Understanding Simple Interest Instalments
This problem involves calculating the annual rate of simple interest (y%) when a series of equal annual instalments are used to pay off a debt over a specific period. The key is to understand that each instalment accrues simple interest from the time it is paid until the end of the debt period.
Defining Variables and Concepts
Let's define the terms involved:
- Total Debt (D): ₹16,310
- Time Period (T): 4 years
- Annual Instalment (I): ₹3,500
- Rate of Simple Interest (R):
y%per annum - Instalment Payment Schedule: End of Year 1, Year 2, Year 3, and Year 4.
The total amount paid through instalments must equal the total debt. This includes the principal amount of the instalments plus the simple interest accumulated on each.
Calculating Interest on Instalments
Since the instalments are paid at the end of each year, they earn simple interest for different durations:
- The 1st instalment (paid at the end of Year 1) earns interest for $4 - 1 = 3$ years.
- The 2nd instalment (paid at the end of Year 2) earns interest for $4 - 2 = 2$ years.
- The 3rd instalment (paid at the end of Year 3) earns interest for $4 - 3 = 1$ year.
- The 4th instalment (paid at the end of Year 4) earns interest for $4 - 4 = 0$ years.
Setting Up the Equation
The formula for Simple Interest (SI) is given by $SI = \frac{P \times R \times T}{100}$, where P is the principal, R is the rate, and T is the time.
In this case, the principal for the interest calculation is the instalment amount (₹3,500), and the rate is y%.
The total value of all instalments at the end of the 4-year period is the sum of the principal amounts plus the total interest earned.
Total Value = (Sum of Instalments) + (Total Simple Interest)
Sum of Instalments = $4 \times ₹3,500 = ₹14,000$
Total Simple Interest = Interest on 1st instalment + Interest on 2nd instalment + Interest on 3rd instalment + Interest on 4th instalment
Total Simple Interest = $ (\frac{3500 \times y \times 3}{100}) + (\frac{3500 \times y \times 2}{100}) + (\frac{3500 \times y \times 1}{100}) + (\frac{3500 \times y \times 0}{100}) $
Total Simple Interest = $ \frac{3500 \times y}{100} \times (3 + 2 + 1 + 0) $
Total Simple Interest = $ 35y \times 6 $
Total Simple Interest = $ 210y $
Now, we set the total value (Sum of Instalments + Total Simple Interest) equal to the Total Debt:
$ ₹14,000 + 210y = ₹16,310 $
Solving for the Interest Rate (y)
We need to solve the equation for y:
- Subtract the sum of instalments from the total debt to find the total interest paid: $ 210y = ₹16,310 - ₹14,000 $ $ 210y = ₹2,310 $
- Calculate the rate
yby dividing the total interest by the factor calculated: $ y = \frac{2310}{210} $ - Simplify the fraction: $ y = \frac{231}{21} $ $ y = 11 $
Therefore, the rate of simple interest is 11% per annum.
Similar Questions
- A man has to discharge a debt of ₹15,600 which is due in 3 years at 4% simple interest per annum. If he pays this amount in equal instalments of annual payment, find the amount for annual payment.
- The annual instalment (in) to discharge a debt of ₹8,432 due in 4 years at 16% simple interest per annum is:
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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?
What annual installment will discharge a debt of ₹5,460 due in 5 years at 10% simple interest per annum?
- A man borrowed money for 2 years and paid back in two equal annual instalments of ₹2,260 at 5% compound interest, compounded annually. What was the sum (in ₹, to the nearest tens) borrowed?
- ₹1,380 has to be paid after 4 years. If four equal instalments are required with a simple interest of 10% on each instalment, then what is the value of each instalment?
- A loan of ₹2,550 is to be paid back in two equal half-yearly instalments. How much is each instalment if the interest is compounded half-yearly at 8% p.a.?
- What annual instalment will discharge a debt of Rs.3,658 due in four years at 12% simple interest per annum?
- Find the amount of equal annual instalment, which will discharge a debt of Rs.40,376 due in 4 years at the rate of 2% p.a. simple interest.
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:
