Before you calculate, look at which amount you know: if it is the whole, you multiply; if it is a part, you divide. Two thirds of a group of 18 students is 12, but if 12 students are three fifths of the class, the class has 20. This guide shows how to decide between the two and how to check each result.
| What you know | What you want | Operation |
|---|---|---|
| The whole and the fraction | The part | Multiply: 2/3 × 18 = 12 |
| A part and the fraction it represents | The whole | Divide: 12 ÷ 3/5 = 20 |
| A fraction of another fraction | The final fraction | Multiply: 1/2 × 1/3 = 1/6 |
For cases with more than one fraction, the guide to fraction of the remainder problems goes into detail, and this page helps you choose the path.
Two solved problems
The first looks for a part and the second looks for the whole. Each line carries its unit so the result keeps its meaning.
Worked example: 2/3 of 18 students
A group has 18 students and two thirds go on a field trip. How many students go?
- Identify the whole. The whole is 18 students, and the fraction is the share that goes on the trip.
- Multiply. 2/3 × 18 students = 36/3 students.
- Simplify. 36 ÷ 3 = 12 students.
Result. 12 students go on the field trip.
Check. 18 − 12 = 6 students stay behind, and 6 is 1/3 of 18, so the two groups together make all three thirds.
Worked example: 12 pages that are three fifths of a chapter
You have read 12 pages, which are three fifths of the chapter. How many pages does the chapter have?
- Identify the part: 12 pages is the known part, and the whole is what is missing.
- Divide by the fraction: 12 pages ÷ 3/5 = 12 × 5/3 pages.
- Calculate: 12 × 5 = 60 and 60 ÷ 3 = 20 pages.
Result. The chapter has 20 pages.
Check. 3/5 × 20 pages = 12 pages, the number you started with. Also, 20 is greater than 12, as it must be when the part is smaller than the whole.
The bar shows why the answer is 20. Three equal parts are worth 12 pages, so each part is worth 4, and the five parts add up to 20.
When two fractions come in a row
There are two possible readings, and the wording decides which one applies. If you spend 1/3 of your money and then 1/4 of the original money, the total spent is 1/3 + 1/4 = 7/12 and 5/12 is left. If the second expense is 1/4 of what was left, then after the first expense two thirds remain and the second is calculated on that part: 2/3 × 3/4 = 1/2. Half of the money is left.
For the second reading, the clue is “the rest” or “what was left.” Each fraction refers to the previous amount, so you multiply the factors that remain. With a 24-gallon tank, 12 gallons remain after those two expenses, and 8 gal + 4 gal + 12 gal = 24 gal accounts for the whole tank.
Practice with six problems
They are mixed on purpose and do not say which operation to use. First write down which amount is known, then calculate with units.
Practice
- A recipe uses two thirds of a cup of flour for 4 servings. How much flour is needed for 6 servings?
- A break lasts 5/6 of an hour. How many minutes is that?
- Half of a third of a garden is watered. What fraction of the garden is watered?
- A board is 2 1/2 ft long and a 3/4 ft piece is cut off. How long is what remains?
- There are 30 students and 2/5 ride the bus. How many ride the bus?
- You have hiked three fifths of a 15-mile trail and one half of a 40-mile trail. Which of the two distances is greater?
Show the answers
- Going from 4 to 6 servings multiplies by 6/4 = 3/2, and 2/3 × 3/2 = 1 cup.
- 5/6 × 60 min = 50 min.
- 1/2 × 1/3 = 1/6 of the garden.
- 5/2 − 3/4 = 10/4 − 3/4 = 7/4 = 1 3/4 ft.
- 30 × 2/5 = 12 students.
- 15 × 3/5 = 9 miles and 40 × 1/2 = 20 miles, so the second is greater.
More practice and tools
Each operation has its own collection of problems with solutions: the add and subtract word problems for parts that are combined or taken away, the multiplication word problems for the model of the first example, and the division word problems for the model of the second. If you just want a single calculation, the fraction of a number tool finds the part and the find the whole from a fraction tool recovers the whole. To train with automatic feedback, open fraction practice, and to review sign and operation slips, see the guide to fraction mistakes.
Sources
Wikipedia, article Fraction, sections “Multiplication” and “Division”: to multiply fractions you multiply the numerators and multiply the denominators, and dividing by a fraction means multiplying by its reciprocal (accessed October 17, 2026).