Adding, subtracting, multiplying, and dividing fractions follow four different rules, and which one applies depends on what the problem asks. With 3/10 and 1/5 you get 1/2, 1/10, 3/50, and 3/2, one per operation. Every word problem hides one of the four, and its wording usually says which. The table links each one to the tool that calculates it and to a checked example; further down, a flour problem uses all four.
| The problem says | Operation | Tool | Example |
|---|---|---|---|
| “in all,” “put together,” “add” | Addition | Add fractions | 3/8 + 1/6 = 13/24 |
| “how much is left,” “difference” | Subtraction | Subtract fractions | 7/10 − 1/4 = 9/20 |
| “of,” “times,” “half of” | Multiplication | Multiply fractions | 3/4 × 2/9 = 1/6 |
| “how many fit,” “split into parts of” | Division | Divide fractions | 3/5 ÷ 1/10 = 6 |
| A mixed number in the data | Whichever the text calls for | Mixed numbers | 1 1/4 × 4/5 = 1 |
| Several operations in a row | Mixed operations | Fraction expressions | 2/3 + 1/6 × 6/5 = 13/15 |
The words give a hint, but context decides: “half of 3/4” is a product even though no × sign appears. If you are unsure, check the size of the result. Adding, and multiplying by a number greater than 1, makes a quantity larger; subtracting, and multiplying by a proper fraction, makes it smaller. Dividing by a proper fraction makes it larger, which surprises people most. And the fraction calculator does any of the four with the steps in view.
A problem that uses all four operations
Worked example: a bakery's flour
A bakery has 3 1/2 lb of flour. It uses 3/4 lb for one batch of dough. Of what is left, 2/5 is used up, and the rest is divided into bags of 1/4 lb. How many full bags can be filled?
- Turn the mixed number into an improper fraction. 3 1/2 = 7/2. There is no operation yet, only a change of notation.
- Subtract what was used. 7/2 − 3/4 = 14/4 − 3/4 = 11/4 lb.
- Multiply to find what is used up. “Of what is left” asks for 2/5 of 11/4: 11/4 × 2/5 = 22/20 = 11/10 lb.
- Subtract again. 11/4 − 11/10 = 55/20 − 22/20 = 33/20 lb.
- Divide by the size of a bag. 33/20 ÷ 1/4 = 33/20 × 4 = 33/5 = 6.6.
That makes 6 full bags, with a part of another left over: 0.6 of a bag (3/20 lb).
To check, 6 bags weigh 6/4 = 3/2 lb, which is 30/20, and 33/20 − 30/20 = 3/20 lb, less than one bag.
The text set the order of the operations, not a priority rule: each calculation used the result of the previous one. When everything sits on a single line, though, the hierarchy in the next section applies.
When to simplify, compare, or sort instead of operating
Simplifying, comparing, and sorting are not operations, because none of them creates a new value. Reducing 12/18 to 2/3 keeps the value and is worth doing at the end of any calculation, with the tool to simplify fractions. Comparing 3/7 and 2/5 with cross-products gives 3 × 5 = 15 against 2 × 7 = 14, so 3/7 is larger; the tool to compare fractions settles it without decimals and, for a long list, order fractions puts them from smallest to largest. To write a quantity with other terms, equivalent fractions multiply the top and bottom by the same number, and finding the missing value completes an equality. If you need a value in between, the fraction between two fractions tool calculates it.
Order of operations with fractions
With several operations on one line, you work parentheses first, then multiplication and division, and finally addition and subtraction. With the same three fractions the result changes depending on the parentheses: 2/3 + 1/6 × 6/5 = 2/3 + 1/5 = 13/15, but (2/3 + 1/6) × 6/5 = 5/6 × 6/5 = 1. The fraction expressions calculator applies this priority and shows each step, and complex fractions cover the case where the numerator or denominator is a full expression.
Other mistakes from the same family, such as adding denominators, are explained in why denominators are not added and in the list of common fraction mistakes. For addition and subtraction with unlike denominators, the common denominator comes from the LCM; the guide on when to use the GCD or LCM clarifies which of the two you need.
Practice each operation
After reading the rule, it helps to work several problems in a row with the answers at hand. There are printable worksheets for adding and subtracting, multiplying, and dividing. The four operations mixed together force you to decide before you calculate, which is exactly the skill in the table above. With the worksheet generator you can build your own sheet, and the full index is printable fraction worksheets. For problems with a solution and a check, see the worked fraction exercises.
What to learn next
If the way the number is written changes, fraction conversions go from a fraction to a decimal, a percent, or a mixed number. To understand why the common denominator and the shared divisor exist, divisibility and denominators explain the basics.
As an outside reference, the English Wikipedia article Fraction (mathematics) states that to multiply fractions you multiply the numerators and multiply the denominators, and that to divide by a fraction you multiply by its reciprocal (accessed October 17, 2026).