A man buy a scooty & paid a cash dawn pay ment for 12,000 & he promise to shopkeeper that he will pay₹ 13,050 after one year and ₹ 23,490 after 2 years at the rate of `12(1)/(2)% `of C.I. Find the amount.

Options

A. ₹41,520
B. ₹43,160
C. ₹42,160
D. ₹41,000

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Problem
A man buys a scooty and makes a down payment of ₹12,000. He promises to pay ₹13,050 after one year and ₹23,490 after two years at a compound interest rate of 12.5% (which is equivalent to \( \frac{1}{8} \)).

### Step 2: Define the Total Amount
Let the total amount (cost of the scooty) be \( X \).

### Step 3: Calculate the Remaining Amount After Down Payment
After paying ₹12,000, the remaining amount to be paid is:
\[ X - 12000 \]

### Step 4: Calculate the Amount After One Year
The amount remaining after one year, considering the interest, is given by:
\[ \text{Amount after 1 year} = (X - 12000) \times \left(1 + \frac{12.5}{100}\right) = (X - 12000) \times \frac{9}{8} \]

### Step 5: Subtract the First Payment
After one year, he pays ₹13,050, so we have:
\[ \text{Remaining amount after first payment} = (X - 12000) \times \frac{9}{8} - 13050 \]

### Step 6: Calculate the Amount After Second Year
This remaining amount will again accrue interest for one more year:
\[ \text{Amount after 2 years} = \left((X - 12000) \times \frac{9}{8} - 13050\right) \times \left(1 + \frac{12.5}{100}\right) = \left((X - 12000) \times \frac{9}{8} - 13050\right) \times \frac{9}{8} \]

### Step 7: Set the Equation for the Second Payment
This amount should equal the second payment of ₹23,490:
\[ \left((X - 12000) \times \frac{9}{8} - 13050\right) \times \frac{9}{8} = 23490 \]

### Step 8: Simplify the Equation
Now, we will simplify the equation:
1. Multiply both sides by 8 to eliminate the fraction:
\[ 9 \left((X - 12000) \times \frac{9}{8} - 13050\right) = 23490 \times 8 \]
\[ 9 \left((X - 12000) \times 9 - 104400\right) = 187920 \]
2. Divide both sides by 9:
\[ (X - 12000) \times 9 - 104400 = 20880 \]
3. Rearranging gives:
\[ (X - 12000) \times 9 = 20880 + 104400 \]
\[ (X - 12000) \times 9 = 125280 \]
4. Divide by 9:
\[ X - 12000 = \frac{125280}{9} = 13920 \]
5. Finally, add ₹12,000 to both sides to find \( X \):
\[ X = 13920 + 12000 = 25920 \]

### Conclusion
The total amount (cost of the scooty) is ₹25,920.

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