If S.P. of a book is 5 times to the discount offered and discount % is equal to the profit % then find ratio of discount offered to C.P.?

Options

A. 5:24
B. 8:25
C. 7:30
D. 7:25

Answer

To solve the problem step by step, we need to define some variables and use the relationships given in the question.

### Step 1: Define Variables
Let:
- \( SP \) = Selling Price
- \( CP \) = Cost Price
- \( D \) = Discount offered
- \( MP \) = Marked Price

According to the question, we have:
1. \( SP = 5D \)
2. The discount percentage is equal to the profit percentage.

### Step 2: Express Discount in Terms of SP and MP
The discount \( D \) can be expressed as:
\[ D = MP - SP \]

### Step 3: Substitute SP in the Discount Formula
From the first equation, we can substitute \( SP \):
\[ D = MP - 5D \]
Rearranging gives:
\[ MP = 6D \]

### Step 4: Calculate Profit
Profit \( P \) can be expressed as:
\[ P = SP - CP \]
Using the relationship between SP and CP, we can express CP in terms of SP:
\[ CP = SP - P \]

### Step 5: Express Profit Percentage
Since the discount percentage is equal to the profit percentage, we can write:
\[ \frac{D}{MP} \times 100 = \frac{P}{CP} \times 100 \]
This simplifies to:
\[ \frac{D}{MP} = \frac{P}{CP} \]

### Step 6: Substitute MP in Terms of D
From \( MP = 6D \), we substitute into the profit percentage equation:
\[ \frac{D}{6D} = \frac{P}{CP} \]
This simplifies to:
\[ \frac{1}{6} = \frac{P}{CP} \]
Thus, we have:
\[ P = \frac{CP}{6} \]

### Step 7: Substitute P in Terms of SP
Now, substituting \( P \) back into the profit formula:
\[ P = SP - CP \]
Substituting for \( P \):
\[ \frac{CP}{6} = SP - CP \]
Rearranging gives:
\[ SP = CP + \frac{CP}{6} \]
Combining the terms:
\[ SP = \frac{6CP + CP}{6} = \frac{7CP}{6} \]

### Step 8: Find the Ratio of Discount to CP
Now, we know:
\[ D = MP - SP = 6D - SP \]
Substituting \( SP \):
\[ D = 6D - \frac{7CP}{6} \]
Rearranging gives:
\[ 6D = D + \frac{7CP}{6} \]
So:
\[ 5D = \frac{7CP}{6} \]
Thus:
\[ D = \frac{7CP}{30} \]

### Step 9: Ratio of Discount to CP
Finally, the ratio of discount offered to CP is:
\[ \frac{D}{CP} = \frac{\frac{7CP}{30}}{CP} = \frac{7}{30} \]

### Final Answer
The ratio of discount offered to CP is:
\[ \frac{7}{30} \]