Question

Simplify the complex fraction, rewriting it by multiplying the numerator and denominator by the LCD

Answer 1

Given the complex fraction:

$\frac{\frac{1}{3}+\frac{4}{x+5}}{\frac{6}{x+5}+\frac{1}{5}}$

You can simplify it as follows:

1. Find the LCD (Least Common Denominator). In this case:

$LCD=(3)(5)(x+5)=15(x+5)$

2. Multiply the numerator and the denominator of the complex fraction by the LCD, in order to get rid of the denominators of the fractions that form the complex fraction:

$=\frac{(\frac{1}{3})\lbrack15(x+5)\rbrack+(\frac{4}{x+5})\lbrack15(x+5)\rbrack}{(\frac{6}{x+5})\lbrack15(x+5)\rbrack+(\frac{1}{5})\lbrack15(x+5)\rbrack}$

Notice that you can simplify multiply and simplify as follows (you can see below the process for simplifying each fraction that forms the complex fraction):

$(\frac{1}{3})\lbrack15(x+5)\rbrack=\frac{\lbrack15(x+5)\rbrack}{3}=5(x+5)$$(\frac{4}{x+5})\lbrack15(x+5)\rbrack=\frac{4\lbrack15(x+5)\rbrack}{x+5}=\frac{60(x+5)}{x+5}=60$$(\frac{6}{x+5})\lbrack15(x+5)\rbrack=\frac{6\lbrack15(x+5)\rbrack}{x+5}=\frac{90(x+5)}{x+5}=90$$(\frac{1}{5})\lbrack15(x+5)\rbrack=\frac{\lbrack15(x+5)\rbrack}{5}=3(x+5)$

Then, you can rewrite the expression as follows:

$=\frac{5(x+5)+60}{90+3(x+5)}$

3. Apply the Distributive Property in the numerator and in the denominator:

$=\frac{(5)(x)+(5)(5)+(60)}{90+(3)(x)+(3)(5)}$$=\frac{5x+25+60}{90+3x+15}$

4. Add the like terms in the numerator and in the denominator:

$=\frac{5x+85}{3x+105}$

Hence, the answer is:

$=\frac{5x+85}{3x+105}$