A sold a watch to B at 20% gain and B sold it to C at a loss of 10%. If C bought the watch for Rs 216, at what price did A purchase it?

Answer

To find out at what price A purchased the watch, we can break down the problem step by step.

### Step 1: Define the Cost Price of A
Let the cost price of the watch for A be Rs. X.

### Step 2: Calculate Selling Price of A
A sold the watch to B at a gain of 20%. Therefore, the selling price (SP) for A can be calculated as:
\[ SP_A = X + 20\% \text{ of } X = X + \frac{20}{100} \times X = X + \frac{1}{5}X = \frac{6X}{5} \]

### Step 3: Calculate Cost Price of B
Now, B sold the watch to C at a loss of 10%. The selling price for B can be calculated as:
\[ SP_B = \frac{6X}{5} - 10\% \text{ of } \frac{6X}{5} = \frac{6X}{5} - \frac{10}{100} \times \frac{6X}{5} = \frac{6X}{5} - \frac{1}{10} \times \frac{6X}{5} \]
Calculating \(10\% \text{ of } \frac{6X}{5}\):
\[ = \frac{6X}{5} - \frac{6X}{50} = \frac{6X}{5} - \frac{3X}{25} \]
To subtract these fractions, we need a common denominator:
\[ = \frac{30X}{25} - \frac{3X}{25} = \frac{27X}{25} \]

### Step 4: Set Selling Price of C
We know that C bought the watch for Rs. 216. Therefore, we can set up the equation:
\[ \frac{27X}{25} = 216 \]

### Step 5: Solve for X
To find X, we multiply both sides by 25:
\[ 27X = 216 \times 25 \]
Calculating \(216 \times 25\):
\[ 216 \times 25 = 5400 \]
Now, divide both sides by 27:
\[ X = \frac{5400}{27} \]
Calculating \(5400 \div 27\):
\[ X = 200 \]

### Conclusion
Thus, the cost price at which A purchased the watch is Rs. 200.

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