weighted average of interest rate

Question

Let say that I have 2 loans. One loan has an interest rate of 5% and the other of 10%. I calculate the weighted average both loans and it is 8% (This is an example). I don't know how can you prove that the total interest yield of both loans with thier original interest rates is the same as if both loans interest rate was 8%. I don't know how to prove it. I would like some intuition.

Answer 1

Set the interests equal to each other to figure the relative size of the loans.

$.08x + .08y = .05x + .10y$

$.03x = .02y$

$x = \frac{2}{3}y$

Therefore the $x$ loan is $\frac{2}{3}$ of the $y$ loan.

Example:

$x = 600; y = 900$

$.08x + .08y = .05x + .10y$

$.08(600) + .08(900) = .05(600) + .10(900)$

$48 + 72 = 30 + 90$

$120 = 120$