A train covers a certain distance in 45 minutes. If its speed is reduced by 5 km/h, it takes 5 minutes more to cover the same distance. The distance (in km) is:

Options

A. 37.5
B. 35
C. 32.5
D. 40

Answer

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

### Step 1: Understand the given information
The train covers a certain distance in 45 minutes. If its speed is reduced by 5 km/h, it takes 5 minutes more to cover the same distance.

### Step 2: Convert time from minutes to hours
Since speed is given in km/h, we need to convert the time from minutes to hours.
- Original time = 45 minutes = 45/60 hours = 3/4 hours
- New time = 45 minutes + 5 minutes = 50 minutes = 50/60 hours = 5/6 hours

### Step 3: Set up the relationship between speed and time
Let the original speed of the train be \( S \) km/h. The distance covered can be expressed as:
- Distance = Speed × Time
- For the original speed: Distance = \( S \times \frac{3}{4} \)
- For the reduced speed (which is \( S - 5 \) km/h): Distance = \( (S - 5) \times \frac{5}{6} \)

### Step 4: Set the distances equal to each other
Since the distance is the same in both cases, we can set the two expressions for distance equal to each other:
\[ S \times \frac{3}{4} = (S - 5) \times \frac{5}{6} \]

### Step 5: Solve the equation
To eliminate the fractions, we can multiply both sides by 12 (the least common multiple of 4 and 6):
\[ 12 \times \left( S \times \frac{3}{4} \right) = 12 \times \left( (S - 5) \times \frac{5}{6} \right) \]
This simplifies to:
\[ 9S = 10(S - 5) \]
Expanding the right side:
\[ 9S = 10S - 50 \]
Rearranging the equation:
\[ 50 = 10S - 9S \]
\[ S = 50 \text{ km/h} \]

### Step 6: Calculate the distance
Now that we have the original speed, we can find the distance:
\[ \text{Distance} = S \times \frac{3}{4} = 50 \times \frac{3}{4} = 37.5 \text{ km} \]

### Final Answer
The distance covered by the train is **37.5 km**.

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