Adding and subtracting fractions with different denominators
Quick answerWhen two fractions share the same denominator, add or subtract the numerators directly. When they don't, find a common denominator first — usually the least common multiple (LCM) of the two denominators — expand each fraction to it, then combine the numerators. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
The reason you can't just add numerators across different denominators is that the "size" of one part depends on how many parts the whole is split into. A half (1/2) and a third (1/3) aren't measured in the same units, so before combining them you have to re-express both as parts of the same whole — the common denominator.
The most reliable common denominator is the least common multiple (LCM) of the two denominators, since it keeps the numbers as small as possible. For 1/4 and 1/6, the LCM of 4 and 6 is 12, so 1/4 becomes 3/12 and 1/6 becomes 2/12, giving 3/12 + 2/12 = 5/12. Subtraction works the same way — find the common denominator, then subtract the numerators instead of adding them.
Worked example (illustrative): say a recipe calls for 3/4 cup of flour for the dough and 1/3 cup for the topping. To find the total flour needed, convert both to twelfths: 3/4 = 9/12 and 1/3 = 4/12, so the total is 13/12 cups, or 1 1/12 cups. This is just an example scenario — check your own recipe for the actual amounts.
Multiplying and dividing fractions
Quick answerMultiplication needs no common denominator — multiply the numerators together and the denominators together: 2/3 × 4/5 = 8/15. Division keeps the first fraction, flips the second into its reciprocal, and multiplies: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12, which simplifies to 5/6.
Multiplying fractions is more straightforward than adding them, precisely because you don't need a shared denominator — you're scaling one fraction by another, not combining like-sized parts. Division is the reverse operation: dividing by a fraction is the same as multiplying by its reciprocal (flip the numerator and denominator), which is why "keep, flip, multiply" is the phrase many people remember from school.
Common fraction operations and their results
| Expression | Result | Decimal |
|---|
| 1/2 + 1/4 | 3/4 | 0.75 |
| 3/4 − 1/3 | 5/12 | 0.41(6) |
| 2/3 × 3/5 | 2/5 | 0.4 |
| 2/3 ÷ 1/6 | 4 | 4 |
Simplifying a fraction with the GCD
Quick answerFind the greatest common divisor (GCD) of the numerator and denominator, then divide both by it. For 8/12, GCD(8,12) = 4, so 8/12 simplifies to 2/3. A fraction is fully simplified once the numerator and denominator share no common factor other than 1.
Improper fraction vs. mixed number: an improper fraction has a numerator equal to or greater than its denominator, like 7/4. Dividing 7 by 4 gives a quotient of 1 and a remainder of 3, so the mixed-number form is 1 3/4. To go the other direction, multiply the whole number by the denominator and add the numerator: 1 3/4 → (1 × 4) + 3 = 7, over the same denominator, giving 7/4.
Whenever fractions are added, subtracted, multiplied, or divided, the result should usually be reported in its simplified (lowest-terms) form. Skipping this step is one of the most common places students lose points on homework, and it's also why a calculator that automatically reduces every result — using exact integer math rather than rounding — saves time and avoids careless errors.
Converting a fraction to a decimal or percent
Quick answerDivide the numerator by the denominator to get the decimal (3/4 = 0.75), then multiply by 100 for the percent (75%). If the division never terminates, the repeating digits are shown in parentheses — for example, 1/3 = 0.(3).
Some fractions convert to clean, terminating decimals (1/4 = 0.25, 3/8 = 0.375) while others repeat forever (1/3 = 0.333..., 1/7 = 0.142857142857...). Writing the repeating block in parentheses, like 0.(3), is a compact way to express the exact value without rounding it off — useful when precision matters, such as double-checking a measurement or a grade calculation. This also works in reverse: any repeating decimal can be converted back into an exact fraction.
Illustrative example: if a survey found that, say, 5 out of 8 respondents preferred option A, that's the fraction 5/8. Converting it gives 5 ÷ 8 = 0.625, or 62.5%. This is a made-up example to show the conversion method, not a real statistic.
Frequently asked questions
How do you add fractions that have different denominators?
Find the least common multiple (LCM) of the two denominators, expand each fraction to that common denominator, then add the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Do you need a common denominator to multiply fractions?
No. Multiplication just multiplies the numerators together and the denominators together, with no need to match denominators first — for example 2/3 × 4/5 = 8/15. A common denominator is only required for addition and subtraction.
How do you know if a fraction is fully simplified?
A fraction is fully simplified when its numerator and denominator share no common factor other than 1 — in other words, their greatest common divisor (GCD) is 1. For 8/12, the GCD is 4, so dividing both by 4 gives the simplified form 2/3.
How do you divide one fraction by another?
Keep the first fraction as is, flip the second fraction to its reciprocal, and multiply the two. For example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12, which simplifies to 5/6.
How do you convert a fraction result into a decimal or percent?
Divide the numerator by the denominator to get the decimal (3/4 = 0.75), then multiply that decimal by 100 to get the percent (75%). If the division never terminates, the repeating digits are written in parentheses, such as 1/3 = 0.(3).