A person increases his speed by 40 % .By doing so . He reaches his office 9 minutes before the usual time .How much time (in minutes)does he take usually to go to his office?

Options

A. `31(1)/(2)`
B. 33
C. 32
D. `30(1)/(2)`

Answer

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

### Step 1: Understand the relationship between speed, time, and distance.
When speed increases, the time taken to cover the same distance decreases. The relationship can be expressed as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
This means that if speed increases, time decreases, and vice versa.

### Step 2: Define the variables.
Let the usual time taken to reach the office be \( T \) minutes. Let the distance to the office be \( D \) and the usual speed be \( S \). Thus, we can express:
\[ D = S \times T \]

### Step 3: Calculate the new speed after the increase.
The speed is increased by 40%, so the new speed \( S' \) can be calculated as:
\[ S' = S + 0.4S = 1.4S \]

### Step 4: Calculate the new time taken with the increased speed.
The new time \( T' \) taken to reach the office with the new speed can be expressed as:
\[ T' = \frac{D}{S'} = \frac{D}{1.4S} \]

### Step 5: Relate the new time to the usual time.
Since the person reaches the office 9 minutes earlier, we can express this relationship as:
\[ T' = T - 9 \]

### Step 6: Substitute the expression for \( T' \) into the equation.
Substituting \( T' \) into the equation gives:
\[ \frac{D}{1.4S} = T - 9 \]

### Step 7: Substitute \( D \) with \( S \times T \).
Now substitute \( D = S \times T \) into the equation:
\[ \frac{S \times T}{1.4S} = T - 9 \]
This simplifies to:
\[ \frac{T}{1.4} = T - 9 \]

### Step 8: Clear the fraction by multiplying through by 1.4.
Multiplying both sides by 1.4 gives:
\[ T = 1.4(T - 9) \]

### Step 9: Distribute and solve for \( T \).
Expanding the right side:
\[ T = 1.4T - 12.6 \]
Now, rearranging gives:
\[ 1.4T - T = 12.6 \]
\[ 0.4T = 12.6 \]

### Step 10: Solve for \( T \).
Dividing both sides by 0.4:
\[ T = \frac{12.6}{0.4} = 31.5 \]

### Conclusion:
The usual time taken to go to the office is \( 31.5 \) minutes.

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