Simplify (15a-b)/(12a)-(a-4b)/(9a)

$\frac{15 a - b}{12 a} - \frac{a - 4 b}{9 a}$

Step 1

To write $\frac{15 a - b}{12 a}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{15 a - b}{12 a} \cdot \frac{3}{3} - \frac{a - 4 b}{9 a}$

Step 2

To write $- \frac{a - 4 b}{9 a}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{15 a - b}{12 a} \cdot \frac{3}{3} - \frac{a - 4 b}{9 a} \cdot \frac{4}{4}$

Step 3

Write each expression with a common denominator of $36 a$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{15 a - b}{12 a}$ by $\frac{3}{3}$ .

$\frac{(15 a - b) \cdot 3}{12 a \cdot 3} - \frac{a - 4 b}{9 a} \cdot \frac{4}{4}$

Multiply $3$ by $12$ .

$\frac{(15 a - b) \cdot 3}{36 a} - \frac{a - 4 b}{9 a} \cdot \frac{4}{4}$

Multiply $\frac{a - 4 b}{9 a}$ by $\frac{4}{4}$ .

$\frac{(15 a - b) \cdot 3}{36 a} - \frac{(a - 4 b) \cdot 4}{9 a \cdot 4}$

Multiply $4$ by $9$ .

$\frac{(15 a - b) \cdot 3}{36 a} - \frac{(a - 4 b) \cdot 4}{36 a}$

Step 4

Combine the numerators over the common denominator.

$\frac{(15 a - b) \cdot 3 - (a - 4 b) \cdot 4}{36 a}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{15 a \cdot 3 - b \cdot 3 - (a - 4 b) \cdot 4}{36 a}$

Multiply $3$ by $15$ .

$\frac{45 a - b \cdot 3 - (a - 4 b) \cdot 4}{36 a}$

Multiply $3$ by $- 1$ .

$\frac{45 a - 3 b - (a - 4 b) \cdot 4}{36 a}$

Apply the distributive property.

$\frac{45 a - 3 b + (- a - (- 4 b)) \cdot 4}{36 a}$

Multiply $- 4$ by $- 1$ .

$\frac{45 a - 3 b + (- a + 4 b) \cdot 4}{36 a}$

Apply the distributive property.

$\frac{45 a - 3 b - a \cdot 4 + 4 b \cdot 4}{36 a}$

Multiply $4$ by $- 1$ .

$\frac{45 a - 3 b - 4 a + 4 b \cdot 4}{36 a}$

Multiply $4$ by $4$ .

$\frac{45 a - 3 b - 4 a + 16 b}{36 a}$

Subtract $4 a$ from $45 a$ .

$\frac{41 a - 3 b + 16 b}{36 a}$

Add $- 3 b$ and $16 b$ .

$\frac{41 a + 13 b}{36 a}$