Question on Compound Interest:

Question

A man borrowed a certain sum of money and paid it back in $2$ years in two equal installments. If the rate of compound interest was $4$% p.a. and he paid back $Rs. 676$ annually, what sum did he borrow?

My Approach:

Here,

   First Installment;

Amount at the end of First year=$Rs. 676$ Rate of Interest=$4$% p.a Time=$1$ year., Now, $$C.A=P(1+\frac {R}{100})^T$$ $$676=P_1(1+\frac {4}{100})^1$$ $$P_1=Rs. 650$$.

Then, what should I do next? Please help.

Answer 1

Let's say he starts with the principal amount $P$. The amount owed at the end of the first year (before any repayment) is $1.04P$. After paying back $676$, the amount owed is reduced to $1.04P - 676$.

That would become the effective principal at the start of the second year of the loan term. The total amount owed at the end of the second year (before repayment) is $(1.04)(1.04P - 676)$. Since the second payment of $676$ settled the loan, you can form the equation:

$$(1.04)(1.04P - 676) = 676$$

and solve it easily.

For a more general formula based approach as @RossMillikan mentions, please see my answer here: Hire purchase problems, which you should be able to adapt.

Answer 2

What that really tells you is that the balance after the first payment was $650$, which grew to $676$ at the end of the second year and was paid off by the second payment. Now you should find the amount borrowed that will leave $650$ at the end of the first year after the payment. How much was the balance just before the payment? Then how much was borrowed to make that the balance after one year?

That was how to continue your approach. You should also be able to find a formula that would work it out for you without going through each year. If you had a $30$ year loan stepping back each year would be a lot of work, though a spreadsheet could automate it.