Question

Jose borrowed some money from his friend at a simple interest rate of 10% and invested the entire amount in stocks. At the end of the first year, he repaid 1/5th of the principal amount. At the end of the second year, he repaid half of the remaining principal amount. At the end of third year, he repaid the entire remaining principal amount. At the end of the fourth year, he paid the last three years’ interest amount. As there was no principal amount left, his friend did not charge any interest in the fourth year. At the end of fourth year, he sold out all his stocks. Later, he calculated that he gained Rs. 97500 after paying principal and interest amounts to his friend. If his invested amount in the stocks became double at the end of the fourth year, how much money did he borrow from his friend?

• A. 250000
• B. 200000
• C. 150000
• D. 125000
• E. None of the above

Answer 1

The Correct Option is D

Let's solve the problem step-by-step:

1. Let the borrowed amount be \(P\) Rs.
2. The simple interest rate is 10% per annum.
3. Interest for the first year = \(\frac{10}{100} \times P = 0.1P\).
4. At the end of the first year, Jose repaid \(\frac{1}{5}\) of the principal amount, which is \(\frac{P}{5}\).
5. Remaining principal after the first year = \(P - \frac{P}{5} = \frac{4P}{5}\).
6. Interest for the second year is calculated on \(\frac{4P}{5}\):
   Interest = \(\frac{10}{100} \times \frac{4P}{5} = \frac{0.4P}{5} = 0.08P\).
7. At the end of the second year, Jose repaid half of the remaining principal amount:
   Repayment = \(\frac{1}{2} \times \frac{4P}{5} = \frac{2P}{5}\).
8. Remaining principal after the second year = \(\frac{4P}{5} - \frac{2P}{5} = \frac{2P}{5}\).
9. Interest for the third year is calculated on \(\frac{2P}{5}\):
   Interest = \(\frac{10}{100} \times \frac{2P}{5} = \frac{0.2P}{5} = 0.04P\).
10. At the end of the third year, Jose repaid the entire remaining principal:
    Complete repayment = \(\frac{2P}{5}\).
11. At the end of the fourth year, he paid the last three years’ interest. Total interest = \(0.1P + 0.08P + 0.04P = 0.22P\).
12. According to the problem, the invested amount in stocks became double at the end of the fourth year.
    Hence, the stock amount at the end of the fourth year is \(2P\).
13. Jose gained Rs. 97500 after paying principal and interest, implying:
    \(2P - P - 0.22P = 97500\)
    Simplifying gives: \(0.78P = 97500\).
14. Solving for \(P\):
    \(P = \frac{97500}{0.78} = 125000\).

Thus, the amount Jose borrowed from his friend is Rs. 125,000.

Answer 2

Let the amount Jose borrowed be P. The simple interest (SI) for one year is calculated as follows:

SI = P × 10% = 0.1P

 

The events occur as follows:

1. At the end of the first year, Jose repaid 1/5th of the principal amount:Remaining principal = P - (P/5) = 4P/5

The interest for the first year is: 0.1P

 

1. At the end of the second year, Jose repaid half of the remaining principal amount:Remaining principal = 4P/5 - (1/2)×(4P/5) = 2P/5

The interest for the second year is: 0.1 × (4P/5) = 0.08P

 

1. At the end of the third year, Jose repaid the entire remaining principal:Remaining principal = 2P/5 - 2P/5 = 0

The interest for the third year is: 0.1 × (2P/5) = 0.04P

 

1. At the end of the fourth year, Jose paid the last three years’ interest amount:Total interest paid ={0.1P + 0.08P + 0.04P} = 0.22P

Since Jose sold his stocks for double the amount he invested:
Investment = P 
Stock value = 2P

Jose's net gain after paying principal and interest: 
(2P) - (P) - (0.22P) = P - 0.22P = 0.78P

Given that Jose gained Rs. 97500:
0.78P = 97500

 

P = 97500 / 0.78

 

P = 125000

 

Thus, the amount Jose borrowed from his friend is Rs. 125000.