A dishonest dealer defrauds to the extent of x % in buying as well as selling his goods by using faulty weight. What will be the gain percent on his outlay?

Options

A. ( 2x%)
B. `(10/x+x^2)`
C. `(2x+x^2/100)`%
D. `(x+x^(2)/100)`

Answer

To find the gain percent on the outlay for a dishonest dealer who defrauds by x% in both buying and selling, we can use the formula for successive percentage changes. Here’s the step-by-step solution:

### Step 1: Understand the Problem
The dealer defrauds by x% when buying and also by x% when selling. This means that he effectively gains from both transactions.

### Step 2: Use the Successive Change Formula
The formula for calculating the overall gain percent when there are two successive changes is given by:
\[ \text{Gain Percent} = a + b + \frac{a \times b}{100} \]
where:
- \(a\) is the gain percent in the first transaction (buying),
- \(b\) is the gain percent in the second transaction (selling).

### Step 3: Substitute Values
In this case, both \(a\) and \(b\) are equal to \(x\) (the percentage by which the dealer defrauds):
\[ \text{Gain Percent} = x + x + \frac{x \times x}{100} \]

### Step 4: Simplify the Expression
Combine the terms:
\[ \text{Gain Percent} = 2x + \frac{x^2}{100} \]

### Step 5: Final Expression
Thus, the gain percent on his outlay is:
\[ \text{Gain Percent} = 2x + \frac{x^2}{100} \]

### Summary
The gain percent on the outlay for the dishonest dealer who defrauds by x% in both buying and selling is given by the formula:
\[ \text{Gain Percent} = 2x + \frac{x^2}{100} \]

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