Question

Vishal has a recurring deposit account in a finance company for 1 year at \(8 \%\) per annum. If he gets Rs 9390 at the time of maturity, then how much amount per month has been invested by Vishal? (1) Rs 650 (2) Rs 700 (3) Rs 750 (4) Rs 800

Short answer

Answer: (3) Rs 750

Step-by-step solution

Rewrite the formula with given values

First, we will rewrite the formula with given values: M = P*((1 + 0.08/12)^(12*1) + ... + (1 + 0.08/12))

Simplify the formula

Now, we simplify the formula further: M = P*( (1 + 0.08/12)^12 + ... + (1 + 0.08/12) )

Solve for P

We have the maturity amount M = 9390. Now, we put the value of M in the formula and solve for P: 9390 = P*( (1 + 0.08/12)^12 + ... + (1 + 0.08/12) ) To solve for P, we first need to find the sum of the terms in parentheses. Since there are 12 terms in the series, we can use the formula for the sum of a geometric series: S = a*(1 - r^n) / (1 - r) Where S is the sum of the series, a is the first term (1 + 0.08/12), r is the common ratio (1 + 0.08/12), and n is the number of terms (12). S = (1 + 0.08/12)*(1 - (1 + 0.08/12)^12) / (1 - (1 + 0.08/12)) Calculating the sum S, we get approximately S = 12.62 Now, we can solve for P: 9390 = P*12.62 P ≈ 9390 / 12.62 P ≈ 744.21 Among the given options, the closest value to Rs 744.21 is Rs 750. Therefore, the correct answer is (3) Rs 750.

Key concepts

Recurring Deposit Calculation

When someone opens a recurring deposit (RD) account, they agree to deposit a fixed amount of money every month for a certain period. This is a type of term deposit offered by banks that helps people with regular incomes save a small amount of their income every month. The recurring deposit calculation involves understanding how much will be the maturity amount after the specified period.

The formula that connects the monthly installment (P), the rate of interest (r), the number of months (n), and the maturity amount (M) typically considers the effect of compound interest. Since the money is invested at the start of every month, each deposit earns interest for a different amount of time. This is where the sum of a geometric series comes into play, to add up the varying amounts of interest earned on each monthly deposit.

To find the monthly installment amount that Vishal invested, we reverse-engineer the formula using the maturity amount received, the interest rate, and the total duration of the RD. By calculating the sum of the interests earned across all deposits and the principal amount, we determine how much was deposited monthly. This reverse calculation ultimately shows us that Vishal had been depositing Rs 750 per month.

Maturity Amount Formula

The maturity amount of a recurring deposit is the sum total of all the deposits made, along with the interest accrued on these deposits over the duration of the term. The compound interest on RDs is calculated on a quarterly basis and is added to the principal for compounding over the next quarter.

To find the maturity amount (M), a specific formula that accounts for the compounding at regular intervals is applied. In our scenario, the formula transformed into a geometric series enabled us to pinpoint the total amount Vishal received at the end of his RD term. Understanding how the maturity formula works is crucial, as it shows the intricate interplay between the monthly installment, interest rate, and the deposit term. Remember that the maturity formula will incorporate the compound interest formula to find the final sum, which is the maturity amount.

The formula used in the given problem is a derivative of the standard compound interest formula but adapted for the recurring deposit scenario, where the installments are made monthly, and the interest is compounded monthly as well.

Geometric Series Sum

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using the formula \( S = a(1 - r^n) / (1 - r) \) where \( S \) is the sum of the series, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In the context of RD calculations, each monthly investment grows at a rate determined by the interest rate, forming a geometric progression of its future values. The sum of these values, due to compound interest, gives us the total amount upon maturity. Using the geometric series sum formula, we can add up the compound interest from all the individual deposits to arrive at the total interest part of the maturity amount.

For Vishal's RD, the geometric series sum helped us calculate the total amount on which his installment of Rs 750 matured to Rs 9390 over the course of the year, factoring in the 8% annual interest rate compounded monthly.