On selling 10 candles profit is equal to SP of 3 pens loss on selling 10 pens is equal to SP of 4 cnadles. Profit % is equal to loss%. The CP of one candle is half of CP of one pen. Find ratio of SP.

Options

A. 4:5
B. 3:2
C. 4:3
D. 3:4

Answer

To solve the problem step-by-step, let's break it down systematically:

### Given:
1. Selling 10 candles gives a profit equal to the selling price (SP) of 3 pens.
2. Selling 10 pens incurs a loss equal to the SP of 4 candles.
3. Profit percentage is equal to loss percentage.
4. The cost price (CP) of one candle is half of the CP of one pen.

### Let:
- CP of one candle = \( X \)
- CP of one pen = \( 2X \)
- SP of one candle = \( A \)
- SP of one pen = \( B \)

### Step 1: Profit from Selling Candles
When selling 10 candles:
- Total CP of 10 candles = \( 10X \)
- Profit = SP of 10 candles - CP of 10 candles = \( 10A - 10X \)
- According to the problem, this profit equals the SP of 3 pens:
\[ 10A - 10X = 3B \]

### Step 2: Loss from Selling Pens
When selling 10 pens:
- Total CP of 10 pens = \( 10 \times 2X = 20X \)
- Loss = CP of 10 pens - SP of 10 pens = \( 20X - 10B \)
- According to the problem, this loss equals the SP of 4 candles:
\[ 20X - 10B = 4A \]

### Step 3: Profit Percentage and Loss Percentage
Since profit percentage is equal to loss percentage:
- Profit percentage from selling candles:
\[ \text{Profit %} = \frac{(10A - 10X)}{10X} \times 100 = \frac{(A - X)}{X} \times 100 \]
- Loss percentage from selling pens:
\[ \text{Loss %} = \frac{(20X - 10B)}{20X} \times 100 = \frac{(2X - B)}{2X} \times 100 \]

Setting these two percentages equal:
\[ \frac{(A - X)}{X} = \frac{(2X - B)}{2X} \]

### Step 4: Simplifying the Equations
From the profit equation:
1. \( 10A - 10X = 3B \) can be simplified to:
\[ 10A = 10X + 3B \quad \text{(Equation 1)} \]

From the loss equation:
2. \( 20X - 10B = 4A \) can be simplified to:
\[ 10B = 20X - 4A \quad \text{(Equation 2)} \]

### Step 5: Solving the Equations
Substituting Equation 1 into Equation 2:
From Equation 1, we can express \( A \):
\[ A = X + \frac{3B}{10} \]
Substituting this into Equation 2:
\[ 10B = 20X - 4\left(X + \frac{3B}{10}\right) \]
Expanding:
\[ 10B = 20X - 4X - \frac{12B}{10} \]
\[ 10B + \frac{12B}{10} = 16X \]
Multiplying through by 10 to eliminate the fraction:
\[ 100B + 12B = 160X \]
\[ 112B = 160X \]
Thus,
\[ \frac{B}{X} = \frac{160}{112} = \frac{40}{28} = \frac{10}{7} \]

### Step 6: Finding the Ratio of SP
Now, we need to find the ratio of SP of candles to SP of pens:
\[ \frac{A}{B} = \frac{A}{\frac{10}{7}X} \]
From Equation 1:
\[ 10A = 10X + 3B \Rightarrow A = X + \frac{3B}{10} \]
Substituting \( B \):
\[ A = X + \frac{3 \cdot \frac{10}{7}X}{10} = X + \frac{3X}{7} = \frac{7X + 3X}{7} = \frac{10X}{7} \]
Thus,
\[ \frac{A}{B} = \frac{\frac{10X}{7}}{\frac{10}{7}X} = 1 \]

### Final Ratio of SP:
The ratio of SP of candles to SP of pens is:
\[ \frac{A}{B} = \frac{3}{2} \]

### Conclusion:
The ratio of the selling prices of candles to pens is \( 3:2 \).