A man cycles at the speed of 8 km/hr and reaches office at 11 am and when he cycles at the speed of 12 km/hr he reaches office at 9 am. At what speed should by cycle so that he reaches his office at 10 am ?

Options

A. 9.6 kmph
B. 10 kmph
C. 11.2 kmph
D. 7.6 kmph

Answer

To solve the problem, we need to determine the distance to the office and then find the speed required to reach there by 10 am. Let's break down the solution step by step.

### Step 1: Define Variables
Let the distance to the office be \( D \) kilometers.

### Step 2: Calculate Time Taken at Different Speeds
1. When the man cycles at 8 km/hr and reaches the office at 11 am:
- Let the time taken to reach the office be \( t_1 \) hours.
- Since he reaches at 11 am, if we denote the time he starts cycling as \( T \), we can express this as:
\[ t_1 = 11 - T \]
- The distance can be expressed as:
\[ D = 8 \times t_1 \]

2. When the man cycles at 12 km/hr and reaches the office at 9 am:
- Let the time taken to reach the office be \( t_2 \) hours.
- Since he reaches at 9 am, we can express this as:
\[ t_2 = 9 - T \]
- The distance can also be expressed as:
\[ D = 12 \times t_2 \]

### Step 3: Set Up Equations
Since both expressions for distance \( D \) are equal, we can set them equal to each other:
\[ 8 \times (11 - T) = 12 \times (9 - T) \]

### Step 4: Solve for \( T \)
Expanding both sides:
\[ 88 - 8T = 108 - 12T \]
Rearranging gives:
\[ 12T - 8T = 108 - 88 \]
\[ 4T = 20 \]
\[ T = 5 \text{ hours} \]

### Step 5: Calculate the Distance \( D \)
Now, substituting \( T \) back to find the distance:
\[ t_1 = 11 - 5 = 6 \text{ hours} \]
\[ D = 8 \times 6 = 48 \text{ km} \]

### Step 6: Determine Required Speed to Arrive at 10 am
To reach the office at 10 am, the man has \( 10 - T = 10 - 5 = 5 \) hours to travel the same distance \( D \):
\[ \text{Required Speed} = \frac{D}{\text{Time}} = \frac{48}{5} = 9.6 \text{ km/hr} \]

### Final Answer
The speed at which the man should cycle to reach the office by 10 am is **9.6 km/hr**.

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