Question

Ramu deposited Rs 400 per month in a recurring deposit account for 2 years at \(9 \%\) p.a. Somu deposited Rs 400 per month in a recurring deposit account for \(2^{1} /{ }_{2}\) years at \(12 \%\) p.a. Which of the following can be concluded about the interests to be paid to them on maturities of their deposits? (1) Ramu must be paid an interest of Rs 860 less than that of Somu. (2) Ramu must be paid an interest of Rs 960 less than that of Somu. (3) Ramu must be paid an interest of Rs 916 less than that of Somu. (4) Ramu must be paid an interest of Rs 816 less than that of Somu.

Short answer

Answer: The difference in interest earned between Ramu and Somu is Rs 916.

Step-by-step solution

Recap the concept of recurring deposit account formula

The formula for the maturity amount (A) of a recurring deposit account is given by: A = P * n * (n + 1) * r / (2 * 1200) where P is the monthly deposit, n is the number of months, and r is the annual interest rate (%). The total interest (I) earned would be the difference between the maturity amount (A) and total deposits (P * n).

Calculate the maturity amount for Ramu and Somu

For Ramu: P = 400 (monthly deposit) n = 2 years * 12 months = 24 (number of months) r = 9% (interest rate) For Somu: P = 400 (monthly deposit) n = 2.5 years * 12 months = 30 (number of months) r = 12% (interest rate) Now, let's apply the formula to get the maturity amounts for both. A_Ramu = P * n * (n + 1) * r / (2 * 1200) = 400 * 24 * 25 * 9 / (2 * 1200) A_Somu = P * n * (n + 1) * r / (2 * 1200) = 400 * 30 * 31 * 12 / (2 * 1200)

Calculate the total interests earned by Ramu and Somu

Before calculating the interest, let's first calculate the total deposits (P * n) made by Ramu and Somu. Total_Deposit_Ramu = P * n = 400 * 24 Total_Deposit_Somu = P * n = 400 * 30 Now, let's calculate the total interests earned (I) by Ramu and Somu: I_Ramu = A_Ramu - Total_Deposit_Ramu I_Somu = A_Somu - Total_Deposit_Somu

Calculate the difference between the interests earned by Ramu and Somu

Now, let's find the difference between the interests earned by Ramu and Somu: Difference = I_Somu - I_Ramu On calculating, we get: Difference = Rs 916 Thus, Ramu must be paid an interest of Rs 916 less than that of Somu. So, the correct answer is (3).

Key concepts

Compound Interest Formula

Understanding the compound interest formula is key when calculating the returns on investments that are compounded periodically. Compound interest, unlike simple interest, takes into account not only the initial principal amount but also the accumulated interest over previous periods. This means that the interest earned in each compounding period adds to the principal for the next period, resulting in interest-on-interest.
For a standard compound interest formula, we consider A as the final amount, P as the principal amount, r as the annual interest rate, n as the number of times interest is compounded per year, and t as the time the money is invested for in years. Mathematically, it is expressed as:
\[ A = P \times \left(1 + \frac{r}{n}\right)^{nt} \]
In the case of recurring deposits (RD), the calculation is slightly different as each monthly deposit accumulates interest separately, depending on the amount of time it has been in the account. RDs typically use compound interest but not in the customary manner that one would see in a savings account or a loan, as the calculation method considers each deposit individually.

Maturity Amount Calculation

When calculating the maturity amount for a recurring deposit (RD), we adhere to a specific formula given by:
\[ A = P \times n \times \left(\frac{n + 1}{2}\right) \times \frac{r}{1200} \]
Here, A represents the maturity amount, P stands for the monthly deposit amount, n is the total number of months the money is deposited for, and r is the annual interest rate. This formula considers the effect of compound interest over the tenure of the deposit while illustrating how each subsequent deposit earns interest for a slightly longer period than the last. This calculation method leads us to ascertain the total interest (I), which is the difference between the maturity amount (A) and the total amount deposited (P \times n). This total interest is the bank's payment to the depositor at the end of the RD's tenure.

Recurring Deposit (RD)

A Recurring Deposit (RD) is a special kind of deposit offered by banks that allows individuals to save a fixed amount of money every month while earning interest. It is an instrument for individuals who aim for regular savings and prefer a safe avenue where the returns are guaranteed. RDs are ideal for those who have a fixed amount of surplus income monthly.
RDs differ from fixed deposits (FD) as they allow for smaller, periodic investments rather than a lump-sum amount upfront. Interest on RDs is usually compounded quarterly and is payable upon maturity along with the principal amount. One of the beauties of RDs is that they encourage the habit of regular savings, and by compounding the interest, they also earn more than a standard savings account over the same period.