Add fractions
Rewrite two fractions over a common denominator, then add their numerators.
Add fractions is one of 3 fractions formulas in the arithmetic section of this library, and it is used at middle school level.
Why add fractions works
You can only add pieces that are the same size. Thirds and quarters are different sizes, so first you cut both into twelfths, which is what multiplying the two bottoms does. Once every piece is a twelfth, you just count them: 8 of them plus 3 of them makes 11 twelfths.
What each symbol means
$a,c$ are numerators and $b,d$ are nonzero denominators.
Add fractions: when it holds
$b\ne0$ and $d\ne0$; simplify the result when numerator and denominator share a factor.
When it stops applying
The formula stops working if a bottom number is 0, because a fraction with 0 underneath has no value. It also does not apply to mixed numbers as written, so change 2 1/3 into 7/3 before you use it.
Add fractions: a worked example
$\frac{2}{3}+\frac{1}{4}=\frac{8+3}{12}=\frac{11}{12}$.
The mistake to avoid
What people do: Students add across the top and across the bottom, turning 2/3 + 1/4 into 3/7.
Why it goes wrong: That answer is about 0.43, which is smaller than the 2/3 you started with. Adding something positive can never make a number shrink.
Do this instead: Give both fractions the same bottom number first: 2/3 becomes 8/12 and 1/4 becomes 3/12, so the total is 11/12, or about 0.92. Estimating first also helps, since 2/3 plus a bit more than a quarter should land just under 1.
Add fractions: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,c$ are numerators and $b,d$ are nonzero denominators.
- Check the conditions before substituting. $b\ne0$ and $d\ne0$; simplify the result when numerator and denominator share a factor.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Fractions
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where add fractions comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- A Practical Guide to Fractions — the lesson behind this formula: compare, combine, multiply, and divide rational quantities.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about add fractions
Do I have to use the smallest common denominator?
No. Multiplying the two bottoms always works and always gives a correct answer, just not always the tidiest one. Adding 1/6 and 1/4 this way gives 10/24, which reduces to the same 5/12 you would get from the smaller denominator.
How do I add a whole number to a fraction?
Write the whole number over 1 first. So 3 becomes 3/1, and then the same rule applies: 3/1 + 1/4 turns into 12/4 + 1/4, which is 13/4.
Why is my answer not in lowest terms?
Because multiplying the two bottoms often makes them bigger than needed. Look for a number that divides both the top and the bottom and cancel it, the way 10/24 becomes 5/12 after you divide both by 2.
Does subtraction use the same rule?
Yes, with one change: the top becomes ad − bc instead of ad + bc. Everything about the common bottom number stays exactly the same, so learning one gives you both.