How to Add and Subtract Fractions Step by Step
Adding and subtracting fractions requires matching denominators before combining numerators. This step-by-step guide explains how to find common denominators, convert fractions, and simplify results so you can handle any fraction problem.
Check If the Denominators Match
The denominator is the bottom number in a fraction, and it tells you what size pieces you are working with. Before you can add or subtract fractions, both fractions must use the same denominator. If they already match, you can proceed directly to combining the numerators without any additional steps.
For example, one-half plus two-halves is straightforward because both fractions already use halves. You simply add the numerators: 1 plus 2 equals 3, and the denominator stays 2, giving you 3/2. The same logic applies to subtraction: 5/8 minus 3/8 equals 2/8, which you can then simplify.
When the denominators do not match, you cannot add or subtract yet. Adding 1/3 and 1/4 directly would be like adding one-third of a pizza and one-quarter of a different pizza and calling it two-sevenths; the piece sizes are different, so you must convert to equal-sized pieces first.
Find the Least Common Denominator
The least common denominator, or LCD, is the smallest number that both denominators divide into evenly. For simple fractions, you can often find it by listing multiples of each denominator until you spot one they share. Multiples of 3 are 3, 6, 9, 12. Multiples of 4 are 4, 8, 12, 16. The first number both lists share is 12, so the LCD of 3 and 4 is 12.
Another method is to multiply the two denominators together. For 1/3 and 1/4, multiplying 3 by 4 gives 12. This always produces a common denominator, though not always the least one. If you use a larger common denominator, you will simply need to simplify more at the end. Both approaches give correct results.
For larger denominators, you can find the LCD using the greatest common divisor. Divide the product of both denominators by their GCD. For denominators of 8 and 12, the GCD is 4. Multiplying 8 by 12 gives 96, and dividing by 4 gives an LCD of 24.
Convert Both Fractions
To convert each fraction to the common denominator, multiply both the numerator and denominator by whatever number makes the denominator equal the LCD. For 1/3 with an LCD of 12, you multiply both parts by 4: the fraction becomes 4/12. For 1/4 with an LCD of 12, you multiply both parts by 3: the fraction becomes 3/12.
Multiplying both the numerator and denominator by the same number does not change the value of the fraction; it only changes how it is expressed. This is because you are multiplying by a form of 1 (since any number divided by itself equals 1). The fractions 1/3 and 4/12 represent the exact same quantity.
After converting, double-check that both fractions now have the same denominator before proceeding. If you are working with three or more fractions, each one must be converted to use the same LCD before any addition or subtraction takes place.
Add or Subtract the Numerators
Once all fractions share the same denominator, add or subtract the numerators while keeping the denominator unchanged. For 4/12 plus 3/12, add the numerators 4 and 3 to get 7, keeping the denominator 12. The result is 7/12.
For subtraction, the same process applies but you subtract instead of add. If the problem is 5/12 minus 3/12, subtract 3 from 5 to get 2, and the result is 2/12. Be careful with subtraction involving mixed numbers or negative results, as you may need to borrow from the whole number portion of a mixed number.
The denominator stays the same throughout the addition or subtraction process. A common mistake is adding the denominators as well as the numerators, which gives a wrong answer. The denominator only changes when you are finding a common denominator in the conversion step, not when you are combining the fractions.
Simplify the Result
After combining the fractions, check whether the result can be simplified. A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, find the greatest common divisor of the numerator and denominator and divide both by it.
For example, the result 2/12 can be simplified because 2 and 12 both share the factor 2. Dividing both by 2 gives 1/6. The fractions 2/12 and 1/6 are equivalent, but 1/6 is the simplified form. If the numerator is larger than the denominator, you have an improper fraction that can be converted to a mixed number.
To convert an improper fraction like 7/4 to a mixed number, divide the numerator by the denominator. Seven divided by 4 is 1 with a remainder of 3, so 7/4 equals 1 and 3/4. Whether to leave a result as an improper fraction or convert to a mixed number often depends on the context of the problem.