Formula · Fractions
Fraction Rules: All Four Operations in One Place
Four operations, four rules, and one idea underneath all of them: you can only combine parts that are the same size.
The four rules, side by side
These are the fraction rules in full. Each one is followed by the reason it works, because a rule you can reconstruct is worth more than one you have memorised.
| Operation | Rule | Example |
|---|---|---|
| Add | match denominators, add numerators | 2/3 + 1/4 = 8/12 + 3/12 = 11/12 |
| Subtract | match denominators, subtract numerators | 7/8 − 1/4 = 7/8 − 2/8 = 5/8 |
| Multiply | numerators together, denominators together | 3/4 × 2/5 = 6/20 = 3/10 |
| Divide | flip the second fraction, then multiply | 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 |
Adding and subtracting: why the denominators must match
You can only combine counts of parts that are the same size. Thirds and quarters are different sizes, so 2/3 + 1/4 means nothing until both are rewritten over a common denominator — here twelfths.
- 2⁄3 + 1⁄4different sized parts
- 8⁄12 + 3⁄12rewrite both as twelfths using equivalent fractions
- 11⁄12now they are the same size, so add the counts
The denominator does not change in the final step, because the size of each part has not changed. Only how many you have.
Multiplying: no common denominator needed
Multiplying fractions asks for a portion of a portion, not for a combined count, so nothing has to match. Multiply straight across.
Half of a third is a sixth: 1/2 × 1/3 = 1/6. Picture cutting a third of a cake in half and the answer is visible — the piece is one of six equal parts of the whole cake.
Dividing: why you flip the second fraction
Dividing by a number is the same as multiplying by its reciprocal — the fraction turned upside down. The rule is not arbitrary; it follows from what division asks.
'6 ÷ 1/2' asks how many halves fit into 6. The answer is 12, which is 6 × 2. Flipping 1/2 to 2/1 and multiplying gives exactly that, and it also explains the result that surprises people: dividing by a fraction smaller than 1 makes the answer larger.
- 2⁄3 ÷ 1⁄4how many quarters fit into two thirds?
- 2⁄3 × 4⁄1multiply by the reciprocal of the second fraction
- 8⁄3which is 2 ⅔ — more than one, as expected
Only the second fraction is flipped. Flipping the first inverts the whole question.
Simplifying to lowest terms
Divide the numerator and denominator by their greatest common factor. 8/12 and 2/3 are the same number, but marked work almost always expects the second.
If you cannot see the greatest common factor at once, divide by any shared factor and repeat. 24/36 → 12/18 → 6/9 → 2/3 reaches the same place as dividing by 12 in one go.
Working with mixed numbers
None of the four rules applies directly to a mixed number, because a mixed number has a whole part sitting outside the fraction. Convert to an improper fraction first, apply the rule, then convert back if the question was set in mixed numbers.
- 1 ½ + 2 ⅓the sum
- 3⁄2 + 7⁄3convert both to improper fractions
- 9⁄6 + 14⁄6common denominator 6
- 23⁄6 = 3 ⅚add, then convert back
For addition and subtraction this is optional but safer; for multiplying and dividing it is not optional at all.
Comparing fractions
To decide which of two fractions is larger, give them a common denominator and compare the numerators. 3/5 and 5/8 become 24/40 and 25/40, so 5/8 is larger by one fortieth.
Cross-multiplying is a faster version of the same comparison: 3 × 8 = 24 against 5 × 5 = 25. The larger product sits above the larger fraction, and it works because it is the common-denominator comparison with the shared denominator left out.
Questions about fraction rules
Do I need a common denominator to multiply fractions?
No. Only addition and subtraction need matching denominators, because only they combine counts of parts. Multiplication takes a portion of a portion and needs no matching.
Why do you flip the second fraction when dividing?
Because dividing by a number is multiplying by its reciprocal. Asking how many quarters fit into a quantity is the same as multiplying that quantity by four.
Should I simplify before or after multiplying?
Before, whenever a numerator and a denominator share a factor. Cancelling first keeps the numbers small and removes the simplification step at the end.
How do I compare two fractions quickly?
Cross-multiply: compare the first numerator times the second denominator against the second numerator times the first denominator. The larger product belongs to the larger fraction.