Simplify (a+2)/(3a)+(b-1)/(5b)

$\frac{a + 2}{3 a} + \frac{b - 1}{5 b}$

Step 1

To write $\frac{a + 2}{3 a}$ as a fraction with a common denominator, multiply by $\frac{5 b}{5 b}$ .

$\frac{a + 2}{3 a} \cdot \frac{5 b}{5 b} + \frac{b - 1}{5 b}$

Step 2

To write $\frac{b - 1}{5 b}$ as a fraction with a common denominator, multiply by $\frac{3 a}{3 a}$ .

$\frac{a + 2}{3 a} \cdot \frac{5 b}{5 b} + \frac{b - 1}{5 b} \cdot \frac{3 a}{3 a}$

Step 3

Write each expression with a common denominator of $15 a b$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a + 2}{3 a}$ by $\frac{5 b}{5 b}$ .

$\frac{(a + 2) (5 b)}{3 a (5 b)} + \frac{b - 1}{5 b} \cdot \frac{3 a}{3 a}$

Multiply $5$ by $3$ .

$\frac{(a + 2) (5 b)}{15 a b} + \frac{b - 1}{5 b} \cdot \frac{3 a}{3 a}$

Multiply $\frac{b - 1}{5 b}$ by $\frac{3 a}{3 a}$ .

$\frac{(a + 2) (5 b)}{15 a b} + \frac{(b - 1) (3 a)}{5 b (3 a)}$

Multiply $3$ by $5$ .

$\frac{(a + 2) (5 b)}{15 a b} + \frac{(b - 1) (3 a)}{15 b a}$

Reorder the factors of $15 b a$ .

$\frac{(a + 2) (5 b)}{15 a b} + \frac{(b - 1) (3 a)}{15 a b}$

Step 4

Combine the numerators over the common denominator.

$\frac{(a + 2) (5 b) + (b - 1) (3 a)}{15 a b}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{(a \cdot 5 + 2 \cdot 5) b + (b - 1) \cdot 3 a}{15 a b}$

Move $5$ to the left of $a$ .

$\frac{(5 \cdot a + 2 \cdot 5) b + (b - 1) \cdot 3 a}{15 a b}$

Multiply $2$ by $5$ .

$\frac{(5 \cdot a + 10) b + (b - 1) \cdot 3 a}{15 a b}$

Apply the distributive property.

$\frac{5 a b + 10 b + (b - 1) \cdot 3 a}{15 a b}$

Apply the distributive property.

$\frac{5 a b + 10 b + (b \cdot 3 - 1 \cdot 3) a}{15 a b}$

Move $3$ to the left of $b$ .

$\frac{5 a b + 10 b + (3 \cdot b - 1 \cdot 3) a}{15 a b}$

Multiply $- 1$ by $3$ .

$\frac{5 a b + 10 b + (3 \cdot b - 3) a}{15 a b}$

Apply the distributive property.

$\frac{5 a b + 10 b + 3 b a - 3 a}{15 a b}$

Add $5 a b$ and $3 b a$ .

Move $b$ .

$\frac{10 b + 5 a b + 3 a b - 3 a}{15 a b}$

Add $5 a b$ and $3 a b$ .

$\frac{10 b + 8 a b - 3 a}{15 a b}$