Question

Bala made a fixed deposit of Rs 30000 in a bank for two years at R\% pa. under compound interest. The maturity value is \(\mathrm{Rs} 35643\). Find \(\mathrm{R}\) (1) 7 (2) 9 (3) 11 (4) 13

Short answer

Question: Bala deposited Rs 30000 in a bank for two years at an interest rate of R% per annum with compound interest. The maturity value after two years is Rs 35643. Determine the interest rate R. Answer: The interest rate, R, is approximately 9% per annum.

Step-by-step solution

Identification of given values

We have the following information given to us: Principal amount (P) = Rs 30000 Duration (n) = 2 years Maturity value (A) = Rs 35643

Apply the compound interest formula

We use the formula for compound interest: \(A = P(1 + \frac{R}{100})^{n}\) Substitute the given values: \(35643 = 30000(1 + \frac{R}{100})^2\)

Solve for R

Now we will solve the equation to find R: \(\frac{35643}{30000} = (1 + \frac{R}{100})^2\) Take the square root on both sides: \(\sqrt{\frac{35643}{30000}} = 1 + \frac{R}{100}\) Now, subtract 1 from both sides: \(\sqrt{\frac{35643}{30000}} - 1 = \frac{R}{100}\) Now, multiply both sides by 100 to get the value of R: \(100 (\sqrt{\frac{35643}{30000}} - 1) = R\) Calculate the value: \(R \approx 9\)

Answer

So, the interest rate R is approximately 9% per annum. The correct option is (2) 9.

Key concepts

Fixed Deposit

A fixed deposit is a financial investment where you deposit a sum of money in a bank for a fixed period, at a predefined interest rate. This is a safe and popular form of investment due to its stable returns. Let's break it down further:

• Principal Amount: This is the initial sum of money deposited by the investor, which in our example, is Rs 30000.
• Tenure: The fixed period during which the money is locked in. For Bala's investment, it is 2 years.
• Interest Rate: This defines the earnings on the principal during the tenure. Bala's rate is what we need to find out through calculation.

During this period, the interest is compounded, which means it is calculated routinely and added back to the principal, increasing the total balance over time. The secured nature of fixed deposits makes them attractive for risk-averse investors.
By the end of the term, the depositor receives the maturity amount, which is the original investment plus the interest earned.

Interest Rate Calculation

Finding the interest rate in compound interest scenarios involves some mathematics but is straightforward once you know the formula. For compound interest, we use:\[ A = P(1 + \frac{R}{100})^n \]Here's how to go about it:

• Maturity Value (A): This is the total amount after the interest is compounded, which for Bala is Rs 35643.
• Principal (P): The original sum, Rs 30000.
• Interest Rate (R): What we're calculating.
• Number of Compounding Periods (n): For Bala, this is 2 years.

To find R, rearrange the formula and substitute the known values. The calculation involves solving for R through algebraic steps—dividing the maturity value by the principal, taking the square root for 2 years, then manipulating the equation to isolate R.
Bala's step-by-step worked problem highlights how this process is carried out, ultimately leading to an interest rate of approximately 9% per annum.

Mathematics Problems

Mathematics problems frequently require comprehension of concepts and careful calculation, especially in interest-related problems. Here is what you should focus on: - Understand the problem entirely: Identify given values and the unknown that needs to be solved. For Bala, we have the principal, time duration, and maturity value. - Use the appropriate formulae: For compound interest, use the standard formula that relates maturity value, principal, rate, and time. - Perform algebraic manipulations: Rearrange the equation logically to isolate and solve for the unknown, such as the interest rate (R) in Bala's case. Mathematics problems often require incremental solving steps, checking each calculation, and being careful with units and conversions if any. Mistakes can easily occur without careful attention at each step.

Financial Mathematics

Financial mathematics involves using mathematical methods to solve problems related to money, finance, and investments. Understanding concepts like compound interest is vital. Financial mathematics includes:

• Investment Analysis: Evaluating different investment options based on returns and risks. Fixed deposits provide a low-risk option.
• Rate of Interest: Different financial products offer varying interest rates. A good understanding helps choose the best investment.
• Compounding: As in compound interest, understanding how interest adds up and grows wealth over time.

In Bala's example, mastering financial mathematics allowed us to determine the interest rate, influencing investment decisions. These calculations guide understanding market trends and personal finance management by knowing how money grows over time using the right financial tools.