Add the following fractions and simplify if possible

Explanation

We begin by expressing each fraction as an equivalent fraction that has the LCD as its denominator. Then we use the rule for adding fractions that have the same denominator.  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:

2 appears once in the fraction of 10

3 appears once in the fraction of 15

5 appears once in the fractions of 15 and 10

15 = 3 • 5

10 = 2 • 5

LCD = 2 • 3 • 5 = 30 

To build \(\frac{7}{15} \) and \(\frac{3}{10}\) so that their denominators are 30.  multiply each by a form of 1.

$\frac{7}{15} + \frac{3}{10} = \frac{7}{15} \cdot \frac22 + \frac{3}{10} \cdot \frac33$

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

$= \frac{14}{30} + \frac{9}{30}$

Add the numerators and write the sum over the common denominator 30.

$= \frac{14 + 9}{30}$

Since 23 and 30 have no common factors other than 1, this fraction is in simplest form

$= \frac{23}{30}$