# How do you evaluate \frac { 25} { 31} \cdot \frac { 62} { 105} + \frac { 3} { 7}?

May 14, 2017

Write out the prime factorization for each number and cancel if possible, then find the LCD to add the fractions.
Answer: $\frac{19}{21}$

#### Explanation:

Evaluate $\frac{25}{31} \cdot \frac{62}{105} + \frac{3}{7}$

First, we can write out the prime factorization for each value:
$= {5}^{2} / 31 \cdot \frac{31 \cdot 2}{3 \cdot 5 \cdot 7} + \frac{3}{7}$

Now, we look to cancel values that appear in the numerator and the denominator, $5$ and $31$:
$= \frac{5}{1} \cdot \frac{2}{3 \cdot 7} + \frac{3}{7}$

We can simplify the multiplication to:
$= \frac{10}{3 \cdot 7} + \frac{3}{7}$

To add the fractions, we notice that lowest common denominator would be $3 \cdot 7 = 21$, which would require us to multiply $\frac{3}{7}$ by $\frac{3}{3}$:
$= \frac{10}{3 \cdot 7} + \frac{3}{7} \cdot \frac{3}{3}$
$= \frac{10}{3 \cdot 7} + \frac{9}{3 \cdot 7}$
$= \frac{10 + 9}{3 \cdot 7}$
$= \frac{19}{21}$