What equal instalment of annual payment (in) will discharge a debt that is due as ₹624 at the end of 5 years at 2% per annum simple interest?
Problem Analysis: Simple Interest Debt Discharge
This problem asks us to find the amount of an equal annual payment that will completely pay off a debt after a certain period, considering simple interest accrues on the outstanding amount.
- Debt Amount: ₹624
- Time Period: 5 years
- Interest Rate: 2% per annum (simple interest)
- Payment Frequency: Annually
- Goal: Find the equal instalment amount.
Since simple interest is involved, each payment made will earn interest for the time remaining until the debt is fully discharged. The sum of all payments, along with the interest earned on each payment, must equal the total debt amount.
Understanding Simple Interest Instalments
When calculating equal instalments to discharge a debt under simple interest, we consider the future value of each instalment. Let the equal annual instalment be represented by $x$. The debt is due in $N$ years at a rate $R$. The instalments are paid at the end of each year.
- The first instalment (paid at the end of year 1) will earn simple interest for $(N-1)$ years.
- The second instalment (paid at the end of year 2) will earn simple interest for $(N-2)$ years.
- ...
- The last instalment (paid at the end of year $N$) will earn simple interest for $0$ years (as it's paid on the due date).
The future value (FV) of each instalment is calculated using the simple interest formula: $FV = P + \text{Interest}$, where $P=x$ and $\text{Interest} = \frac{x \times R \times T}{100}$.
The sum of the future values of all instalments must equal the total debt amount.
Step-by-Step Calculation for Equal Instalment
We are given:
- Total Debt = ₹624
- Number of years, $N = 5$
- Rate of simple interest, $R = 2\%$ per annum
- Let the equal annual instalment be $x$.
We calculate the future value of each of the 5 annual instalments:
- 1st Instalment (end of year 1): Earns interest for $5 - 1 = 4$ years.
Future Value = $x + \frac{x \times 2 \times 4}{100} = x \left( 1 + \frac{8}{100} \right) = 1.08x$ - 2nd Instalment (end of year 2): Earns interest for $5 - 2 = 3$ years.
Future Value = $x + \frac{x \times 2 \times 3}{100} = x \left( 1 + \frac{6}{100} \right) = 1.06x$ - 3rd Instalment (end of year 3): Earns interest for $5 - 3 = 2$ years.
Future Value = $x + \frac{x \times 2 \times 2}{100} = x \left( 1 + \frac{4}{100} \right) = 1.04x$ - 4th Instalment (end of year 4): Earns interest for $5 - 4 = 1$ year.
Future Value = $x + \frac{x \times 2 \times 1}{100} = x \left( 1 + \frac{2}{100} \right) = 1.02x$ - 5th Instalment (end of year 5): Earns interest for $5 - 5 = 0$ years.
Future Value = $x$
The sum of the future values of these instalments must equal the debt amount:
$$1.08x + 1.06x + 1.04x + 1.02x + x = 624$$Combine the terms involving $x$:
$$x(1.08 + 1.06 + 1.04 + 1.02 + 1) = 624$$Sum the coefficients:
$$1.08 + 1.06 + 1.04 + 1.02 + 1 = 5.20$$So the equation becomes:
$$5.20x = 624$$Now, solve for $x$:
$$x = \frac{624}{5.20}$$To simplify the division, multiply the numerator and denominator by 100:
$$x = \frac{624 \times 100}{5.20 \times 100} = \frac{62400}{520}$$Simplify the fraction by dividing both by 10:
$$x = \frac{6240}{52}$$Performing the division:
$$x = 120$$Conclusion
Therefore, the equal instalment of the annual payment required to discharge the debt of ₹624 in 5 years at 2% simple interest is ₹120.
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