A shopkeeper sells an item for Rs 348, making a profir equal to `(1)/(5)` of the cost price. Find the gain per cent and the cost price.

Answer

To solve the problem step by step, we will follow the logical sequence outlined in the video transcript.

### Step 1: Identify the given information
- Selling Price (SP) = Rs 348
- Profit = (1/5) of Cost Price (CP)

### Step 2: Let the Cost Price be Rs X
- We denote the Cost Price as X.

### Step 3: Express Profit in terms of Cost Price
- According to the problem, Profit = (1/5) * CP = (1/5) * X

### Step 4: Use the relationship between Selling Price, Cost Price, and Profit
- The formula for Selling Price is:
\[ SP = CP + Profit \]
- Substituting the known values:
\[ 348 = X + \frac{1}{5}X \]

### Step 5: Combine like terms
- To combine the terms on the right side, we need a common denominator:
\[ 348 = X + \frac{1}{5}X = \frac{5X}{5} + \frac{1X}{5} = \frac{6X}{5} \]

### Step 6: Solve for Cost Price
- Now, we can solve for X:
\[ 348 = \frac{6X}{5} \]
- Multiply both sides by 5 to eliminate the fraction:
\[ 348 * 5 = 6X \]
\[ 1740 = 6X \]
- Divide both sides by 6:
\[ X = \frac{1740}{6} = 290 \]
- Therefore, the Cost Price (CP) is Rs 290.

### Step 7: Calculate the Profit
- Now, we can find the Profit:
\[ Profit = \frac{1}{5} * CP = \frac{1}{5} * 290 = 58 \]

### Step 8: Calculate the Gain Percent
- The formula for Gain Percent is:
\[ Gain\% = \left(\frac{Profit}{CP}\right) * 100 \]
- Substituting the values:
\[ Gain\% = \left(\frac{58}{290}\right) * 100 \]
- Simplifying:
\[ Gain\% = \frac{58 * 100}{290} = \frac{5800}{290} = 20\% \]

### Final Answers
- Cost Price (CP) = Rs 290
- Gain Percent = 20%