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?
5,400
Understanding the Problem: Debt Discharge and Instalments
The question asks about the amount of a debt that can be discharged by a series of equal payments over a specific period, with interest applied annually. The key elements provided are the instalment amount, the number of instalments, the time period (which matches the number of instalments), and the annual interest rate.
The term "debt that will be discharged in 6 equal instalments" typically refers to the principal amount of a loan or debt that is paid off over time through these regular payments. This concept usually involves calculating the present value of an annuity, where the debt is the lump sum amount borrowed now, and the instalments are the future payments made to repay it.
However, upon reviewing the options, the results from a standard present value calculation do not match. Let's perform the calculation for both Present Value (PV) and Future Value (FV) of the annuity to see which result is closer to the given options.
Financial Concepts: Annuities
An annuity is a series of equal payments made at regular intervals over a specified period. In this case, we have 6 equal instalments of Rs. 800 each, paid annually over 6 years at a 5% interest rate.
- Ordinary Annuity: Payments are made at the end of each period. This is the most common type for loan repayments.
- Annuity Due: Payments are made at the beginning of each period.
Given the typical nature of instalment payments discharging a debt, we assume this is an ordinary annuity.
Step-by-Step Calculation Analysis
We are given:
- Instalment Payment (PMT) = Rs. 800
- Number of periods (n) = 6 years
- Interest rate per period (r) = 5% per annum = 0.05
Let's calculate the Present Value (PV) of this ordinary annuity:
The formula for the Present Value of an ordinary annuity is:
$\text{PV} = \text{PMT} \times \frac{1 - (1 + r)^{-n}}{r}$
Substituting the values:
$\text{PV} = 800 \times \frac{1 - (1 + 0.05)^{-6}}{0.05}$
$\text{PV} = 800 \times \frac{1 - (1.05)^{-6}}{0.05}$
$\text{PV} = 800 \times \frac{1 - 0.746215}{0.05}$ (approximate value of $(1.05)^{-6}$)
$\text{PV} = 800 \times \frac{0.253785}{0.05}$
$\text{PV} = 800 \times 5.0757$ (approximate value of the annuity factor)
$\text{PV} \approx 4060.56$
This value (approximately Rs. 4060.56) is not close to any of the provided options.
Now, let's calculate the Future Value (FV) of this ordinary annuity. While the wording "debt discharged" usually relates to present value, let's consider if the question implies the accumulated value of these payments.
The formula for the Future Value of an ordinary annuity is:
$\text{FV} = \text{PMT} \times \frac{(1 + r)^n - 1}{r}$
Substituting the values:
$\text{FV} = 800 \times \frac{(1 + 0.05)^6 - 1}{0.05}$
$\text{FV} = 800 \times \frac{(1.05)^6 - 1}{0.05}$
$\text{FV} = 800 \times \frac{1.3400956 - 1}{0.05}$ (approximate value of $(1.05)^6$)
$\text{FV} = 800 \times \frac{0.3400956}{0.05}$
$\text{FV} = 800 \times 6.801912$ (approximate value of the annuity factor)
$\text{FV} \approx 5441.53$
Comparing Calculation Results with Options
Our calculated Present Value is approximately Rs. 4060.56. Our calculated Future Value is approximately Rs. 5441.53.
Let's look at the options:
- Option 1: Rs. 6,600
- Option 2: Rs. 7,500
- Option 3: Rs. 8,000
- Option 4: Rs. 5,400
The Future Value calculation result (Rs. 5441.53) is closest to Option 4 (Rs. 5,400). The difference is approximately Rs. 41.53, which could be due to rounding in the interest factor used to derive the options, or a slightly simplified assumption made when creating the question/options.
Although the standard interpretation of "debt discharged" points towards present value, the proximity of the future value calculation to one of the options suggests that the question might be asking for the future value of the stream of payments, equating this accumulated value to the debt amount.
Conclusion
Based on the calculation that yields a result closest to the given options, the future value of the annuity payments is the most likely intended amount of the debt. The future value of 6 annual instalments of Rs. 800 at 5% per annum is approximately Rs. 5441.53, which is closest to Rs. 5,400.
The amount of debt that will be discharged, interpreted as the future value of the instalment stream, is approximately Rs. 5441.53.
| Calculation Type | Formula | Result (Approx.) |
|---|---|---|
| Present Value of Ordinary Annuity | $\text{PV} = \text{PMT} \times \frac{1 - (1 + r)^{-n}}{r}$ | Rs. 4060.56 |
| Future Value of Ordinary Annuity | $\text{FV} = \text{PMT} \times \frac{(1 + r)^n - 1}{r}$ | Rs. 5441.53 |
Revision Table: Key Financial Terms
| Term | Definition | Relevance to Problem |
|---|---|---|
| Debt | An amount of money owed. | The principal amount being discharged or related to the instalment payments. |
| Instalment | One of a number of payments into which an amount is divided. | The regular Rs. 800 payments made to discharge the debt. |
| Annuity | A series of equal payments made at fixed intervals. | The sequence of 6 equal Rs. 800 instalments. |
| Present Value (PV) | The current worth of a future stream of payments, discounted at a specific interest rate. | Typically represents the original debt amount discharged by the payments. |
| Future Value (FV) | The value of an asset or cash at a specified date in the future, based on a given rate of growth. | Represents the accumulated value of the instalment payments at the end of the term. |
Additional Information on Time Value of Money
This problem is a good example of the time value of money principle. This principle states that a sum of money is worth more now than the same sum will be at a future date due to its potential earning capacity. Interest rates are used to adjust cash flows for their timing.
In the context of debt and instalments:
- When you take out a loan, the bank gives you the Present Value (PV) lump sum now. You repay it with future instalments whose total future value, considering interest, equals the future value of the initial loan amount.
- The payments you make include both principal and interest over time.
- Calculating the PV of future payments tells you what that stream of payments is worth *today*.
- Calculating the FV of future payments tells you what that stream of payments will be worth *in the future*, accumulated with interest.
While "debt discharged" strongly implies PV, the provided options sometimes require considering alternative interpretations or rounding conventions.
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?
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:
To buy a car, Sahil borrowed some amount at 10% simple interest rate. He has paid some loan amount but Rs. 1,81,500 is due which he has to pay in 3 years. What will be the annual installment (in Rs.)?
