Question

In Exercises 29-32, Perform the indicated operations.

(a)$\frac{3}{10}+\frac{4}{15}$

(b)$\frac{1}{4}+\frac{1}{5}$

Short answer

(a) Sum of the fraction is $1/2$

(b) Sum of the fraction is $9/20$

Step-by-step solution

Part a. Step 1. Find the least common denominator.

Factoring each denominator into prime factors give

$10=2\cdot 5$

$15=3\cdot 5$

LCD can be given as the product of all the prime factors that occur in these factorizations.

$2\cdot 3\cdot 5=30$

Part a. Step 2. Use a common denominator.

Multiply the denominators by the required number to get the LCD. And then multiple the numerator with the same number.

$\begin{array}{l}\frac{3}{10}+\frac{4}{15}\\ =\frac{3\cdot 3}{10\cdot 3}+\frac{4\cdot 2}{15\cdot 2}\\ =\frac{9}{30}+\frac{8}{30}\end{array}$

Add the numerator

$\frac{15}{30}$

Part a. Step 3. Cancel numbers that are common factors in numerator and denominator.

Express the numerator and denominator into prime factors in order to simplify them.

$\begin{array}{l}\frac{15}{30}\\ =\frac{5\cdot 3}{5\cdot 3\cdot 2}\\ =\frac{1}{2}\end{array}$

Part b. Step 1. Find the least common denominator.

Factoring each denominator into prime factors give

$4=1\cdot 4$

$5=1\cdot 5$

LCD can be given as the product of all the prime factors that occur in these factorizations.

Thus, the LCD is $1\cdot 4\cdot 5=20$.

Part b. Step 2. Use a common denominator.

Multiply the denominators by the required number to get the LCD. And then multiple the numerator with the same number.

$\begin{array}{l}\frac{1}{4}+\frac{1}{5}\\ =\frac{1\cdot 5}{4\cdot 5}+\frac{1\cdot 4}{5\cdot 4}\\ =\frac{5}{20}+\frac{4}{20}\end{array}$

Part b. Step 3. Adding fractions with the same denominator.

Adding the numerator

$\frac{9}{20}$