MBA Nexus - CAT & OMETS 2024, [31-10-2024 08:51]
Q7 A shopkeeper marks up the price of an article by 100% and then offers three successive discounts of a%, b% and c%, where a, b and c are positive integers. If a + b + c = 30, then find the maximum profit that he can earn.

MBA Nexus - CAT & OMETS 2024, [31-10-2024 08:52]
[ Poll : Ans ]
- 40%
- 45.8%
- 70%
- 56.4%     use assumption method

To solve the problem, we can break it down step by step.

1. **Initial Price and Markup**:
   Let the cost price (CP) of the article be \( x \).
   After marking up by 100%, the marked price (MP) becomes:
   \[
   MP = x + 100\% \text{ of } x = 2x
   \]

2. **Discount Calculation**:
   The shopkeeper offers three successive discounts of \( a\% \), \( b\% \), and \( c\% \) such that \( a + b + c = 30\). The selling price (SP) after these discounts can be calculated using the formula for successive discounts:
   \[
   SP = MP \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100})
   \]

3. **Profit Calculation**:
   The profit can be calculated as:
   \[
   Profit = SP - CP
   \]
   To express it as a percentage of the cost price:
   \[
   Profit \% = \frac{SP - CP}{CP} \times 100 = \frac{SP - x}{x} \times 100
   \]

4. **Combining the Equations**:
   Substitute the SP:
   \[
   Profit \% = \frac{(2x \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100})) - x}{x} \times 100
   \]
   Simplifying this:
   \[
   Profit \% = (2 \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100}) - 1) \times 100
   \]

5. **Expressing \( a, b, c \)**:
   Given \( a + b + c = 30 \), we can try different combinations to maximize the profit.

To maximize profit, the discounts should be structured as follows:
- \( a = b = 10\% \) and \( c = 10\% \) 
or 
- \( a = 12\%, b = 12\%, c = 6\% \) 

### Calculation of Profits for Different Combinations

Let's calculate for \( a = 10, b = 10, c = 10 \):
\[
SP = 2x \times (1 - 0.1) \times (1 - 0.1) \times (1 - 0.1) = 2x \times 0.9^3 = 2x \times 0.729 = 1.458x
\]
Thus,
\[
Profit = 1.458x - x = 0.458x
\]
Profit percentage:
\[
Profit \% = \frac{0.458x}{x} \times 100 = 45.8\%
\]

### Conclusion:
Thus, the maximum profit percentage he can earn is **45.8%**.

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How does changing the discount percentages affect the overall profit?
2. What would be the profit percentage if only one discount was applied?
3. How do successive discounts differ from a single discount?
4. Can the shopkeeper still profit if the cost price increases?
5. What is the impact of the initial markup on profit margins?

**Tip:** Always calculate your selling price after discounts to determine the actual profit before making pricing decisions.

A shopkeeper marks up the price of an article by 100% and then offers three successive discounts of a%, b% and c%, where a, b and c are positive integers. If a + b + c = 30, then find the maximum profit that he can earn.

Explore how a shopkeeper can maximize profit by marking up an article by 100% and applying three successive discounts where their sum equals 30%. This problem illustrates key concepts in profit calculation and percentage discounts, suitable for Grades 9-12.

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