A shopkeeper promise to sell his goods at x% profit but he uses 20% less weight thus gain 37`1/2`%. Find x.

Options

A. A. 0.1
B. B. 0.09
C. C. 0.11
D. D. 0.15

Answer

To solve the problem step by step, we need to find the value of \( x \) given that a shopkeeper promises to sell his goods at \( x\% \) profit but actually gains \( 37.5\% \) profit by using \( 20\% \) less weight.

### Step 1: Understand the relationship between cost price, selling price, and profit.
The profit percentage can be expressed as:
\[ \text{Profit} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100 \]

### Step 2: Define the variables.
Let:
- \( p \) = cost price
- Selling Price (SP) when selling at \( x\% \) profit:
\[ SP = p \left(1 + \frac{x}{100}\right) \]

### Step 3: Calculate the effective cost price when using less weight.
Since the shopkeeper uses \( 20\% \) less weight, the effective cost price becomes:
\[ \text{Effective Cost Price} = p \times (1 - 0.20) = 0.8p \]

### Step 4: Set up the equation for actual profit.
The actual profit gained is given as \( 37.5\% \). Thus, we can set up the equation:
\[ \frac{SP - \text{Effective Cost Price}}{\text{Effective Cost Price}} \times 100 = 37.5 \]

Substituting the values we have:
\[ \frac{p \left(1 + \frac{x}{100}\right) - 0.8p}{0.8p} \times 100 = 37.5 \]

### Step 5: Simplify the equation.
Cancelling \( p \) from the equation (assuming \( p \neq 0 \)):
\[ \frac{1 + \frac{x}{100} - 0.8}{0.8} \times 100 = 37.5 \]
\[ \frac{0.2 + \frac{x}{100}}{0.8} \times 100 = 37.5 \]
\[ \frac{0.2 + \frac{x}{100}}{0.8} = 0.375 \]

### Step 6: Cross-multiply to eliminate the fraction.
\[ 0.2 + \frac{x}{100} = 0.375 \times 0.8 \]
Calculating \( 0.375 \times 0.8 \):
\[ 0.375 \times 0.8 = 0.3 \]
So we have:
\[ 0.2 + \frac{x}{100} = 0.3 \]

### Step 7: Isolate \( x \).
Subtract \( 0.2 \) from both sides:
\[ \frac{x}{100} = 0.3 - 0.2 \]
\[ \frac{x}{100} = 0.1 \]
Multiplying both sides by \( 100 \):
\[ x = 10 \]

### Final Answer:
The value of \( x \) is \( 10\% \).

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