A train takes 1 hour 10 min to travel between two stations P and Q. If it travels at `(5)/(7)` of its usual speed, what time will it take to travel between P and Q?

Options

A. 1 hr, 40 min
B. 1 hr, 50 min
C. 1 hr, 38 min
D. 50 min

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given information
The train takes 1 hour and 10 minutes to travel between stations P and Q. We need to find out how long it will take if it travels at \( \frac{5}{7} \) of its usual speed.

### Step 2: Convert time into minutes
First, convert the time taken from hours and minutes into just minutes:
- 1 hour = 60 minutes
- 10 minutes = 10 minutes
- Therefore, 1 hour 10 minutes = \( 60 + 10 = 70 \) minutes.

### Step 3: Assume a speed for easier calculation
Let's assume the usual speed of the train is 7 units (this is an arbitrary choice to simplify calculations).

### Step 4: Calculate the decreased speed
If the usual speed is 7 units, then the speed at \( \frac{5}{7} \) of its usual speed is:
\[ \text{Decreased speed} = \frac{5}{7} \times 7 = 5 \text{ units}. \]

### Step 5: Determine the ratio of speeds
The ratio of the usual speed to the decreased speed is:
\[ \text{Speed ratio} = 7 : 5. \]

### Step 6: Determine the ratio of times
Since speed is inversely proportional to time, the ratio of times will be the inverse of the speed ratio:
\[ \text{Time ratio} = 5 : 7. \]

### Step 7: Relate the time taken to the time ratio
We know that the time taken at the decreased speed (5 units) is 70 minutes. This corresponds to the first part of the ratio (5 units).

### Step 8: Calculate the time for 7 units
Now, we need to find the time taken for 7 units. Since 5 units corresponds to 70 minutes, we can find the time for 1 unit:
\[ \text{Time for 1 unit} = \frac{70 \text{ minutes}}{5} = 14 \text{ minutes}. \]
Now, for 7 units:
\[ \text{Time for 7 units} = 14 \text{ minutes} \times 7 = 98 \text{ minutes}. \]

### Step 9: Convert minutes back to hours and minutes
To convert 98 minutes back to hours and minutes:
- 98 minutes = 1 hour and 38 minutes.

### Final Answer
Thus, the time taken to travel between stations P and Q at \( \frac{5}{7} \) of its usual speed is **1 hour and 38 minutes**.