A shopkeeper promises to sell his goods at 10% less but uses x% less weight thus gains 20%. Find x?

Options

A. `20%`
B. `10%`
C. `15%`
D. `25%`

Answer

To solve the problem step by step, we will analyze the situation of the shopkeeper and calculate the percentage of weight he is actually reducing (x%).

### Step 1: Understand the Selling Price and Profit
The shopkeeper claims to sell his goods at 10% less than the original price. This means if the original price is considered as 100 units, he sells it for:
\[ \text{Selling Price} = 100 - 10\% \text{ of } 100 = 100 - 10 = 90 \text{ units} \]
He also makes a profit of 20%. Therefore, if we assume the cost price (CP) is 100 units, the selling price (SP) can be calculated as:
\[ \text{SP} = \text{CP} + 20\% \text{ of CP} = 100 + 20 = 120 \text{ units} \]

### Step 2: Relate Selling Price to Weight
Since the shopkeeper is selling 90 units (after the 10% reduction) but is actually gaining a profit, we need to find out how much weight he is actually selling.

### Step 3: Calculate the Cost Price per Unit
Given that he is selling 90 units for a selling price of 120 units, we can find the cost price per unit. Let’s denote the cost price for the goods as 100 units. The cost price per unit is:
\[ \text{Cost Price per unit} = \frac{100 \text{ units}}{100} = 1 \text{ unit} \]

### Step 4: Determine the Effective Selling Price per Unit
Now, since he sells 90 units for 120 units, the effective selling price per unit becomes:
\[ \text{Effective SP per unit} = \frac{120 \text{ units}}{90 \text{ units}} = \frac{120}{90} = \frac{4}{3} \text{ units} \]

### Step 5: Calculate the Cost Price for 90 Units
The cost price for 90 units (since the cost price for 100 units is 100 units) is:
\[ \text{Cost Price for 90 units} = \frac{90}{100} \times 100 = 90 \text{ units} \]

### Step 6: Calculate the Gain
The gain can be calculated as:
\[ \text{Gain} = \text{Selling Price} - \text{Cost Price} = 120 - 90 = 30 \text{ units} \]

### Step 7: Determine the Percentage Reduction in Weight
Now, we need to find out how much weight he is actually selling. If he claims to sell 100 units but sells 90 units, the percentage reduction in weight (x%) can be calculated as:
\[ x\% = \frac{100 - 90}{100} \times 100 = 10\% \]

### Step 8: Solve for x
Since we are given that he gains 20% profit, we can equate:
\[ \text{Gain} = \text{Selling Price} - \text{Cost Price} = 30 \text{ units} \]
Using the values we derived, we can set up the equation:
\[ x = 100 - \left(\frac{75}{100} \times 100\right) = 25\% \]

### Final Answer
Thus, the value of x is:
\[ \boxed{25\%} \]