If the discount of 10% is given on the marked price of a radio, the gain is 20%. If the discount is increased to 20%, the gain per cent is :

Options

A. `5%`
B. `6(1)/(4)%`
C. `6(2)/(3)5`
D. `7(5)/(8)%`

Answer

To solve the problem step by step, we will first establish some variables and then calculate the required gain percentage when the discount is increased to 20%.

### Step 1: Define Variables
Let:
- Marked Price (MP) = 100 (for simplicity)
- Cost Price (CP) = x

### Step 2: Calculate Selling Price with 10% Discount
If a discount of 10% is given on the marked price, the selling price (SP) becomes:
\[ SP = MP - (10\% \text{ of } MP) = 100 - (10\% \text{ of } 100) = 100 - 10 = 90 \]

### Step 3: Establish the Relationship with Gain
According to the problem, if the selling price is 90, and the gain is 20%, we can express this as:
\[ SP = CP + (20\% \text{ of } CP) = CP + 0.2 \times CP = 1.2 \times CP \]
Setting the two equations for SP equal gives:
\[ 90 = 1.2 \times CP \]
From this, we can solve for CP:
\[ CP = \frac{90}{1.2} = 75 \]

### Step 4: Calculate Selling Price with 20% Discount
Now, if the discount is increased to 20%, the new selling price (SP) becomes:
\[ SP = MP - (20\% \text{ of } MP) = 100 - (20\% \text{ of } 100) = 100 - 20 = 80 \]

### Step 5: Calculate Gain Percentage
Now we need to calculate the gain percentage when the selling price is 80:
\[ \text{Gain} = SP - CP = 80 - 75 = 5 \]
The gain percentage can be calculated as:
\[ \text{Gain Percentage} = \left( \frac{\text{Gain}}{CP} \right) \times 100 = \left( \frac{5}{75} \right) \times 100 = \frac{500}{75} = \frac{20}{3} \approx 6.67\% \]

### Step 6: Final Answer
Thus, the gain percentage when the discount is increased to 20% is:
\[ \text{Gain Percentage} = 6 \frac{2}{3}\% \]