Loan Instalment Calculation Explained
This problem asks us to find the amount of each equal half-yearly instalment needed to repay a loan of ₹2,550, considering that interest is compounded half-yearly at a rate of 8% per annum.
Understanding Loan Instalment Formula
When a loan is repaid in equal instalments at regular intervals, we can use the concept of the present value of an annuity. The total loan amount must be equal to the sum of the present values of all the future instalments.
The formula used is:
$$ P = \frac{x}{(1+r)^1} + \frac{x}{(1+r)^2} + ... + \frac{x}{(1+r)^n} $$
Where:
- \( P \) is the principal loan amount.
- \( x \) is the amount of each equal instalment.
- \( r \) is the interest rate per compounding period.
- \( n \) is the total number of compounding periods.
In this case, the loan is paid in two equal half-yearly instalments.
Step-by-Step Calculation
Let's break down the calculation:
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Identify Given Values:
- Principal Loan Amount, \( P = ₹2,550 \)
- Annual Interest Rate, \( R = 8\% \) p.a.
- Number of instalments = 2
- Instalment frequency = Half-yearly
-
Calculate Rate per Period:
Since interest is compounded half-yearly, we need the rate for a half-year period.
Rate per period, \( r = \frac{\text{Annual Rate}}{2} = \frac{8\%}{2} = 4\% \)
Converting the percentage to a decimal:
\( r = \frac{4}{100} = 0.04 \)
-
Determine the Number of Periods:
The loan is paid in two half-yearly instalments, so there are 2 periods.
Number of periods, \( n = 2 \)
-
Apply the Instalment Formula:
The formula simplifies for two periods:
$$ P = \frac{x}{(1+r)^1} + \frac{x}{(1+r)^2} $$
Substitute the known values:
$$ ₹2,550 = \frac{x}{(1+0.04)^1} + \frac{x}{(1+0.04)^2} $$
$$ ₹2,550 = \frac{x}{1.04} + \frac{x}{(1.04)^2} $$
$$ ₹2,550 = \frac{x}{1.04} + \frac{x}{1.0816} $$
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Solve for Instalment Amount (x):
Factor out \( x \):
$$ ₹2,550 = x \left( \frac{1}{1.04} + \frac{1}{1.0816} \right) $$
Calculate the values inside the parenthesis:
\( \frac{1}{1.04} \approx 0.961538 \)
\( \frac{1}{1.0816} \approx 0.924556 \)
Add these values:
\( 0.961538 + 0.924556 \approx 1.886094 \)
Now substitute this back into the equation:
$$ ₹2,550 = x (1.886094) $$
Isolate \( x \):
$$ x = \frac{₹2,550}{1.886094} $$
$$ x \approx ₹1,351.999 $$
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Final Answer:
Rounding the result to the nearest rupee, each half-yearly instalment is ₹1,352.
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)
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