Before you pick a calculator, decide what the problem asks for: a part, the total, a comparison, a recipe at a different size or a measurement in another unit. Each question points to a different tool, and the table below tells you which one. For longer cases, read fractions in everyday life.
| What the problem asks | Tool | Example |
|---|---|---|
| A part of an amount | Fraction of a number | 3/4 of 80 items = 60 items |
| The total, if you know a part | Find the whole from a fraction | 18 lb is 3/5 of 30 lb |
| What fraction one amount is of another | What fraction is it | 18 of 30 = 3/5 |
| A percentage | Percentages | 9 of 12 = 75% |
| Reducing a ratio | Simplify ratios | 6 : 9 = 2 : 3 |
| Sharing by weights | Proportional sharing | 100 in the ratio 2 : 3 = 40 and 60 |
| A recipe for more servings | Scale recipes | 3/4 cup × 3/2 = 1 1/8 cups |
| A measurement in another unit | Inches to mm | 5/8 in = 15.875 mm |
| Part of an hour | Fraction of an hour to minutes | 7/8 h = 52 min 30 s |
Read the table by the question, not by the numbers. Two problems with the same figures can ask for different things, and that decides whether you multiply, divide or convert a unit. The first three rows work with concrete amounts; the others add a ratio, a length factor or the 60 minutes in an hour.
Why the operation changes
A fraction acts as a factor when you look for the part and as a divisor when you look for the total. It is the same relationship read in two directions. If 3/4 of 80 is 60, then 60 divided by 3/4 gives back 80.
The most common mistake is to multiply every time. Before you calculate, write down which amount you know and which one you want, with its unit. If what you know is the part, you divide; if what you know is the whole, you multiply.
Worked example: recovering the total
At a bakery, 18 lb of flour is three fifths of everything that was bought. How many pounds were bought in all?
- Identify what you know. 18 lb is the part, and the fraction 3/5 says what share of the total it is.
- Divide the part by the fraction. 18 ÷ 3/5 = 18 × 5/3 = 30.
- Write the unit. 30 lb.
30 lb were bought in all.
Check. 3/5 of 30 is 18, the part you started with. If you had multiplied 18 × 3/5 you would have gotten 10.8 lb, which is less than the part and cannot be the total.
The find the whole from a fraction tool does this calculation, and fraction of a number goes the opposite way.
Recipes, measurements and time
Scaling a recipe means multiplying every amount by the same ratio of new servings to original servings. To go from 4 to 6 servings, that ratio is 6/4 = 3/2, and 3/4 cup becomes 9/8, which is 1 1/8 cups. The unit does not change, but a cup is not an exact measure: it is roughly 240 ml, so 3/4 cup is about 180 ml, and it is better to use grams if you have them. Not everything scales the same way, because baking time and yeast do not follow the same proportion as the flour.
A measurement in another unit does need its own factor. One inch is exactly 25.4 mm, so 5/8 inch is 15.875 mm. Time works in a similar way, because an hour has 60 minutes and 7/8 of an hour is 52.5 minutes, or 52 min 30 s. A fraction of an hour is a duration, not a time of day.
Ratios and sharing
A ratio compares two amounts without saying what the total is. The ratio 2 : 3 splits 100 into 40 and 60 because 2 + 3 = 5 parts and each part is worth 20. If the units cannot be divided, such as people or items, the split leaves a remainder and you have to decide who gets it. To find out whether two ratios form an equality, use proportions or the rule of three.
What to learn next
Keep the unit next to the result and check whether the problem wants an exact value or a rounded one, as in rounding fractions. If you first need to add or divide fractions, do it in the fraction calculator. To go from a ratio to a fraction, use ratio to fraction and, for several values, average fractions.
- Fraction word problems with worked solutions: problems about parts, totals, recipes and sharing for you to practice.
- Proportional sharing with fractions: follows each share and what happens to the remainder.
- Fractions in everyday life: worked cases of recipes, inches and hours in more detail.
- Ratio, proportion and fraction: explains when two ratios can be set equal.
- Probability with fractions: favorable outcomes over possible outcomes.