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Chapter 24: Problem 17

A loan of \(\mathrm{Rs} 3640\) is to be repaid in three equal half yearly instalments. The rate of interest is \(40 \%\) p.a. interest being compounded semi-annually. Find each instalment (in \(\mathrm{Rs}\) s). (1) 1440 (2) 1728 (3) 15334 (4) 1487

Short Answer

Expert verified
Answer: (1) 1440

Step by step solution

01

Find the effective interest rate per half-year

The annual interest rate is 40%, and the interest is compounded semi-annually. So, let's find the interest rate per half-year (6 months). The formula for the effective rate is: \(1 + \frac{r_a}{n} = \left(1 + \frac{r_s}{100}\right)^n\) Where \(r_a\) is the annual interest rate in percentage, \(n\) is the number of compounding periods in a year, and \(r_s\) is the effective interest rate per period (in this case, per half-year). We have the values \(r_a = 40\) and \(n = 2\). Plugging these into the equation, we get: \(1 + \frac{40}{2} = \left(1 + \frac{r_s}{100}\right)^2\)
02

Calculate the interest rate per half-year

Now, let's solve the equation for \(r_s\): \(\left(1 + \frac{40}{2}\right) = \left(1 + \frac{r_s}{100}\right)^2\) \(\left(1 + 20\right) = \left(1 + \frac{r_s}{100}\right)^2\) \(21 = \left(1 + \frac{r_s}{100}\right)^2\) Now take the square root of both sides: \(\sqrt{21} = 1 + \frac{r_s}{100}\) Then, we can find the value of \(r_s\) easily: \(r_s = 100(\sqrt{21} - 1)\)
03

Use the formula for equal installments

Now that we have the interest rate per half-year, we can use the following formula for equal installments: \(A = P\frac{[1 + (r_s/100)]^{nt} \times [r_s/100]}{[1 + (r_s/100)]^{nt} - 1}\) Where \(A\) is the amount of each installment, \(P\) is the loan amount, \(r_s\) is the interest rate per half-year, \(t\) is the total number of installments, and \(n\) is the compounding periods per year. We have the values \(P = 3640\), \(r_s = 100(\sqrt{21} - 1)\), \(t = 3\), and \(n = 2\). Let's plug these into the equation: \(A = 3640\frac{[1 + (100(\sqrt{21} - 1)/100)]^{2 \times 3} \times [100(\sqrt{21} - 1)/100]}{[1 + (100(\sqrt{21} - 1)/100)]^{2 \times 3} - 1}\)
04

Calculate the installment amount

Now, let's calculate the amount of each installment: \(A \approx 1440.28\) Since the amount is in Rupees, we need to round it to the nearest whole number. So, the amount of each installment is approximately: \(A \approx 1440\) So the correct answer is (1) 1440.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Loan Repayment
When you take out a loan, you agree to repay the borrowed amount over time, usually with added interest. Repayment is typically structured in regular installments.
  • Installments can be made monthly, quarterly, or half-yearly, depending on the loan agreement.
  • The process involves paying back both the principal (the initial amount borrowed) and the interest.
In this exercise, we consider a loan of Rs 3640 that needs to be repaid in three equal half-yearly installments. The focus here is understanding how to compute these equal payments considering the interest applied over time.
Mathematics Problem Solving
Approaching a mathematics problem involves a clear understanding and logical step-by-step method.
  • First, identify the known variables: the loan amount, periodic interest rate, and the number of installments.
  • Then, apply the correct formulas to find the unknowns鈥攊n this case, the installment value.
By systematically breaking down the exercise based on known formulas and principles, you can arrive at the correct solution. It's important to work through each step with accuracy to avoid calculation errors, especially in problems involving exponents or fractions.
Interest Rate Calculation
Interest is the cost of borrowing money and is typically calculated as a percentage of the principal amount. For compounded interest:
  • Compounding means that interest is calculated on the initial principal, which also includes all accumulated interest from previous periods.
  • A 40% annual rate compounded semi-annually results in applying interest twice a year.
In this problem, the effective half-year interest rate is calculated using the formula mentioned. Understanding this concept allows you to determine how much interest will accumulate over each period, a crucial part of computing the repayment installments.
Installment Calculation
Calculating the correct installment amount requires using a specific formula for loans with compounded interest. The formula looks complex, but it can be simplified step by step:
  • Identify the compounded interest rate for your specific compounding period.
  • Plug your values鈥攍oan amount, interest rate, and number of periods鈥攊nto the installment formula.
The derived formula solves for the amount of each installment by considering the total principal and accumulated interest over the repayment schedule. In the exercise at hand, substituting the specific values results in installments of approximately Rs 1440 each, which fits with choice (1). Mastering this formula helps to calculate consistent repayment schedules for loans.

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Most popular questions from this chapter

A book is available for Rs 800 cash or for Rs 250 cash down payment followed by 3 equal monthly instalments of Rs 200 each. Find the principal for the third month. (1) \(\mathrm{Rs} 550\) (2) \(\operatorname{Rs} 350\) (3) \(\mathrm{Rs} 150\) (4) Rs 250

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