Ajay closes his account in an investment scheme of HDFC bank by withdrawing Rs. 10000/-. One year ago he had withdrawn 6000/-. Two years ago he had withdrawn Rs.
5000/-. Three years ago he had not withdrawn any money How much money had he deposited approximately at the time of opening the account 4 years ago, if the annual
interest is 10%.
 
A. 15700 
B. 18,500 
C. 15,500 
D. None of these

To solve this problem, we need to determine the initial deposit amount made by Ajay four years ago, given that he has withdrawn various amounts over the years and the interest rate is 10% per annum.

### Let's Analyze the Withdrawals:

1. **Withdrawal after 3 years**: Rs. 5000 (withdrawn 2 years ago)
2. **Withdrawal after 4 years**: Rs. 6000 (withdrawn 1 year ago)
3. **Final Withdrawal after 5 years**: Rs. 10000 (withdrawn now)

### Step-by-Step Calculation:

We need to find the initial deposit, \( P \), such that the amount grows with interest to meet the total withdrawals Ajay made over time.

Let's denote:

- \( P \) = Initial deposit
- \( r = 10\% = 0.1 \) (interest rate per annum)
- Interest compounds annually

The amount Ajay would have after 3 years, considering he did not withdraw anything until then, can be expressed as:

\[
A_3 = P \times (1 + r)^3
\]

#### Calculating Withdrawals:

Ajay withdrew money in the following order:

1. **After 3 years (2 years ago)**:  
   Withdrawal: Rs. 5000  
   This amount would have earned interest for 2 years.

   So, the effective amount at the time of closing after 5 years is:

\[
5000 \times (1 + r)^2 = 5000 \times 1.1^2 = 5000 \times 1.21 = 6050
\]

2. **After 4 years (1 year ago)**:  
   Withdrawal: Rs. 6000  
   This amount would have earned interest for 1 year.

   So, the effective amount at the time of closing after 5 years is:

\[
6000 \times (1 + r) = 6000 \times 1.1 = 6600
\]

3. **At the end of 5 years (now)**:  
   Withdrawal: Rs. 10000  
   This amount is already the final withdrawal.

   So, the total amount at the end of the 5 years would be:

\[
10000
\]

#### Total Amount Required:

The total accumulated amount that must be available at the end of 5 years to cover all these withdrawals is:

\[
\text{Total amount} = 6050 + 6600 + 10000 = 22650
\]

#### Find the Initial Deposit \( P \):

Now, we know that after 3 years, the accumulated amount \( A_3 \) should be:

\[
A_3 = P \times (1 + r)^3 = 22650
\]

Solving for \( P \):

\[
P = \frac{22650}{(1.1)^3}
\]

Let's calculate this value:

\[
P \approx \frac{22650}{1.331} \approx 17015.77
\]

### Approximation to the Closest Option:

The closest option to Rs. 17015.77 is **Rs. 18,500 (Option B)**.

### Conclusion:

**Answer: B. 18,500**

Would you like further details, or do you have any questions?

### Additional Questions to Expand Understanding:

1. How would the solution change if the interest rate was compounded semi-annually?
2. What would the initial deposit be if Ajay had withdrawn different amounts at different intervals?
3. How does compound interest differ from simple interest in such problems?
4. What formula would we use if interest were compounded monthly instead of annually?
5. How can we set up an equation to determine the exact time at which the total withdrawal amount exceeds the deposit?

### Tip:

Always check the compounding frequency in interest rate problems, as it affects the total amount differently depending on whether interest compounds annually, semi-annually, or monthly.

Ajay closes his account in an investment scheme of HDFC bank by withdrawing Rs. 10000/-. One year ago he had withdrawn 6000/-. Two years ago he had withdrawn Rs. 5000/-. Three years ago he had not withdrawn any money. How much money had he deposited approximately at the time of opening the account 4 years ago, if the annual interest is 10%. A. 15700 B. 18,500 C. 15,500 D. None of these

Learn how to determine Ajay's initial deposit in an investment scheme with a 10% annual interest rate. This problem involves calculating compound interest and adjusting for withdrawals over time. Suitable for students in Grades 10-12.

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