When an article is sold for `Rs. 768`, the profit percentage is `x%`. When the same article is sold for `Rs. 896`, the profit percentage is `(x+20)%` . What is the value of `x` ?

Options

A. 30
B. 25
C. 15
D. 20

Answer

To solve the problem step by step, we will use the formula for profit percentage and set up equations based on the given selling prices and profit percentages.

### Step 1: Define the Variables
Let the Cost Price (CP) of the article be \( CP \).

### Step 2: Set Up the First Equation
When the article is sold for Rs. 768, the profit percentage is \( x\% \).
Using the profit percentage formula:
\[ \text{Profit Percentage} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100 \]
We can write the first equation as:
\[ x = \frac{768 - CP}{CP} \times 100 \]
Rearranging gives:
\[ \frac{768 - CP}{CP} = \frac{x}{100} \]
Multiplying both sides by \( CP \times 100 \):
\[ 768 - CP = \frac{x}{100} \times CP \]
Thus, we have:
\[ 768 = CP + \frac{x}{100} \times CP \]
This can be simplified to:
\[ 768 = CP \left(1 + \frac{x}{100}\right) \]
(Equation 1)

### Step 3: Set Up the Second Equation
When the article is sold for Rs. 896, the profit percentage is \( (x + 20)\% \).
Using the same profit percentage formula, we can write the second equation as:
\[ x + 20 = \frac{896 - CP}{CP} \times 100 \]
Rearranging gives:
\[ \frac{896 - CP}{CP} = \frac{x + 20}{100} \]
Multiplying both sides by \( CP \times 100 \):
\[ 896 - CP = \frac{x + 20}{100} \times CP \]
Thus, we have:
\[ 896 = CP + \frac{x + 20}{100} \times CP \]
This can be simplified to:
\[ 896 = CP \left(1 + \frac{x + 20}{100}\right) \]
(Equation 2)

### Step 4: Solve the Two Equations
Now we have two equations:
1. \( 768 = CP \left(1 + \frac{x}{100}\right) \)
2. \( 896 = CP \left(1 + \frac{x + 20}{100}\right) \)

We can express \( CP \) from both equations:
From Equation 1:
\[ CP = \frac{768}{1 + \frac{x}{100}} \]
From Equation 2:
\[ CP = \frac{896}{1 + \frac{x + 20}{100}} \]

Setting these two expressions for \( CP \) equal to each other:
\[ \frac{768}{1 + \frac{x}{100}} = \frac{896}{1 + \frac{x + 20}{100}} \]

### Step 5: Cross-Multiply and Simplify
Cross-multiplying gives:
\[ 768 \left(1 + \frac{x + 20}{100}\right) = 896 \left(1 + \frac{x}{100}\right) \]
Expanding both sides:
\[ 768 + \frac{768(x + 20)}{100} = 896 + \frac{896x}{100} \]
Multiplying through by 100 to eliminate the fractions:
\[ 76800 + 768(x + 20) = 89600 + 896x \]
Distributing:
\[ 76800 + 768x + 15360 = 89600 + 896x \]
Combining like terms:
\[ 92160 + 768x = 89600 + 896x \]
Rearranging gives:
\[ 92160 - 89600 = 896x - 768x \]
\[ 2560 = 128x \]
Dividing both sides by 128:
\[ x = \frac{2560}{128} = 20 \]

### Final Answer
The value of \( x \) is \( 20\% \).