes in the numerator and the whole goes in the denominator: 18 ÷ 60 = 0.30 = 30%.
Percent Increase and Percent Decrease
Percent change compares the amount of change with the original amount. The original amount is always the denominator, even when the new amount is larger.
Percent Change = |New − Original| ÷ Original × 100%
Subtract
Find the positive difference between the new amount and original amount.
Divide
Divide the amount of change by the original reference amount.
Write a Percent
Multiply the decimal quotient by 100 and state increase or decrease.
Percent Change Is Directional
A change from 80 to 100 is a 25% increase because 20 ÷ 80 = 0.25. A change from 100 to 80 is a 20% decrease because 20 ÷ 100 = 0.20. Reversing the direction changes the original amount and therefore changes the percentage.
Reverse-Percent Problems
A reverse-percent problem gives an amount after an increase, discount, tax, or withholding and asks for the original amount. The given result represents more or less than 100% of the original.
| Situation | Result Represents | Original Amount |
|---|---|---|
| 20% discount | 80% of the original | Sale price ÷ 0.80 |
| 15% increase | 115% of the original | New amount ÷ 1.15 |
| 8% tax added | 108% of the original | Total price ÷ 1.08 |
| 12% withheld | 88% of the original | Remaining amount ÷ 0.88 |
Do Not Divide by the Discount Rate
If a $68 sale price follows a 15% discount, $68 is 85% of the original price—not 15%. Divide by 0.85 to recover the original.
Successive Percent Changes
When changes happen one after another, each new percent applies to the amount produced by the previous step. Use a multiplier for each change and apply them in order.
Increase r%: multiply by (1 + r ÷ 100)
|
Decrease r%: multiply by (1 − r ÷ 100)
Why the Percentages Do Not Simply Cancel
Suppose $100 increases by 20% and then decreases by 20%. The increase produces $120. The decrease is then 20% of $120, which is $24.
$100 × 1.20 × 0.80 = $96
Equal percent increases and decreases do not cancel because they use different reference amounts.
Worked Percent Word Problems
Restaurant Tip
A restaurant bill is $52.50. Find an 18% tip.
- The bill is the whole.
- Convert 18% to 0.18.
- Multiply 52.50 × 0.18 = 9.45.
The tip is $9.45.
Original Price
A $24 discount is 30% of an item’s original price. Find the original price.
- The $24 discount is the part.
- Convert 30% to 0.30.
- Divide 24 ÷ 0.30 = 80.
The original price was $80.
Quiz Score
A student answers 36 of 48 questions correctly. Find the percent correct.
- The correct answers are the part.
- The 48 total questions are the whole.
- Compute 36 ÷ 48 × 100 = 75%.
The student answered 75% correctly.
Membership Decrease
Membership falls from 80 members to 68 members. Find the percent decrease.
- The decrease is 80 − 68 = 12.
- Divide by the original amount: 12 ÷ 80 = 0.15.
- Convert 0.15 to 15%.
Membership decreased by 15%.
Price Before a Discount
After a 15% discount, an item costs $68. Find the original price.
- After the discount, 85% of the price remains.
- Convert 85% to 0.85.
- Divide 68 ÷ 0.85 = 80.
The original price was $80.
Increase Then Discount
A $120 price increases by 20%, then decreases by 10%. Find the final price.
- Apply the increase: 120 × 1.20 = 144.
- Apply the decrease to $144: 144 × 0.90 = 129.60.
- The changes do not combine into a simple 10% increase.
The final price is $129.60.
Common Percent Word Problem Mistakes
Using the Percent as a Whole Number
Use 0.25—not 25—when multiplying by 25%. Divide the percent by 100 before using it in the equation.
Choosing the Wrong Operation
Multiply to find a part. Divide by the decimal percent to find the whole. Divide the part by the whole to find the percent.
Using the New Amount as the Percent-Change Base
Percent increase or decrease compares the change with the original amount, so the original belongs in the denominator.
Confusing the Change With the Final Amount
A $20 discount and an $80 sale price answer different questions. Read whether the problem asks how much changed or what remains.
Using the Discount Rate in a Reverse Problem
After a 20% discount, the sale price represents 80% of the original. Divide by 0.80, not 0.20.
Adding Successive Percent Changes
Each change uses a new reference amount. Apply the percent multipliers one at a time in the order given.
Rounding Too Early
Keep the unrounded calculator value during intermediate steps and round money to the nearest cent or percentages as directed at the end.
How to Solve Percent Word Problems
1. Identify the question
Determine whether the problem asks for the part, whole, percent, amount of change, or final amount.
2. Label the quantities
Identify the part, the 100% reference amount, and the percent or rate of change.
3. Convert the percent
Divide by 100 before multiplying or dividing in the percent equation.
4. Choose the relationship
Multiply for the part, divide for the whole, or compare the part with the whole to find the percent.
5. Calculate carefully
For multiple changes, apply each multiplier in order and avoid rounding intermediate results unnecessarily.
6. Interpret and check
Attach the correct unit and confirm that the answer represents what the question actually requested.
A complete answer states both the number and its meaning, such as “The discount is $24” or “Membership decreased by 15%.”
Practice Percent Word Problems
Practice finding the part, whole, and percent, then move into percent change, reverse percentages, and multi-step problems with randomized questions, immediate feedback, hints, and explanations.
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