A and B started at the same time from the same place for a certain destination. B walking at `(5)/(6)` of A's speed reached the destination 1 hour 15 minutes after A Breached the destination in

Options

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

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understand the relationship between A's and B's speeds
Let the speed of A be \( v \) (in any unit, say km/h). Then, the speed of B is given as:
\[ \text{Speed of B} = \frac{5}{6}v \]

### Step 2: Define the time taken by A to reach the destination
Let the time taken by A to reach the destination be \( t_A \) hours. Therefore, the distance to the destination can be expressed as:
\[ \text{Distance} = v \times t_A \]

### Step 3: Define the time taken by B to reach the destination
Since B reaches the destination 1 hour and 15 minutes after A, we can express this time as:
\[ t_B = t_A + 1.25 \quad \text{(since 1 hour 15 minutes = 1.25 hours)} \]

### Step 4: Express the distance in terms of B's speed and time
Using B's speed, we can also express the distance as:
\[ \text{Distance} = \left(\frac{5}{6}v\right) \times t_B \]

### Step 5: Set the two distance equations equal to each other
Since both expressions represent the same distance, we can set them equal:
\[ v \times t_A = \left(\frac{5}{6}v\right) \times (t_A + 1.25) \]

### Step 6: Simplify the equation
We can cancel \( v \) from both sides (assuming \( v \neq 0 \)):
\[ t_A = \frac{5}{6}(t_A + 1.25) \]

### Step 7: Solve for \( t_A \)
Distributing \( \frac{5}{6} \):
\[ t_A = \frac{5}{6}t_A + \frac{5}{6} \times 1.25 \]
\[ t_A = \frac{5}{6}t_A + \frac{25}{36} \]

Now, isolate \( t_A \):
\[ t_A - \frac{5}{6}t_A = \frac{25}{36} \]
\[ \frac{1}{6}t_A = \frac{25}{36} \]
\[ t_A = \frac{25}{36} \times 6 = \frac{150}{36} = \frac{25}{6} \text{ hours} \]

### Step 8: Calculate \( t_B \)
Now we can find \( t_B \):
\[ t_B = t_A + 1.25 = \frac{25}{6} + \frac{5}{4} \]

To add these fractions, we need a common denominator:
\[ \frac{25}{6} = \frac{100}{24} \quad \text{and} \quad \frac{5}{4} = \frac{30}{24} \]
\[ t_B = \frac{100}{24} + \frac{30}{24} = \frac{130}{24} = \frac{65}{12} \text{ hours} \]

### Step 9: Convert \( t_B \) into hours and minutes
To convert \( \frac{65}{12} \) hours into hours and minutes:
\[ \frac{65}{12} = 5 \text{ hours and } \frac{5}{12} \text{ of an hour} \]
Calculating \( \frac{5}{12} \) of an hour in minutes:
\[ \frac{5}{12} \times 60 = 25 \text{ minutes} \]
Thus, \( t_B = 5 \text{ hours and } 25 \text{ minutes} \).

### Final Answer
B reaches the destination in **5 hours and 25 minutes**.

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