Simplify (b+2)/(15b)-(3c-5)/(45c)

$\frac{b + 2}{15 b} - \frac{3 c - 5}{45 c}$

Step 1

To write $\frac{b + 2}{15 b}$ as a fraction with a common denominator, multiply by $\frac{3 c}{3 c}$ .

$\frac{b + 2}{15 b} \cdot \frac{3 c}{3 c} - \frac{3 c - 5}{45 c}$

Step 2

To write $- \frac{3 c - 5}{45 c}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{b + 2}{15 b} \cdot \frac{3 c}{3 c} - \frac{3 c - 5}{45 c} \cdot \frac{b}{b}$

Step 3

Write each expression with a common denominator of $45 b c$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{b + 2}{15 b}$ by $\frac{3 c}{3 c}$ .

$\frac{(b + 2) (3 c)}{15 b (3 c)} - \frac{3 c - 5}{45 c} \cdot \frac{b}{b}$

Multiply $3$ by $15$ .

$\frac{(b + 2) (3 c)}{45 b c} - \frac{3 c - 5}{45 c} \cdot \frac{b}{b}$

Multiply $\frac{3 c - 5}{45 c}$ by $\frac{b}{b}$ .

$\frac{(b + 2) (3 c)}{45 b c} - \frac{(3 c - 5) b}{45 c b}$

Reorder the factors of $45 c b$ .

$\frac{(b + 2) (3 c)}{45 b c} - \frac{(3 c - 5) b}{45 b c}$

Step 4

Combine the numerators over the common denominator.

$\frac{(b + 2) (3 c) - (3 c - 5) b}{45 b c}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{(b \cdot 3 + 2 \cdot 3) c - (3 c - 5) b}{45 b c}$

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

$\frac{(3 \cdot b + 2 \cdot 3) c - (3 c - 5) b}{45 b c}$

Multiply $2$ by $3$ .

$\frac{(3 \cdot b + 6) c - (3 c - 5) b}{45 b c}$

Apply the distributive property.

$\frac{3 b c + 6 c - (3 c - 5) b}{45 b c}$

Apply the distributive property.

$\frac{3 b c + 6 c + (- (3 c) - - 5) b}{45 b c}$

Multiply $3$ by $- 1$ .

$\frac{3 b c + 6 c + (- 3 c - - 5) b}{45 b c}$

Multiply $- 1$ by $- 5$ .

$\frac{3 b c + 6 c + (- 3 c + 5) b}{45 b c}$

Apply the distributive property.

$\frac{3 b c + 6 c - 3 c b + 5 b}{45 b c}$

Subtract $3 c b$ from $3 b c$ .

Move $c$ .

$\frac{6 c + 3 b c - 3 b c + 5 b}{45 b c}$

Subtract $3 b c$ from $3 b c$ .

$\frac{6 c + 0 + 5 b}{45 b c}$

Add $6 c$ and $0$ .

$\frac{6 c + 5 b}{45 b c}$