Understanding the Loan Repayment Calculation
This question asks us to find the initial amount of money borrowed (the principal sum). The loan was taken for a duration of 2 years. It was repaid with two equal annual instalments, each amounting to ₹2,260. The loan accrued 5% compound interest, compounded annually. Our goal is to calculate the total sum borrowed and round it to the nearest tens.
Applying the Present Value Concept for Instalments
When a loan is repaid in equal instalments over a specific period, the original amount borrowed is equivalent to the sum of the present values of all the future instalments. The present value accounts for the time value of money, meaning money received in the future is worth less than money received today due to potential earnings from interest.
Let's define the variables:
- $S$: The sum borrowed (the principal amount we need to find).
- $P$: The amount of each equal annual instalment. Given $P = \text{₹}2,260\).
- $r$: The annual interest rate. Given $r = 5\% = 0.05\).
- $n$: The total number of years for the loan. Given $n = 2\).
The instalments are paid annually at the end of each year. The first instalment is paid at the end of year 1, and the second instalment is paid at the end of year 2. To find their value today (present value), we need to discount them back using the given compound interest rate.
The formula to calculate the present value ($PV$) of a single future payment is:
$$ PV = \frac{\text{Future Value}}{(1+r)^t} $$
Where $t$ is the number of years until the payment is made.
Therefore, the total sum borrowed $S$ is the sum of the present values of the two instalments:
$$ S = \frac{P}{(1+r)^1} + \frac{P}{(1+r)^2} $$
Step-by-Step Calculation of the Borrowed Sum
We will now substitute the given values into the formula and perform the calculations:
Given values:
- $P = \text{₹}2,260\)
- $r = 0.05\)
- Number of years = 2
First, let's calculate the value of \(1+r\):
\( 1+r = 1 + 0.05 = 1.05 \)
Now, we calculate the present value of the first instalment, which is paid at the end of year 1:
$$ PV_1 = \frac{2260}{1.05} $$
Next, we calculate the present value of the second instalment, which is paid at the end of year 2:
$$ PV_2 = \frac{2260}{(1.05)^2} $$
Let's calculate $ (1.05)^2 $:
\( (1.05)^2 = 1.1025 \)
Now, we find the numerical values for $PV_1$ and $PV_2$:
\( PV_1 = \frac{2260}{1.05} \approx 2152.38095 \)
\( PV_2 = \frac{2260}{1.1025} \approx 2049.88662 \)
The total sum borrowed $S$ is the sum of these two present values:
$$ S = PV_1 + PV_2 $$
\( S \approx 2152.38095 + 2049.88662 \)
\( S \approx 4202.26757 \)
Alternatively, we can calculate this by factoring out $P$:
$$ S = P \times \left[ \frac{1}{(1+r)^1} + \frac{1}{(1+r)^2} \right] $$
\( S = 2260 \times \left[ \frac{1}{1.05} + \frac{1}{1.1025} \right] \)
\( S = 2260 \times [0.95238095... + 0.90702948...] \)
\( S = 2260 \times 1.85941043... \)
\( S \approx 4202.26757 \)
Rounding to the Nearest Tens
The question requires the final answer to be rounded to the nearest tens. Our calculated sum borrowed is approximately ₹4202.27.
To round ₹4202.27 to the nearest tens:
- Identify the digit in the tens place, which is 0.
- Look at the digit immediately to its right (the units place), which is 2.
- Since 2 is less than 5, we keep the tens digit as it is (0) and change all digits to its right (units and decimals) to zero.
Therefore, ₹4202.27 rounded to the nearest tens is ₹4,200.
Similar Questions
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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:
