A sells an article to B at profit of 20% . B sells it to C at 10% profit and C sells it to D at `12(1)/(2)%` profit. If D has paid Rs 29.70 then at what price A had purchased it ?

Answer

To find out at what price A had purchased the article, we will work backwards from the final selling price paid by D. Let's break it down step by step.

### Step 1: Understand the Selling Price from D
D pays Rs 29.70 for the article. This price includes a profit of 12.5% (which is the same as 12(1/2)%).

### Step 2: Calculate the Cost Price for C
Let the cost price for C be \( CP_C \).
The selling price (SP) for C is given by:
\[ SP_C = CP_C + 12.5\% \text{ of } CP_C \]
\[ SP_C = CP_C \times \left(1 + \frac{12.5}{100}\right) \]
\[ SP_C = CP_C \times 1.125 \]

Since we know that \( SP_C = 29.70 \):
\[ 29.70 = CP_C \times 1.125 \]

Now, we can solve for \( CP_C \):
\[ CP_C = \frac{29.70}{1.125} \]
\[ CP_C = 26.40 \]

### Step 3: Calculate the Cost Price for B
Let the cost price for B be \( CP_B \).
B sells to C at a profit of 10%.
The selling price for B is given by:
\[ SP_B = CP_B + 10\% \text{ of } CP_B \]
\[ SP_B = CP_B \times \left(1 + \frac{10}{100}\right) \]
\[ SP_B = CP_B \times 1.10 \]

Since we know that \( SP_B = CP_C = 26.40 \):
\[ 26.40 = CP_B \times 1.10 \]

Now, we can solve for \( CP_B \):
\[ CP_B = \frac{26.40}{1.10} \]
\[ CP_B = 24.00 \]

### Step 4: Calculate the Cost Price for A
Let the cost price for A be \( CP_A \).
A sells to B at a profit of 20%.
The selling price for A is given by:
\[ SP_A = CP_A + 20\% \text{ of } CP_A \]
\[ SP_A = CP_A \times \left(1 + \frac{20}{100}\right) \]
\[ SP_A = CP_A \times 1.20 \]

Since we know that \( SP_A = CP_B = 24.00 \):
\[ 24.00 = CP_A \times 1.20 \]

Now, we can solve for \( CP_A \):
\[ CP_A = \frac{24.00}{1.20} \]
\[ CP_A = 20.00 \]

### Conclusion
Thus, A had purchased the article for Rs 20.00.

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