A and B start to cover a distance at their usual speeds at the same time. But B takes 1 hour 15 minutes more than A to reach the destination if he walks `(5)/(6)` th of A's speed. The time taken by B is

Options

A. 6 hours 45 min.
B. 7 hours 15 min.
C. 7 hours 30 min.
D. 8 hours 15 min.

Answer

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

### Step 1: Define Variables
Let the speed of A be \( y \) km/h. Since B walks at \( \frac{5}{6} \) of A's speed, the speed of B will be:
\[ \text{Speed of B} = \frac{5}{6}y \text{ km/h} \]

### Step 2: Define Time Variables
Let the time taken by A to reach the destination be \( x \) minutes. Since B takes 1 hour 15 minutes more than A, the time taken by B will be:
\[ \text{Time of B} = x + 75 \text{ minutes} \]

### Step 3: Calculate Distances
The distance covered by both A and B is the same. The distance covered by A can be expressed as:
\[ \text{Distance of A} = \text{Speed of A} \times \text{Time of A} = y \cdot x \text{ km} \]
The distance covered by B can be expressed as:
\[ \text{Distance of B} = \text{Speed of B} \times \text{Time of B} = \left(\frac{5}{6}y\right) \cdot (x + 75) \text{ km} \]

### Step 4: Set Distances Equal
Since the distances are equal, we can set the two distance equations equal to each other:
\[ yx = \left(\frac{5}{6}y\right)(x + 75) \]

### Step 5: Simplify the Equation
To eliminate \( y \) from the equation, we can divide both sides by \( y \) (assuming \( y \neq 0 \)):
\[ x = \frac{5}{6}(x + 75) \]

### Step 6: Solve for \( x \)
Now, multiply both sides by 6 to eliminate the fraction:
\[ 6x = 5(x + 75) \]
Expanding the right side:
\[ 6x = 5x + 375 \]
Subtract \( 5x \) from both sides:
\[ x = 375 \text{ minutes} \]

### Step 7: Convert Minutes to Hours
Now, convert \( x \) from minutes to hours:
\[ 375 \text{ minutes} = 6 \text{ hours } 15 \text{ minutes} \]

### Step 8: Calculate Time Taken by B
Since B takes 1 hour 15 minutes more than A:
\[ \text{Time taken by B} = x + 75 = 375 + 75 = 450 \text{ minutes} \]
Convert this back to hours:
\[ 450 \text{ minutes} = 7 \text{ hours } 30 \text{ minutes} \]

### Final Answer
Thus, the time taken by B is:
\[ \boxed{7 \text{ hours } 30 \text{ minutes}} \]