Atul is a dishonest shopkeeper uses a faulty weight while selling items to his customers. He sold 'x' grams to Varun instead of 'x + 120' grams and '2x - 1000' grams to Sachin instead of 'x + 340' grams. Then what is the profit percent earned by Atul?

Let's analyze the problem step by step.

### 1. Understanding the Problem
Atul is using a faulty weight, which means he is giving less quantity of the product than he claims. Let's denote the actual price of the product per gram as \( p \) (which we will assume is constant for both transactions).

- **Varun's Transaction:**
  - Atul claims to sell \( x + 120 \) grams but actually sells only \( x \) grams.
  - The revenue Atul earns from Varun = \( (x + 120) \times p \).
  - The actual value of the goods given to Varun = \( x \times p \).

- **Sachin's Transaction:**
  - Atul claims to sell \( x + 340 \) grams but actually sells only \( 2x - 1000 \) grams.
  - The revenue Atul earns from Sachin = \( (x + 340) \times p \).
  - The actual value of the goods given to Sachin = \( (2x - 1000) \times p \).

### 2. Total Revenue and Cost
- **Total Revenue:**
  \[
  \text{Total Revenue} = (x + 120) \times p + (x + 340) \times p = (2x + 460) \times p
  \]

- **Total Cost:**
  \[
  \text{Total Cost} = x \times p + (2x - 1000) \times p = (3x - 1000) \times p
  \]

### 3. Profit Calculation
Profit \( P \) is given by:
\[
P = \text{Total Revenue} - \text{Total Cost}
\]
\[
P = \left[(2x + 460) \times p \right] - \left[(3x - 1000) \times p \right]
\]
\[
P = \left[(2x + 460) - (3x - 1000)\right] \times p
\]
\[
P = \left[-x + 1460\right] \times p
\]
\[
P = (1460 - x) \times p
\]

### 4. Profit Percent Calculation
The profit percent is calculated using the formula:
\[
\text{Profit Percent} = \left(\frac{\text{Profit}}{\text{Total Cost}}\right) \times 100
\]

Substituting the values:
\[
\text{Profit Percent} = \left(\frac{(1460 - x) \times p}{(3x - 1000) \times p}\right) \times 100 = \left(\frac{1460 - x}{3x - 1000}\right) \times 100
\]

### Final Expression
The profit percent earned by Atul is:
\[
\text{Profit Percent} = \left(\frac{1460 - x}{3x - 1000}\right) \times 100
\]

If you have a specific value for \( x \), we can substitute it to get the exact profit percentage. 

Do you need further details or have any questions?

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Here are 5 related questions:
1. What happens to the profit percentage if \( x \) is very large?
2. How would the profit change if Atul increased the faulty weight by a larger amount?
3. What is the break-even point where Atul earns zero profit?
4. How would the profit percentage change if the costs per gram were different for each customer?
5. If Atul's profit margin was capped at 20%, what is the maximum value of \( x \) for Varun?

**Tip:** Always check for hidden factors (like faulty scales) when analyzing profit scenarios in real-life transactions.

Explore how to calculate the profit percent earned by Atul, a dishonest shopkeeper using a faulty weight. Understand the mathematical analysis involving profit, revenue, and cost in the context of Varun and Sachin's transactions.

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