What is the cost price of the cassette ?
(i) The percent profit made when the cassette is sold for Rs. 78 is twice as much as when it is sold for Rs. 69
(ii) If the price of the assette is marked at `20%` above the cost price and discount of `10%` is offfered on the marked price, the seller gain `8%`

Answer

To determine the cost price of the cassette, we will analyze the two statements provided in the question step by step.

### Step-by-Step Solution:

**Let the cost price (CP) of the cassette be denoted as X.**

### Analyzing Statement (i):
1. **Selling Price (SP) when sold for Rs. 78:**
- Profit when sold for Rs. 78 = SP - CP = 78 - X
- Profit Percentage when sold for Rs. 78 = \(\frac{(78 - X)}{X} \times 100\)

2. **Selling Price (SP) when sold for Rs. 69:**
- Profit when sold for Rs. 69 = SP - CP = 69 - X
- Profit Percentage when sold for Rs. 69 = \(\frac{(69 - X)}{X} \times 100\)

3. **According to the statement, the profit percentage when sold for Rs. 78 is twice that when sold for Rs. 69:**
- \(\frac{(78 - X)}{X} \times 100 = 2 \times \frac{(69 - X)}{X} \times 100\)

4. **Simplifying the equation:**
- \(78 - X = 2(69 - X)\)
- \(78 - X = 138 - 2X\)
- Rearranging gives us: \(X = 138 - 78\)
- Thus, \(X = 60\)

### Conclusion from Statement (i):
- The cost price of the cassette is Rs. 60.

### Analyzing Statement (ii):
1. **Marked Price (MP) is 20% above the cost price:**
- MP = \(X + 0.2X = 1.2X\)

2. **Discount of 10% on the marked price:**
- Selling Price after discount = \(MP - 10\% \text{ of } MP = 1.2X - 0.1(1.2X) = 1.2X - 0.12X = 1.08X\)

3. **According to the statement, the seller gains 8%:**
- Gain = Selling Price - Cost Price = \(1.08X - X = 0.08X\)
- Gain Percentage = \(\frac{0.08X}{X} \times 100 = 8\%\)

### Conclusion from Statement (ii):
- This statement confirms that the cost price of the cassette is also Rs. 60.

### Final Conclusion:
Both statements (i) and (ii) are sufficient to determine the cost price of the cassette, which is Rs. 60.