Marked price of an article is 40% above its cost price and when shopkeeper allows d% discount then shopkeeper make a profit of ___% and where shopkeeper allows '2d'% discount then he make a profit of ____%. Which of the following options possible to fill both respective blanks?
(A) (22.5,5) (B) (33,26) (C) (19,2) (D) (26, 12) (E) (23.2, 6.4)

Options

A. Only option A, B, C possible
B. Only option B, D & E possible
C. Only option A,B, & E possible
D. All of above options are possible

Answer

To solve the problem step by step, we need to analyze the information given in the question.

### Step 1: Understand the Marked Price and Cost Price
- Let the Cost Price (CP) of the article be \(100\).
- Since the Marked Price (MP) is 40% above the CP, we can calculate it as follows:
\[ MP = CP + 40\% \text{ of } CP = 100 + 40 = 140 \]

### Step 2: Determine the Selling Price for Profit of \(d\%\)
- When the shopkeeper allows a discount of \(d\%\), the Selling Price (SP) can be expressed as:
\[ SP = MP - d\% \text{ of } MP = 140 - \frac{d}{100} \times 140 \]
This simplifies to:
\[ SP = 140 \left(1 - \frac{d}{100}\right) \]

### Step 3: Calculate Profit Percentage for \(d\%\) Discount
- The profit made when selling at this price is given by:
\[ \text{Profit} = SP - CP = 140 \left(1 - \frac{d}{100}\right) - 100 \]
- The profit percentage can be calculated as:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{CP} \times 100 = \frac{140 \left(1 - \frac{d}{100}\right) - 100}{100} \times 100 \]
Simplifying:
\[ \text{Profit Percentage} = 140 \left(1 - \frac{d}{100}\right) - 100 \]
\[ = 40 - \frac{140d}{100} \]
\[ = 40 - 1.4d \]

### Step 4: Determine Selling Price for Profit of \(2d\%\)
- When the shopkeeper allows a discount of \(2d\%\), the Selling Price becomes:
\[ SP = 140 \left(1 - \frac{2d}{100}\right) \]
- The profit made in this case is:
\[ \text{Profit} = 140 \left(1 - \frac{2d}{100}\right) - 100 \]
- The profit percentage is:
\[ \text{Profit Percentage} = \frac{140 \left(1 - \frac{2d}{100}\right) - 100}{100} \times 100 \]
Simplifying:
\[ = 140 \left(1 - \frac{2d}{100}\right) - 100 \]
\[ = 40 - 2.8d \]

### Step 5: Set Up the Equations
- From the problem, we know:
\[ 40 - 1.4d = \text{Profit Percentage for } d\% \]
\[ 40 - 2.8d = \text{Profit Percentage for } 2d\% \]

### Step 6: Solve for \(d\) and Profit Percentages
- Let's denote the profit percentages as \(x\) and \(y\):
\[ x = 40 - 1.4d \]
\[ y = 40 - 2.8d \]
- We can express \(d\) in terms of \(x\):
\[ d = \frac{40 - x}{1.4} \]
- Substitute \(d\) into the equation for \(y\):
\[ y = 40 - 2.8 \left(\frac{40 - x}{1.4}\right) \]
Simplifying this gives:
\[ y = 40 - 2 \times (40 - x) = 40 - 80 + 2x = 2x - 40 \]

### Step 7: Check Options
We need to find pairs \((x, y)\) from the options provided that satisfy the equations derived.

1. **Option A: (22.5, 5)**
- \(y = 2(22.5) - 40 = 45 - 40 = 5\) (Valid)

2. **Option B: (33, 26)**
- \(y = 2(33) - 40 = 66 - 40 = 26\) (Valid)

3. **Option C: (19, 2)**
- \(y = 2(19) - 40 = 38 - 40 = -2\) (Invalid)

4. **Option D: (26, 12)**
- \(y = 2(26) - 40 = 52 - 40 = 12\) (Valid)

5. **Option E: (23.2, 6.4)**
- \(y = 2(23.2) - 40 = 46.4 - 40 = 6.4\) (Valid)

### Conclusion
The valid options that satisfy both blanks are:
- **Option A: (22.5, 5)**
- **Option B: (33, 26)**
- **Option D: (26, 12)**
- **Option E: (23.2, 6.4)**