The bus fare between two cities is increased in the ratio `7 : 9`. Find the increase in the fare, if :
(i) the original fare is `"Rs." 245`,
(ii) the increased fare is `"Rs." 207`.

Answer

To solve the problem step by step, we will address both parts of the question separately.

### Part (i): Original fare is Rs. 245

1. **Understand the ratio of increase**: The bus fare is increased in the ratio of 7:9. This means that if the original fare is represented as 7 parts, the increased fare will be represented as 9 parts.

2. **Set up the equation**: Let the original fare be \( F = 245 \) Rs. The new fare can be calculated using the ratio:
\[ \text{New Fare} = \frac{9}{7} \times \text{Original Fare} \]

3. **Calculate the new fare**:
\[ \text{New Fare} = \frac{9}{7} \times 245 \]
First, calculate \( \frac{245}{7} \):
\[ \frac{245}{7} = 35 \]
Now, multiply by 9:
\[ \text{New Fare} = 9 \times 35 = 315 \text{ Rs.} \]

4. **Calculate the increase in fare**:
\[ \text{Increase} = \text{New Fare} - \text{Original Fare} = 315 - 245 = 70 \text{ Rs.} \]

### Part (ii): Increased fare is Rs. 207

1. **Understand the ratio of increase**: Again, the ratio of the original fare to the increased fare is 7:9.

2. **Set up the equation**: Let the increased fare be \( F' = 207 \) Rs. The original fare can be calculated using the ratio:
\[ \text{Original Fare} = \frac{7}{9} \times \text{Increased Fare} \]

3. **Calculate the original fare**:
\[ \text{Original Fare} = \frac{7}{9} \times 207 \]
First, calculate \( \frac{207}{9} \):
\[ \frac{207}{9} = 23 \text{ (approximately)} \]
Now, multiply by 7:
\[ \text{Original Fare} = 7 \times 23 = 161 \text{ Rs.} \]

4. **Calculate the increase in fare**:
\[ \text{Increase} = \text{Increased Fare} - \text{Original Fare} = 207 - 161 = 46 \text{ Rs.} \]

### Summary of the Answers:
- (i) The increase in fare when the original fare is Rs. 245 is **Rs. 70**.
- (ii) The increase in fare when the increased fare is Rs. 207 is **Rs. 46**.