Compound Interest problem , loan amount

Question

A lent £80,000 to B with the assurance that B would return £50,000 every year for two years. What is the approximate rate of compound interest per annum that A is charging B?

(A) 11.8 %

(B) 25 %

(C) 18.75 %

(D) 16.25 %

My approach:-

My approach is to find the Future value of the installments at the end of year 2, and equate it with the future value of the amount to be discharged, that is £80,000.

With that: $$80000(1+r)^2=50000(1+r)^1 + 50000(1+r)^0.$$

Solving for $r$, I get 0.162 %.

Where am I going wrong ?

Answer 1

I believe you were on the right track. I also solved for $r$ and got $D$ as my answer. After expanding the formula you used there, simplifying by $10,000$ for ease of calculations, I came to the following equation $$8+16r+8r^2=5r+10.$$ Then, combining them together, I got the quadratic equation $$8r^2+11r-2=0.$$ After solving this quadratic equation, I found two values for $r$: $$r_1=\frac{-11+ \sqrt{11^2-4*8*(-2)}}{2*8}=0.1625,$$ or $16.25\%.$

The other solution was $$r_2=-1.5375,$$ which didn't look like any of the options, and generally speaking, negative interest rates are not used for the level and kind of questions you are looking at. So, the actual answer is 16.25%, option D.