Fractions trip up a lot of people, but every operation follows a small set of reliable rules. Once you know when to find a common denominator and when to just multiply straight across, the rest is arithmetic. Here is the whole toolkit, with worked examples you can check by hand.
Adding and subtracting fractions
To add or subtract two fractions, they must share the same denominator. If they already match, just add (or subtract) the numerators and keep the denominator:
- 2/7 + 3/7 = 5/7
When the denominators differ, find a common denominator — the easiest reliable choice is the product of the two denominators (you can simplify afterward).
Worked example: 1/4 + 1/6
- Multiply denominators: 4 × 6 = 24.
- Convert each fraction: 1/4 = 6/24 and 1/6 = 4/24.
- Add the numerators: 6/24 + 4/24 = 10/24.
- Simplify (see below): 10/24 = 5/12.
Subtraction works identically — just subtract the numerators after matching denominators: 3/4 − 1/6 = 18/24 − 4/24 = 14/24 = 7/12.
Multiplying and dividing fractions
Multiplication is the simplest operation: multiply across the top and across the bottom.
- 2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15
Division means multiply by the reciprocal — flip the second fraction, then multiply:
- 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
A handy memory aid: "keep, change, flip." Keep the first fraction, change ÷ to ×, and flip the second.
Simplifying with the GCD
A fraction is in lowest terms when its numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common divisor (GCD).
Worked example: simplify 18/24.
- Factors of 18: 1, 2, 3, 6, 9, 18.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- The largest shared factor (GCD) is 6.
- Divide both: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4.
If you cannot spot the GCD immediately, divide by any common factor repeatedly: 18/24 → 9/12 → 3/4. You reach the same answer.
How to use it
You can check any of these by hand, but it is faster to let the Fraction Calculator do the matching, multiplying, and simplifying for you. Enter two fractions, pick the operation, and you get the reduced result plus its decimal value in one step.
Fractions also connect to other everyday math. A fraction like 3/4 is just 75%, which you can confirm with the Percentage Calculator. And when you need to summarize a set of fractional scores or measurements, the Average Calculator turns a list of values into a single mean. Together these tools cover most of the quick math you run into at school, work, or in the kitchen.
Keep the rules in mind: match denominators to add or subtract, multiply straight across, flip to divide, and reduce with the GCD. Practice a few by hand and the patterns stick fast.