An article was sold at `11(1)/(9)%` profit. Had it purchased at Rs 1300 less and sold at Rs 3000 less there would have been loss of `9(1)/(11)%` Find C.P. of article?

Answer

To solve the problem, we need to find the cost price (C.P.) of the article based on the given profit and loss percentages. Let's break it down step by step.

### Step 1: Convert the given profit and loss percentages into fractions.
The profit percentage is given as \( 11\frac{1}{9}\% \). We can convert this into a fraction:
\[ 11\frac{1}{9}\% = \frac{100}{9}\% \]
This means that the selling price (S.P.) is \( \frac{100 + 100/9}{100} \) times the cost price (C.P.).

Calculating this:
\[ \text{Profit} = \frac{100}{9}\% \Rightarrow \text{S.P.} = C.P. \times \left(1 + \frac{100}{900}\right) = C.P. \times \frac{109}{100} \]

### Step 2: Set up the equation for the selling price.
Let the cost price be \( x \). The selling price can be expressed as:
\[ \text{S.P.} = x \times \frac{109}{100} \]

### Step 3: Calculate the new cost price and selling price when the adjustments are made.
If the article was purchased at Rs 1300 less, the new cost price would be:
\[ \text{New C.P.} = x - 1300 \]
If it was sold at Rs 3000 less, the new selling price would be:
\[ \text{New S.P.} = x \times \frac{109}{100} - 3000 \]

### Step 4: Set up the equation for the loss percentage.
The loss percentage is given as \( 9\frac{1}{11}\% \). We convert this into a fraction:
\[ 9\frac{1}{11}\% = \frac{100}{11}\% \]
This means that the new selling price is \( \frac{100 - 100/11}{100} \) times the new cost price.

Calculating this:
\[ \text{Loss} = \frac{100}{11}\% \Rightarrow \text{New S.P.} = \text{New C.P.} \times \left(1 - \frac{100}{1100}\right) = \text{New C.P.} \times \frac{90}{100} \]

### Step 5: Set up the equation based on the loss.
From the loss percentage, we have:
\[ x \times \frac{109}{100} - 3000 = (x - 1300) \times \frac{90}{100} \]

### Step 6: Solve the equation.
Expanding both sides:
\[ x \times \frac{109}{100} - 3000 = \frac{90x - 117000}{100} \]
Multiplying through by 100 to eliminate the fraction:
\[ 109x - 300000 = 90x - 117000 \]
Rearranging gives:
\[ 109x - 90x = 300000 - 117000 \]
\[ 19x = 183000 \]
\[ x = \frac{183000}{19} = 9631.58 \]

### Step 7: Conclusion
Thus, the cost price (C.P.) of the article is approximately Rs 9631.58.