A trader earns a `25%` profit by selling his goods at the cost price while using a faulty balance. How much per cent does his balance weigh less than what it should weigh?

Options

A. 25
B. 20
C. 16.67
D. 22.22

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understanding the Profit
The trader earns a profit of 25% by selling goods at the cost price. This means that the selling price (SP) is 125% of the cost price (CP).

### Step 2: Setting Up the Equation
Let’s denote the cost price as \( CP \). Then, the selling price can be expressed as:
\[ SP = CP + 0.25 \times CP = 1.25 \times CP \]

### Step 3: Using the Faulty Balance
The trader uses a faulty balance, which means he is selling less quantity than he claims. Let’s denote the actual weight he should sell as \( W \) and the weight he actually sells as \( W' \). Since he claims to sell the goods at cost price, we can assume:
\[ W' = W - x \quad \text{(where \( x \) is the weight he is shorting)} \]

### Step 4: Relating Profit to Weight
Since he makes a 25% profit, we can set up the following relationship based on the weights:
\[ \frac{SP}{CP} = \frac{W'}{W} = \frac{125}{100} = \frac{5}{4} \]

### Step 5: Cross Multiplication
Cross multiplying gives us:
\[ 5W = 4W' \]
Substituting \( W' \) from our earlier expression:
\[ 5W = 4(W - x) \]

### Step 6: Simplifying the Equation
Expanding the equation:
\[ 5W = 4W - 4x \]
Rearranging gives:
\[ 5W - 4W = -4x \implies W = -4x \]

### Step 7: Finding the Percentage Shortage
The percentage shortage can be calculated as:
\[ \text{Percentage Shortage} = \left(\frac{x}{W}\right) \times 100 \]
Since \( W = 4x \), substituting gives:
\[ \text{Percentage Shortage} = \left(\frac{x}{4x}\right) \times 100 = \frac{1}{4} \times 100 = 25\% \]

### Conclusion
Thus, the balance weighs 25% less than what it should weigh.