A shopkeeper sells an item for Rs 875, making a profit equal to `(1)/(4)` of the cost price. Find the gain per cent and the cost price.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the given information
- Selling Price (SP) = Rs 875
- Profit (P) = (1/4) of Cost Price (CP)

### Step 2: Write the relationship between Profit and Cost Price
From the information given, we can express the profit in terms of the cost price:
\[ P = \frac{1}{4} \times CP \]

### Step 3: Write the formula for Profit
Profit can also be expressed as:
\[ P = SP - CP \]
Substituting the expression for Profit from Step 2 into this formula gives us:
\[ \frac{1}{4} \times CP = SP - CP \]

### Step 4: Substitute the Selling Price
Now, substitute the Selling Price (SP = 875) into the equation:
\[ \frac{1}{4} \times CP = 875 - CP \]

### Step 5: Rearrange the equation
To solve for CP, we can rearrange the equation:
\[ \frac{1}{4} \times CP + CP = 875 \]
This can be simplified to:
\[ \frac{1}{4} \times CP + \frac{4}{4} \times CP = 875 \]
\[ \frac{5}{4} \times CP = 875 \]

### Step 6: Solve for Cost Price (CP)
Now, multiply both sides by 4 to eliminate the fraction:
\[ 5 \times CP = 875 \times 4 \]
\[ 5 \times CP = 3500 \]
Now, divide both sides by 5:
\[ CP = \frac{3500}{5} = 700 \]

### Step 7: Calculate the Profit
Now that we have the Cost Price (CP), we can calculate the Profit (P):
\[ P = SP - CP = 875 - 700 = 175 \]

### Step 8: Calculate the Gain Percent (Profit Percent)
The formula for Gain Percent is:
\[ \text{Gain Percent} = \left( \frac{P}{CP} \right) \times 100 \]
Substituting the values we have:
\[ \text{Gain Percent} = \left( \frac{175}{700} \right) \times 100 \]

### Step 9: Simplify the Gain Percent
Calculating the fraction:
\[ \frac{175}{700} = \frac{1}{4} \]
Now, multiply by 100:
\[ \text{Gain Percent} = \frac{1}{4} \times 100 = 25\% \]

### Final Answers
- Cost Price (CP) = Rs 700
- Gain Percent = 25%