To determine the minimum amount Leila should deposit now to achieve her car purchase goal, we need to calculate the present value of her future investment. This problem involves understanding compound interest, where the interest earned is added to the principal, and subsequent interest is calculated on the new, larger principal.

Investment Goal Analysis

Leila's goal is to buy a car worth Rs. 10,00,000 after 5 years. This target amount represents the future value of her investment. The bank offers an annual interest rate of 10%, compounded annually. To find the minimum deposit now, we need to work backward from the future value to find the present value required today.

Compound Interest Formula

The compound interest formula helps us calculate the future value (FV) of an investment given its present value (PV), interest rate (r), and the number of compounding periods (n). The standard formula is:

$$FV = PV \times (1 + r)^n$$

In this problem, we know the future value (FV) and need to find the present value (PV). We can rearrange the formula to solve for PV:

$$PV = \frac{FV}{(1 + r)^n}$$

Identifying Financial Values

Let's list the known values from the problem statement:

• Future Value (FV): The desired amount Leila wants to have after 5 years for her car, which is Rs. 10,00,000.
• Annual Interest Rate (r): The rate offered by the bank, which is 10%. When used in calculations, this is converted to a decimal: 0.10.
• Number of Years (n): The time period over which the investment will grow, which is 5 years.
• Compounding Frequency: The interest is compounded annually, meaning the interest is calculated and added to the principal once a year.

Calculating Present Deposit Amount

Now, let's substitute the identified values into the rearranged present value formula to find the minimum amount Leila should deposit now:

$$PV = \frac{10,00,000}{(1 + 0.10)^5}$$

First, calculate the value of $(1 + r)^n$:

$$ (1 + 0.10)^5 = (1.10)^5 $$

Let's compute $(1.10)^5$ step-by-step:

• $1.10^1 = 1.10$
• $1.10^2 = 1.10 \times 1.10 = 1.21$
• $1.10^3 = 1.21 \times 1.10 = 1.331$
• $1.10^4 = 1.331 \times 1.10 = 1.4641$
• $1.10^5 = 1.4641 \times 1.10 = 1.61051$

Now, substitute this calculated value back into the PV formula:

$$PV = \frac{10,00,000}{1.61051}$$

Performing the division:

$$PV \approx 620921.32$$

Rounding this amount to the nearest whole rupee, Leila should deposit approximately Rs. 6,20,921. Comparing this calculated present value with the given options, the closest amount is Rs. 6,21,000.