Computations With Fractions

Here is a cheat sheet, a basic outline of what you need to know about fractions when you are required to perform computations that involve fractions. In a nonscientific sense, the word computations refers to problems involving addition, subtraction, multiplication, and division. You should have an understanding of simplifying fractions and calculating common denominators before adding, subtracting, multiplying, and dividing fractions.

Multiplying

Once you learn that the numerator refers to the top number and the denominator refers to the bottom number of a fraction, you are on your way to being able to multiply fractions. To do so, you multiply the numerators and then multiply the denominators. You will be left with an answer that might require one additional step: simplifying.

Let's try one:

1/2 x 3/4
1 x 3 = 3 (multiply the numerators)
2 x 4 = 8 (multiply the denominators)
The answer is 3/8

Dividing

Again, you need to know that the numerator refers to the top number and the denominator to the bottom number. You also need to know that in dividing fractions, the first fraction is referred to as the dividend and the second is called the divisor. In the division of fractions, invert the divisor and then multiply it by the dividend. Put simply, turn the second fraction upside down (called the reciprocal) and then multiply the numerators and the denominators:

1/2 ÷ 1/6
1/2 x 6/1 (the result of flipping 1/6)
1 x 6 = 6 (multiply the numerators)
2 x 1 = 2 (multiply the denominators)
6/2 = 3
The answer is 3

Adding

Unlike multiplying and dividing fractions, adding and subtracting fractions sometimes requires that you calculate a like, or common, denominator. That's not necessary when you're adding fractions with the same denominator; you simply leave the denominator as it is and add the numerators:

3/4 + 10/4 = 13/4

The numerator is larger than the denominator, so you simplify by dividing and the result is a mixed number:
3 1/4

However, when adding fractions with unlike denominators, a common denominator must be found before adding the fractions.

Let's try one:

2/3 + 1/4

The lowest common denominator is 12; that's the smallest number each of the two denominators can be divided into with a whole number as a result.

3 goes into 12 4 times, so you multiply both the numerator and denominator by 4 and get 8/12. 4 goes into 12 3 times, so you multiply both the numerator and denominator by 3 and get 3/12.

8/12 + 3/12 = 11/12

Subtracting

When subtracting fractions with the same denominator, leave the denominator as it is and subtract the numerators:
9/4 - 8/4 = 1/4

When subtracting fractions without the same denominator, a common denominator must be found before subtracting the fractions:
For example:

1/2 - 1/6

The lowest common denominator is 6.

2 goes into 6 3 times, so you multiply both the numerator and denominator by 3 and get 3/6.

The denominator in the second fraction is already 6, so that doesn't need to be changed.

3/6 - 1/6 = 2/6, which can be reduced to 1/3.