Guide

How to Add and Subtract Fractions Step by Step

A fraction names a part of a whole, or a ratio of two integers. The top number is the numerator. The bottom number is the denominator. You cannot add or subtract those two roles as if they were independent. 1/2 + 1/3 is not 2/5. The pieces have different sizes until you rewrite them with a common denominator. An adding fractions calculator does that rewriting with exact integers, then simplifies.

What fractions represent

The fraction a/b means a copies of a unit that is 1/b of a whole, provided b is not zero. 3/4 means three fourths. −1/2 means the opposite of one half. 0/5 is zero. 5/4 is more than a whole; it is still a fraction, just an improper one. Arithmetic with fractions is arithmetic with rational numbers: each value is a ratio of two integers.

Numerators and denominators

The denominator sets the size of each piece. The numerator counts how many pieces you have. Changing the denominator without changing the numerator changes the value. Changing both by the same nonzero factor does not. That is why 2/4 equals 1/2. Addition has to respect piece size, which is why unlike denominators need extra work.

Like denominators

When two fractions already share a denominator, add or subtract the numerators and keep that denominator. 2/5 + 3/5 = 5/5 = 1. 4/7 − 1/7 = 3/7. Then simplify if a common factor remains. This is the case people remember from early lessons, and it is still the last arithmetic step after unlike fractions have been rewritten.

Unlike denominators

1/2 and 1/3 do not share a denominator. Halves and thirds are different units. You need equivalent fractions with a common denominator before the numerators can meet. Any common multiple of 2 and 3 would work, including 6, 12, or 18. The least common denominator is the least common multiple of the two denominators, here 6. Using 6 keeps the numbers smaller than using 2 × 3 only when that product is not already the least common multiple. For 1/4 + 1/6 the product is 24, but the least common multiple is 12.

Least common denominator

The least common multiple of two positive integers can be computed as |a b| / gcd(a, b). gcd(4, 6) = 2, so lcm(4, 6) = 24 / 2 = 12. The adding fractions calculator uses that LCD in its steps rather than always multiplying denominators. That matches classroom work and still yields an exact rational after simplification.

Equivalent fractions

To rewrite a/b with denominator m, multiply numerator and denominator by m/b, which is an integer when m is a multiple of b. 1/4 becomes 3/12 because 12/4 = 3. 1/6 becomes 2/12. The new fractions name the same amounts as the old ones.

Worked addition example

Add 1/2 and 1/3. LCD = 6. 1/2 = 3/6. 1/3 = 2/6. 3 + 2 = 5. Result 5/6, already simplest. Try it on the fraction calculator, which defaults to this pair. A second example: 1/4 + 1/6. LCD 12. 3/12 + 2/12 = 5/12. Open the adding fractions calculator with those inputs.

Subtracting fractions

Subtraction uses the same rewrite, then subtracts numerators. 3/4 − 1/6: LCD 12. 9/12 − 2/12 = 7/12. The subtracting fractions calculator keeps the sign. 1/3 − 2/3 = −1/3. 1/2 − 1/2 = 0, stored as 0/1. Swapping the two fractions changes the answer because subtraction is not commutative: 1/6 − 3/4 = −7/12.

Worked subtraction example

Subtract 1/3 from 3/4, which is 3/4 − 1/3. LCD 12. 9/12 − 4/12 = 5/12. That is example 3 on the homepage list. It is a different problem from 3/4 − 1/6.

Simplifying results

After adding or subtracting, divide numerator and denominator by their GCD. 2/5 + 3/5 = 5/5, GCD 5, result 1. Leaving 5/5 is equivalent but not simplest. The site always reduces the stored rational. See simplifying with the GCD.

Negative answers

A smaller fraction minus a larger one is negative. Do not flip the sign to make the homework look friendlier. The calculator also accepts negative inputs: −1/2 + 1/3 = −1/6. A negative denominator is normalized: 1/−2 is stored as −1/2 before arithmetic.

Common mistakes

Adding numerators and denominators separately produces 1/2 + 1/3 = 2/5, which is smaller than both addends in a way that cannot be right for positive fractions. Forgetting a common denominator is the same error in slower motion. The homepage mistakes list walks through that 2/5 trap. Another slip is using 24 as the denominator for 1/4 + 1/6 and then forgetting to simplify 10/24 to 5/12. The LCD path lands on 5/12 directly after adding 3 and 2.

Continue with multiplication and division or the formula reference.