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 computer is available for Rs. 39,000 on cash payment or Rs. 19,000 as cash payment followed by five monthly instalments of Rs. 4,200 each. What is the rate of interest per annum under the instalment plan?
What is the amount (in Rs.) of debt that will be discharged in 6 equal instalments of Rs. 800 each, if the debt is due in 6 years at 5% per annum?
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 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 computer is available for ₹75,300 cash or for ₹25,740 cash down payment and two equal half-yearly instalments. If the dealer charges interest at 20% p.a., compounded half-yearly, then the total interest to be paid by a customer who buys it in instalment scheme is:
