Solving for B's Purchase Price

This problem involves comparing the buying and selling prices of an article between two people, A and B, considering profit percentages and price differences. We need to determine the price at which B bought the article.

Understanding the Transactions

• A's Transaction: A buys an article at a certain price and sells it making a 10% profit.
• B's Transaction: B buys the same article at a price 10% less than A's purchase price. B then sells it ₹18 less than A's selling price, achieving a 20% gain.

Step-by-Step Calculation

Let's use variables to represent the unknown prices:

• Let $P_A$ be the price A paid for the article (A's Cost Price).

Calculating A's Selling Price ($SP_A$)

A sold the article at a 10% profit. The selling price is calculated as:

$SP_A = P_A + (10\% \text{ of } P_A)$
$SP_A = P_A + \frac{10}{100} \times P_A$
$SP_A = P_A + 0.10 P_A$
$SP_A = 1.10 P_A$

Calculating B's Cost Price ($CP_B$)

B bought the article at a price 10% lesser than A's purchase price ($P_A$).

$CP_B = P_A - (10\% \text{ of } P_A)$
$CP_B = P_A - \frac{10}{100} \times P_A$
$CP_B = P_A - 0.10 P_A$
$CP_B = 0.90 P_A$

Calculating B's Selling Price ($SP_B$)

B sold the article at ₹18 lesser than A's selling price ($SP_A$).

$SP_B = SP_A - 18$
Substituting the expression for $SP_A$:
$SP_B = (1.10 P_A) - 18$

Using B's Profit Percentage

B made a 20% gain on his purchase price ($CP_B$). This means:

$SP_B = CP_B + (20\% \text{ of } CP_B)$
$SP_B = CP_B + \frac{20}{100} \times CP_B$
$SP_B = CP_B + 0.20 CP_B$
$SP_B = 1.20 CP_B$

Setting up the Equation to Solve for $P_A$

Now we equate the two expressions for $SP_B$ and substitute the expression for $CP_B$ in terms of $P_A$:

$1.10 P_A - 18 = 1.20 \times (0.90 P_A)$
$1.10 P_A - 18 = 1.08 P_A$

Now, we solve this equation for $P_A$:

$1.10 P_A - 1.08 P_A = 18$
$0.02 P_A = 18$
$P_A = \frac{18}{0.02}$
$P_A = \frac{1800}{2}$
$P_A = 900$

So, A's purchase price was ₹900.

Finding B's Purchase Price

The question asks for the price at which B bought the article ($CP_B$). We calculated this earlier:

$CP_B = 0.90 P_A$
$CP_B = 0.90 \times 900$
$CP_B = 810$

Therefore, B bought the article at ₹810.

Final Answer Check

If $P_A = 900$:

• $SP_A = 1.10 \times 900 = 990$.
• $CP_B = 0.90 \times 900 = 810$.
• $SP_B = SP_A - 18 = 990 - 18 = 972$.
• B's profit = $SP_B - CP_B = 972 - 810 = 162$.
• B's profit percentage = $\frac{\text{Profit}}{CP_B} \times 100 = \frac{162}{810} \times 100 = \frac{1}{5} \times 100 = 20\%$.

This matches the information given in the problem.