Question:
A man takes a loan of 1000 and repays it by paying 530 at the end of the first year, and 594 at the end of the second year. Find the rate of interest per annum.
A man takes a loan of 1000 and repays it by paying 530 at the end of the first year, and 594 at the end of the second year. Find the rate of interest per annum.
Show Hint
In loan installment problems with multiple-choice options, back-solving by plugging in the given interest rates is almost always faster than solving the algebraic equation.
Updated On: Nov 30, 2025
- 8%
- 10%
- 12%
- 9%
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The problem describes a loan that is paid back in two installments over two years. The interest is compounded annually. We need to find the annual interest rate. The principle is that the present value of all the repayments must equal the initial loan amount.
Step 2: Key Formula or Approach:
The present value (PV) of a future amount (FV) paid after \(n\) years at an annual interest rate \(r\) is given by the formula:
\[ PV = \frac{FV}{(1+r)^n} \] The total loan amount must be equal to the sum of the present values of all installments.
\[ \text{Loan Amount} = \frac{\text{Installment}_1}{(1+r)^1} + \frac{\text{Installment}_2}{(1+r)^2} \] Step 3: Detailed Explanation:
Given:
- Loan Amount = 1000
- Installment at the end of year 1 = 530
- Installment at the end of year 2 = 594
Let the annual interest rate be \(r\). The equation is:
\[ 1000 = \frac{530}{(1+r)} + \frac{594}{(1+r)^2} \] Solving this equation directly would result in a quadratic equation, which can be time-consuming. A more efficient method for competitive exams is to test the given options.
Testing Option (A) 8%:
If \(r = 8% = 0.08\), then \(1+r = 1.08\).
Let's calculate the sum of the present values of the installments:
\[ PV = \frac{530}{1.08} + \frac{594}{(1.08)^2} \] \[ PV = \frac{530}{1.08} + \frac{594}{1.1664} \] \[ PV = 490.7407... + 509.259... \] \[ PV \approx 999.999... \] This value is approximately equal to 1000. Therefore, the interest rate is 8%.
Testing Option (B) 10%:
If \(r = 10% = 0.10\), then \(1+r = 1.1\).
\[ PV = \frac{530}{1.1} + \frac{594}{(1.1)^2} = \frac{530}{1.1} + \frac{594}{1.21} \] \[ PV = 481.81 + 490.90 = 972.71 \] This is not equal to 1000.
The other options will also not yield 1000.
Step 4: Final Answer
By substituting the options into the present value formula, we find that an interest rate of 8% correctly equates the present value of the repayments to the loan amount.
The problem describes a loan that is paid back in two installments over two years. The interest is compounded annually. We need to find the annual interest rate. The principle is that the present value of all the repayments must equal the initial loan amount.
Step 2: Key Formula or Approach:
The present value (PV) of a future amount (FV) paid after \(n\) years at an annual interest rate \(r\) is given by the formula:
\[ PV = \frac{FV}{(1+r)^n} \] The total loan amount must be equal to the sum of the present values of all installments.
\[ \text{Loan Amount} = \frac{\text{Installment}_1}{(1+r)^1} + \frac{\text{Installment}_2}{(1+r)^2} \] Step 3: Detailed Explanation:
Given:
- Loan Amount = 1000
- Installment at the end of year 1 = 530
- Installment at the end of year 2 = 594
Let the annual interest rate be \(r\). The equation is:
\[ 1000 = \frac{530}{(1+r)} + \frac{594}{(1+r)^2} \] Solving this equation directly would result in a quadratic equation, which can be time-consuming. A more efficient method for competitive exams is to test the given options.
Testing Option (A) 8%:
If \(r = 8% = 0.08\), then \(1+r = 1.08\).
Let's calculate the sum of the present values of the installments:
\[ PV = \frac{530}{1.08} + \frac{594}{(1.08)^2} \] \[ PV = \frac{530}{1.08} + \frac{594}{1.1664} \] \[ PV = 490.7407... + 509.259... \] \[ PV \approx 999.999... \] This value is approximately equal to 1000. Therefore, the interest rate is 8%.
Testing Option (B) 10%:
If \(r = 10% = 0.10\), then \(1+r = 1.1\).
\[ PV = \frac{530}{1.1} + \frac{594}{(1.1)^2} = \frac{530}{1.1} + \frac{594}{1.21} \] \[ PV = 481.81 + 490.90 = 972.71 \] This is not equal to 1000.
The other options will also not yield 1000.
Step 4: Final Answer
By substituting the options into the present value formula, we find that an interest rate of 8% correctly equates the present value of the repayments to the loan amount.
Was this answer helpful?
0
0
Top Questions on SI & CI
- Ramesh bought a mobile from a local store. He paid 1/6 of the price via UPI and 1/3 of the price via cash. He agreed to pay the balance amount a year later. While paying back the balance amount, Ramesh paid 10% interest on the balance amount. If the interest paid was Rs. 6000, what was the original price of the mobile?
- A sum of money triples itself in 8 years at simple interest. In how many years will it become five times itself at the same rate?
- Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually.If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is 25 : 36, then the minimum number of years required for the value of the investment to exceed Rs 20000 is
- Anil invests Rs 22000 for 6 years in a scheme with 4% interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for 5 years, and then reinvests the entire amount he receives at the end of 5 years, for one year at 10% simple interest. If the amounts received by both at the end of 6 years are equal, then the initial investment, in rupees, made by Sunil is
- An amount of Rs 10000 is deposited in bank A for a certain number of years at a simple interest of 5% per annum. On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of 6% per annum. If the interests received from bank A and bank B are in the ratio 10 : 13, then the investment period, in years, in bank A is
View More Questions
Questions Asked in CAT exam
- Arun, Tarun and Varun work for 24, 21 and 15 days respectively and get paid 2160, 2400 and 2160 rupees respectively. They get paid the same even if they work for a partial day. If the work has to be completed within 10 days or less, what is the minimum amount that has to be paid to complete the entire task?
- CAT - 2025
- Time and Work
If \((2m+n) + (2n+m)=27\), find the maximum value of \((2m-3)\), assuming m and n are positive integers.
- CAT - 2025
- Algebra
- A shopkeeper sells an item at a profit of 20%. If he had bought it at 10% less and sold it for 60 more, his profit would become 60%. What is the cost price of the item?
- CAT - 2025
- Profit & Loss
- A container has a mixture of milk and water in the ratio 5 : 2. If 14 liters of this mixture is removed and replaced with water, the ratio becomes 5 : 4. What is the initial quantity of the mixture?
- CAT - 2025
- Ratio and Proportion
- Two circles of radii 6 cm and 6 cm intersect each other. The distance between their centers is 8 cm. What is the length of their common chord?
- CAT - 2025
- Geometry
View More Questions