An umbrella is sold for Rs. 198 at a gain of 10%. What is the cost price of the umbrella? At what price must it be sold to gain 25%?

Answer

To solve the problem step by step, let's break it down into two parts: finding the cost price of the umbrella and then determining the selling price for a 25% gain.

### Step 1: Find the Cost Price (CP)

1. **Identify the given values**:
- Selling Price (SP) = Rs. 198
- Gain Percentage = 10%

2. **Use the formula for Selling Price**:
\[ SP = CP + \text{Gain} \]
The gain can be expressed as:
\[ \text{Gain} = \frac{\text{Gain Percentage}}{100} \times CP \]
Therefore, we can rewrite the Selling Price formula as:
\[ SP = CP + \frac{10}{100} \times CP \]
This simplifies to:
\[ SP = CP \left(1 + \frac{10}{100}\right) = CP \left(1 + 0.1\right) = CP \times 1.1 \]

3. **Substituting the Selling Price**:
\[ 198 = CP \times 1.1 \]

4. **Solve for CP**:
\[ CP = \frac{198}{1.1} = \frac{1980}{11} = 180 \]
So, the Cost Price (CP) of the umbrella is Rs. 180.

### Step 2: Find the Selling Price for a 25% Gain

1. **Calculate the new Selling Price (SP')**:
\[ SP' = CP + \text{New Gain} \]
The new gain for a 25% profit can be expressed as:
\[ \text{New Gain} = \frac{25}{100} \times CP \]
Thus, we can rewrite the new Selling Price formula as:
\[ SP' = CP + \frac{25}{100} \times CP = CP \left(1 + \frac{25}{100}\right) = CP \times 1.25 \]

2. **Substituting the Cost Price**:
\[ SP' = 180 \times 1.25 \]

3. **Calculate SP'**:
\[ SP' = 180 \times 1.25 = 225 \]
So, the Selling Price for a 25% gain is Rs. 225.

### Final Answers
- Cost Price of the umbrella = Rs. 180
- Selling Price for a 25% gain = Rs. 225