Subtract and simplify the following two fractions

Explanation

We begin by expressing each fraction as an equivalent fraction that has the LCD for its denominator. Then we use the rule for subtracting fractions with like denominators.   To add (or subtract) fractions, the fractions must have like denominators.

To find the LCD, we find the prime factorization of both denominators and use each prime factor the greatest number of times it appears in any one factorization:

28 = 2 • 2 • 7

21 = 3 • 7

2 appears twice in the factorization of 28.

3 appears once in the factorization of 21.

7 appears once in the factorizations of 28 and 21.

We will compare the prime factorizations of 28, 21, and the prime factorization of the LCD, 84, to determine what forms of 1 to use to build equivalent fractions for \(\frac{13}{28}\) and \(\frac{1}{21}\) with a denominator of 84.

LCD =  2 • 2 • 3 • 7

Cover the prime factorization of 28 .  Since 3 is left uncovered, use \(\frac33\) to build \(\frac{13}{28}\)

LCD =  2 • 2 • 3 • 7

Cover the prime factorization of 21.  Since 2 • 2 = 4 is left covered, use \(\frac44\) to build \(\frac{1}{21}\)

To build \(\frac{13}{28}\) and \(\frac{1}{21}\) so that their denominators are 84, multiply each by a form of 1.

$\frac{13}{28} - \frac{1}{21} = \frac{13}{28} \cdot \frac33 - \frac{1}{21} \cdot \frac44$

Multiply the numerators.  Multiply the denominators.  The denominators are now the same.

$= \frac{39}{84} - \frac{4}{84}$

Subtract the numerators and write the difference over the common denominator.

$= \frac{39 - 4}{84}$

This fraction is not in simplest form.

$= \frac{35}{84}$

To simplify, factor 35 and 84.  Then remove the common factor of 7 from the numerator and denominator.

$\require{cancel} = \frac{5 \cdot \cancel{7}^1}{2 \cdot 2 \cdot 3 \cdot \cancel{7}_1}$

Multiply the remaining factors in the numerator: 5 • 1 = 5.  Multiply the remaining factors in the denominator: 2 • 2 • 3 • 1 = 12.