Addition and Subtraction of Fractions: Overview

If a candy bar is broken into two halves to share, it is not accurate to say, “I want the bigger half.” If it is truly broken into two halves, then both parts are equal. One half of the candy bar combined with the other half would equal the whole candy bar. The fraction means one of two equal parts of the whole. The 1 is the numerator, and the 2 is the denominator of the fraction.

Before computing with fractions, it's important to understand the meanings of the numerator and denominator of a fraction. Figure 1 shows a rectangle divided into four equal parts.

Figure 2 displays three of the four parts shaded. The fraction represents three of four equal parts. The 3, or numerator, tells you how many parts are shaded, while the 4, or denominator, shows how many equal parts the whole is divided into.

When adding fractions such as + , you add only the numerators, while keeping the same denominator ( + = ). A pictorial demonstration of addition of fractions with like denominators provides a much clearer explanation of why we do not add the denominators. Figure 3 shows a rectangle divided into six equal parts.

Figure 4 shows the same rectangle with two of six equal parts shaded.
Figure 5 shows three parts of the whole (six parts) shaded. When adding + , the result is , which means five of six equal parts are shaded, as shown in Figure 6.

It is important to notice that + = and is a fraction with a denominator of 6. Since you have added 2 of six equal parts to 3 of six equal parts your result is 5 of six equal parts. Similarly, in subtraction, if you were to start with 5 parts shaded, then erase the shading from 2 parts, 3 shaded parts would remain. Thus, − = .