Adding and Subtracting Fractions

Examples with solutions:

Example 1:

Perform the subtraction and express the final answer in simplest form.

solution:

9 = 3 ²

and

21 = 3 ¹ × 7 ¹

so

LCD = 3 ² × 7 ¹ = 63

Thus

The factorization shown in brackets indicates that the numerator and denominator of this answer have no common factors, so no further simplification is possible. Thus, in simplest form

This procedure works when more than two fractions are involved. In such cases, the LCD must be formed for all fractions present. We illustrate with an example.

 

Example 2:

Perform the arithmetic and express the final answer in simplest form.

solution:

First write each of the three denominators as products of prime factors:

7 = 7 ¹

21 = 3 ¹ × 7 ¹

35 = 5 ¹ × 7 ¹

Thus, for these three fractions,

LCD = 3 ¹ × 5 ¹ × 7 ¹ = 105.

Thus,

Now,

122 = 2 × 61 (and 61 is prime)

105 = 3 × 5 × 7

Therefore, the numerator and denominator of this answer have no prime factors in common and so no further simplification is possible. So our final answer here is:

 

Converting Mixed Numbers to Pure Fractions

We now have the tools to justify the method used previously for converting mixed numbers to pure fractions. For example,

This example also illustrates how we can add or subtract whole numbers and fractions.