Solving for the Value of xy

We are given the equations:

$ \frac{x-y}{3} = \frac{x+y}{5} = \frac{xy}{8} $

We need to find the value of xy.

Step 1: Equate First Two Ratios

Set the first two parts of the proportion equal:

$ \frac{x-y}{3} = \frac{x+y}{5} $

Cross-multiply:

$ 5(x-y) = 3(x+y) $

$ 5x - 5y = 3x + 3y $

Rearrange the terms to solve for x in terms of y:

$ 5x - 3x = 3y + 5y $

$ 2x = 8y $

$ x = 4y $

Step 2: Substitute and Equate Ratios

Now, substitute $x = 4y$ into the second and third parts of the proportion:

$ \frac{x+y}{5} = \frac{xy}{8} $

$ \frac{(4y)+y}{5} = \frac{(4y)y}{8} $

$ \frac{5y}{5} = \frac{4y^2}{8} $

Simplify the equation:

$ y = \frac{y^2}{2} $

Step 3: Solve for y

Rearrange the equation to solve for $y$:

$ 2y = y^2 $

$ y^2 - 2y = 0 $

Factor out $y$:

$ y(y-2) = 0 $

This gives two possible values for $y$: $y=0$ or $y=2$.

Step 4: Determine x and Calculate xy

Case 1: If $y=0$, then $x = 4(0) = 0$. This gives $xy = 0 \times 0 = 0$. This is a valid solution to the equations but not among the options.

Case 2: If $y=2$, then $x = 4(2) = 8$. This gives $xy = 8 \times 2 = 16$.

Verify this solution with the original proportions:

• $ \frac{x-y}{3} = \frac{8-2}{3} = \frac{6}{3} = 2 $
• $ \frac{x+y}{5} = \frac{8+2}{5} = \frac{10}{5} = 2 $
• $ \frac{xy}{8} = \frac{8 \times 2}{8} = \frac{16}{8} = 2 $

Since all parts equal 2, the values $x=8$ and $y=2$ are correct.

Therefore, the value of xy is 16.