Adding and Subtracting Fractions

When we want to add or subtract numbers that are written as fractions, we distinguish between two different cases: one where the fractions have the same denominator and one where they have different denominators.

I. The denominator is the same

In figure 1, we see that $\frac{3}{7}$ (read three-sevenths) of the circle is colored red and $\frac{2}{7}$ is green.
Without specifying the colors, we can say that $\frac{5}{7}$ of the circle is colored.


This is written $\frac{3}{7}+\frac{2}{7}=\frac{5}{7}$.


Rule:
To add two numbers written as fractions with the same denominator:
keep the common denominator;
add the numerators.


To subtract two numbers written as fractions with the same denominator:
keep the common denominator;
subtract the numerators.


In other words, using letters (a, b, and d representing signed numbers; d ≠ 0):  $\frac{a}{d}+\frac{b}{d}=\frac{a+b}{d}$ and $\frac{a}{d}-\frac{b}{d}=\frac{a-b}{d}$.


Examples:

II. The denominators are different

A. Establishing that two fractions are equal

Rule: The value of a quotient is not changed if we multiply its numerator and its denominator by the same number other than zero.
In other words, using letters (a, c, and k representing numbers; c  ≠ 0 and k ≠  0):  $\frac{a}{c}=\frac{ka}{kc}$


Examples:

B. Adding or subtracting

Rule: To add (or subtract) numbers that do not have the same denominator, first replace them with quotients that have the same denominator (and represent the same numbers); then apply the rule from Section I.


Examples:

Specific cases:


If the denominators are powers of 10 (1, 10, 100 …), these numbers can be written in decimal form before adding or subtracting them.

If the sum includes numbers written in decimal form, we can convert them into fractions with the denominators being powers of 10.