Simplify: $\frac{x^2 - y^2}{(x + y)^2} \div \frac{(x - y)^2}{(3x + 3y)}$

Step 1: Analyze the question We are dividing one algebraic fraction by another. To divide fractions, we multiply by the reciprocal of the second fraction. Factorising first makes cancelling easy.

Step 2: Key concepts

• Difference of two squares: x² − y² = (x − y)(x + y)
• Common factor: 3x + 3y = 3(x + y)
• Dividing by a fraction means multiplying by its reciprocal.

Step 3: Solution

Flip the second fraction and multiply:

Now cancel common factors:

• (x + y) in the first numerator cancels with one (x + y) in the denominator (x + y)², leaving (x + y) in the denominator. Then the 3(x + y) in the second numerator cancels this remaining (x + y).
• One (x − y) in the numerator cancels with one of the (x − y) in (x − y)².

This leaves:

Step 4: Answer The result is 3/(x − y), which is option C.

Tip: In simplification questions, factorise everything first (difference of two squares, common factors) before cancelling. Also, always change ÷ to × with the reciprocal before you cancel.