Loan Instalment Calculation Using Compound Interest
The problem asks us to find the value of each equal yearly instalment for a loan amount ($P$) of $3,825$, repaid over two years with a compound interest rate ($R$) of 4% per annum, compounded annually.
Problem Variables
- Principal Loan Amount ($P$): $3,825$
- Annual Interest Rate ($R$): $4\%$ or $0.04$
- Number of Instalments ($n$): $2$
- Instalment Amount: Let this be $x$
Instalment Formula
When a loan is repaid in equal instalments ($x$) over $n$ periods at a compound interest rate $R$, the principal amount $P$ is equal to the sum of the present values of all instalments. The formula is:
$ P = \frac{x}{(1+R)^1} + \frac{x}{(1+R)^2} + \dots + \frac{x}{(1+R)^n} $For $n=2$, the formula becomes:
$ P = \frac{x}{1+R} + \frac{x}{(1+R)^2} $Calculation Steps
- Substitute the known values into the formula: $ 3825 = \frac{x}{1+0.04} + \frac{x}{(1+0.04)^2} $
- Simplify the denominators: $ 3825 = \frac{x}{1.04} + \frac{x}{(1.04)^2} $ $ 3825 = \frac{x}{1.04} + \frac{x}{1.0816} $
- Factor out $x$: $ 3825 = x \left( \frac{1}{1.04} + \frac{1}{1.0816} \right) $
- Find a common denominator for the terms inside the parenthesis: $ 3825 = x \left( \frac{1.0816 + 1.04}{1.04 \times 1.0816} \right) $ $ 3825 = x \left( \frac{2.1216}{1.124864} \right) $ Alternatively, use the simplified formula derived from the annuity formula: $P = x \times \frac{1 - (1+R)^{-n}}{R}$ is for ordinary annuities, but for loan redemption it's often simpler to use the sum of present values. Let's stick to the sum of present values approach as it's more direct here. Re-calculating step 3's simplification: $ 3825 = x \left( \frac{1.04 + 1}{(1.04)^2} \right) $ $ 3825 = x \left( \frac{2.04}{1.0816} \right) $
- Solve for $x$: $ x = 3825 \times \frac{1.0816}{2.04} $ $ x = 3825 \times 0.530196... $ $ x \approx 2028.00 $
Therefore, the value of each equal yearly instalment is approximately $2,028$.
Important Questions from Installments
A loan of Rs. 1,50,000 is availed with compound interest rate of 10% per annum for two years compounded annually. It is to be paid in equal yearly installments, and the installment is to be paid at the end of each year. The value of the equal yearly installment is : (Rounded off to two places of decimal)
A sum of Rs. P was borrowed and paid back in two equal yearly instalments, each of Rs. 35,280. If the rate of interest was 5% per annum and interest is compounding annually, then the value of P is ________.
A sum of Rs. 16400 is borrowed to be paid back in 2 years by equal payments allowing 5% compound interest. Find the annual payment.
A sum of Rs. 1100 was taken as a loan. This is to be paid in two equal installments. If the rate of interest is 20% per annum, compounded annually, find the amount payable in each installment.
The formula for finding the annual installment, when A is the amount taken on loan, where r% is the rate of interest, n is the number of installments, is:
