You reach a problem that says a jacket costs $48 after a 20% discount, and your first instinct is to add 20% to $48. The question actually gives you the sale price, so the discount was calculated from a different amount. To solve percentage word problems, identify what counts as 100%, write the relationship in words and solve for the missing number. The same approach works for ordinary parts, price changes and questions that ask you to work backward.
Why percentage word problems are hard
The word percent tells you the scale of the rate, but it does not tell you which quantity the rate applies to. “20% of $48” and “$48 after 20% off” contain the same numbers. In the first case $48 is the base. In the second, it is the result after a reduction.
Separate three quantities when the question describes a part of a whole: the part, the base and the rate. The basic equation is part = decimal rate × base. When the question describes a change, distinguish the amount added or removed from the amount left at the end. OpenStax works through both kinds of application. OpenStax percent applications
A short table can help you decide what to write before doing any arithmetic.
| What the question asks for | First relationship to write |
|---|---|
| The amount represented by a percent | Part = decimal rate × base |
| A percent given a part and a base | Decimal rate = part ÷ base |
| The base given a part and a percent | Base = part ÷ decimal rate |
| A price after a discount | Final = original × fraction remaining |
| The original price after a known discount | Original = final ÷ fraction remaining |
The following problems are invented for practice. Dollar amounts are just convenient units for the calculations.
7 ways to solve percentage word problems
1. Identify what counts as 100%
Read the question once for the situation and again for the quantity the percentage refers to. Write that quantity beside “100%.” It may be known or unknown. This step matters because dividing or multiplying using the wrong base can still produce an answer that looks plausible.
Suppose 18 of the 60 seats in a room are empty. The question asks what percentage of seats are empty. The whole room provides the base, so 60 seats represents 100%. The empty seats provide the part, so 18 ÷ 60 = 0.30, or 30%.
If you divide 18 by the number of occupied seats instead, you answer a different question: how the empty count compares with the occupied count. Both are meaningful ratios, but only one matches the question.
2. Convert the rate and find the part
A percent means a number per hundred. For a calculation, convert 12% to 12 ÷ 100, or 0.12. Then multiply by the base when you want the corresponding part.
Imagine a workshop has 75 attendees and 12% are in the morning group. The morning group represents 0.12 × 75 = 9 attendees. The unit stays “attendees” because a rate multiplied by a count gives another count.
Use a quick estimate before you trust the result. Ten percent of 75 is 7.5, so 12% should be a little higher. If you got 90 or 900, check whether you used 1.2 or 12 instead of 0.12.
This estimate is a check, not the final method. It helps locate mistakes without requiring you to repeat the full calculation.
3. Find the whole when only the part is given
Suppose 14 members represent 35% of a club. If the question asks for the total membership, write 14 = 0.35 × total. Now divide by 0.35 to get 40 members.
Some readers see “35%” and multiply 14 by 0.35. That would find 35% of the known part. The equation prevents the mistake by showing that 14 is already the result of multiplying a different number by the rate.
Check the answer in the original wording. Thirty-five percent of 40 equals 14. A second check is that 14 is only part of the club, so the total should exceed it in this example. When the rate is above 100%, that second check changes. Use the wording, not a blanket rule about which value must be larger.
4. Separate the change from the final amount
A jacket originally costs $60 and is marked down 20%. The discount is 0.20 × $60 = $12. Subtract $12 from $60 to get a sale price of $48. The question might ask for either number, so label both.
A shortcut is to calculate the part that remains. After 20% off, 80% of the original price remains. Multiply $60 by 0.80 and you again get $48. For an increase of 20%, the corresponding factor would be 1.20 because the full original amount stays and another 20% is added.
These forms solve the same problem. The separate subtraction makes the discount amount visible. The single factor makes the link between original and final price easy to reverse. OpenStax discount examples
5. Reverse a change with the factor that produced it
Return to the opening problem. The sale price is $48 after a 20% discount. Because 80% remains, write $48 = 0.80 × original. Divide $48 by 0.80 to recover the original price of $60.
Adding 20% to $48 gives $57.60. That operation uses the sale price as the base, although the original discount used the original price. A quick forward check exposes the difference: taking 20% off $57.60 does not produce $48.
Write the forward equation even if the question asks you to work backward. Then isolate the unknown. This works for increases too. If a quantity ends at 132 after increasing 10%, divide by 1.10 and check that 120 increased by 10% gives 132.
6. Use the starting amount to find percent change
Suppose attendance rises from 40 to 50. The change is 10 attendees. Divide the change by the starting attendance: 10 ÷ 40 = 0.25, so attendance increased by 25%.
If attendance later drops from 50 back to 40, the difference is still 10, but 50 is now the base. The decrease is 10 ÷ 50 = 0.20, or 20%. Equal changes in units can therefore have unequal percentage rates.
You can check either answer by applying the rate to its own starting value. Twenty-five percent of 40 is 10. Twenty percent of 50 is also 10. This pair is useful practice if you tend to divide by whichever number is bigger.
7. Give each successive change its own base
Suppose an item costs $100, goes up 10% and then goes down 10%. The first change produces $110. The second is calculated from $110, so the reduction is $11 and the final price is $99.
The same calculation is $100 × 1.10 × 0.90 = $99. The factors combine to 0.99, which means the final amount is 99% of the original. The net change is a 1% decrease.
Before combining percentages, ask whether the second one applies to the original amount or the updated amount. The question's wording controls that decision. Mark intermediate values on paper so you can see when the base changes.
When the equation isn't enough
An equation cannot settle an ambiguous question. If the problem says “20% more” without identifying what it is more than, write the comparison you are assuming. If a price problem mentions two discounts, determine whether each discount uses the original price or follows the previous one. Ask for clarification if the wording leaves the distinction open.
When checking your own work, state the answer with its unit and meaning. “12” is incomplete if the question asks whether that number is dollars saved or dollars paid. Try a second problem that uses the same figures but moves the unknown to another position.
For regular practice, use the math study schedule to spread questions across sessions. If you want another explanation after your first attempt, the iPhone math solver guide describes tools for reviewing a setup.
How Math Solver can help
Math Solver is an iPhone app whose supplied description includes scanning maths questions and showing step-by-step explanations. For practice, attempt the setup on paper first. Then compare the first line of the explanation with your own choice of base and unknown.
Focus on the decision that caused a mismatch. If your answer differs because you used the sale price as 100%, write that correction before trying a fresh problem. Verify the arithmetic independently and follow your school's rules about using AI on assessed work. The aim is to reproduce the reasoning when the numbers change.
Frequently asked questions
How do you know whether to multiply or divide in a percent problem?
Write part equals decimal rate times base. Multiply when the rate and base are given and the part is missing. Divide the part by the known rate or base when you need the missing quantity.
How do you find the original price after a discount?
Subtract the discount rate from 100% to find the percentage remaining. Convert that to a decimal and divide the sale price by it. A sale price of $48 after 20% off gives $48 divided by 0.80, or $60.
What number goes on the bottom for percent change?
Use the starting amount as the denominator and the difference between the final and starting amounts as the numerator. If a quantity rises from 40 to 50, divide 10 by 40 to get a 25% increase.
Do a 10% increase and a 10% decrease cancel out?
They do not cancel when the decrease applies to the updated amount. A 10% increase takes 100 to 110. A 10% decrease of 110 removes 11, leaving 99. The two operations use different bases.
Conclusion
Read the question for the base before deciding whether to multiply or divide. Name the part, the rate and the whole, or label the original, change and final amount when the problem describes an increase or decrease. Write the forward relationship even when you need to reverse it. Finish by checking your answer in the original wording and reporting the number with the right unit.