When an article is sold at 40% discount, a loss of 10% occurs but when it sold at 20% discount then there is a profit of Rs.7.5. What is the cost price of the article?

Options

A. A)37.5Rs
B. B)40Rs
C. C)66.67 Rs
D. D)48 Rs

Answer

To find the cost price of the article based on the given conditions, we can follow these steps:

### Step 1: Understand the relationship between Cost Price (CP), Marked Price (MP), and Selling Price (SP)
When an article is sold at a discount, the selling price can be calculated using the marked price and the discount percentage. The relationship can be expressed as:
\[ SP = MP \times \left(1 - \frac{\text{Discount}}{100}\right) \]

### Step 2: Set up the equations based on the discounts
1. **For a 40% discount with a 10% loss:**
- Loss means that the selling price is less than the cost price.
- Therefore, we can express this as:
\[ SP_1 = CP \times \left(1 - \frac{10}{100}\right) = CP \times 0.9 \]
- The selling price with a 40% discount:
\[ SP_1 = MP \times (1 - 0.4) = MP \times 0.6 \]

2. **For a 20% discount with a profit of Rs. 7.5:**
- Profit means that the selling price is more than the cost price.
- Therefore, we can express this as:
\[ SP_2 = CP + 7.5 \]
- The selling price with a 20% discount:
\[ SP_2 = MP \times (1 - 0.2) = MP \times 0.8 \]

### Step 3: Establish the relationship between CP and MP
From the first case, we have:
\[ MP \times 0.6 = CP \times 0.9 \]
Rearranging gives us:
\[ \frac{CP}{MP} = \frac{0.6}{0.9} = \frac{2}{3} \]
This means:
\[ CP = \frac{2}{3} MP \]

### Step 4: Substitute CP in the second equation
Now, substitute \( CP \) in the second equation:
\[ SP_2 = CP + 7.5 \]
\[ MP \times 0.8 = \frac{2}{3} MP + 7.5 \]

### Step 5: Solve for MP
Rearranging the equation:
\[ MP \times 0.8 - \frac{2}{3} MP = 7.5 \]
To combine the terms, we need a common denominator:
\[ \frac{8}{10} MP - \frac{2}{3} MP = 7.5 \]
Converting \( \frac{8}{10} \) to a fraction with a denominator of 30:
\[ \frac{24}{30} MP - \frac{20}{30} MP = 7.5 \]
This simplifies to:
\[ \frac{4}{30} MP = 7.5 \]
Multiplying both sides by 30:
\[ 4 MP = 225 \]
Dividing by 4:
\[ MP = 56.25 \]

### Step 6: Find the Cost Price (CP)
Now that we have the marked price, we can find the cost price:
\[ CP = \frac{2}{3} MP = \frac{2}{3} \times 56.25 = 37.5 \]

### Final Answer
The cost price of the article is **Rs. 37.5**.

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