What annual installment will discharge a certain debt?

Question

What annual installment will discharge a debt of $\$ 717.60$ due in $4$ years at $20\%$ p.a. simple interest, if the installments are paid at the each end of each year?

I tried the following:

$\$717.60$ corresponds to $\textrm{SI} = (717.60 * 4 * 20)/100$ simple interest. So, each installment should be $\textrm{SI}/4$. I do not understand what is wrong with my solution.

Answer to the question is $\$138$.

Answer 1

In my previous answer I've assumed compound interest, but it is simple interest. I reworked my answer. After $x$ years a capital of $C_0$ is $C_x=(1+i\cdot x)\cdot C_0$

We have sum the 4 payments, which are made on different years.

1. The first payment bears interest for 3 years. The factor is $1+i\cdot 3$
2. The second payment bears interest for 2 years. The factor is $1+i\cdot 2$
3. The third payment bears interest for 1 years. The factor is $1+i\cdot 1$
4. The fourth payment bears no interest. The factor is $1+i\cdot 0=1$

The sum is $4+6\cdot i=4+6\cdot 0.2=4+1.2=5.2$. Thus the equation is

$$C_0\cdot 5.2=717.6$$

$$C_0=\frac{717.6}{5.2}=138 \ \ \color{\limegreen}{\checkmark}$$

Now it´s fine.

Answer 2

See my answer to this question. Use the bracketed formula at the end (preferably, once you make sure you understand how it is derived) with $R=1\mathord. 2$, $T=4$, and $v_0=\$717\mathord.60/1.2^4$, where $v_0$ is the present value of the future debt. This works because owing $\$717\mathord.60$ in four years is essentially the same thing as owing $\$717\mathord.60/1.2^4$ today.