A piece of land came to a person through three middleman each gaining 20%. If the person purchased the land for 3,45,600the original cost of the land was

Options

A. 1,00,000
B. 1,50,000
C. 1,75,800
D. 2,00,000

Answer

To find the original cost of the land before it passed through the three middlemen, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Gain of Each Middleman**:
Each middleman gains 20%. This means that when the land is sold, it is sold for 120% of the cost price. In fraction form, this is:
\[ 120\% = \frac{120}{100} = \frac{6}{5} \]

2. **Calculating the Total Gain Through Three Middlemen**:
Since there are three middlemen, the cost price of the land will be multiplied by \( \left(\frac{6}{5}\right)^3 \) to find the final selling price. This can be expressed as:
\[ \text{Final Selling Price} = x \times \left(\frac{6}{5}\right)^3 \]
where \( x \) is the original cost of the land.

3. **Setting Up the Equation**:
We know the final selling price is 3,45,600. Therefore, we can set up the equation:
\[ x \times \left(\frac{6}{5}\right)^3 = 3,45,600 \]

4. **Calculating \( \left(\frac{6}{5}\right)^3 \)**:
First, calculate \( \left(\frac{6}{5}\right)^3 \):
\[ \left(\frac{6}{5}\right)^3 = \frac{6^3}{5^3} = \frac{216}{125} \]

5. **Substituting Back into the Equation**:
Now substitute this back into the equation:
\[ x \times \frac{216}{125} = 3,45,600 \]

6. **Solving for \( x \)**:
To isolate \( x \), multiply both sides by the reciprocal of \( \frac{216}{125} \):
\[ x = 3,45,600 \times \frac{125}{216} \]

7. **Calculating \( x \)**:
Now perform the multiplication:
\[ x = 3,45,600 \times \frac{125}{216} = 3,45,600 \div 216 \times 125 \]
First, calculate \( 3,45,600 \div 216 \):
\[ 3,45,600 \div 216 = 1,600 \]
Then multiply by 125:
\[ 1,600 \times 125 = 2,00,000 \]

8. **Final Answer**:
Thus, the original cost of the land was:
\[ \boxed{2,00,000} \]