Walking at `(3)/(4)` of his usual speed,a man is `1(1)/(2)` hours late. His usual time to cover the same distance, in hours, is

Options

A. `4(1)/(2) `
B. `4`
C. `5(1)/(2)`
D. 5

Answer

To solve the problem, let's break it down step by step.

### Step 1: Understand the relationship between speed, time, and distance
We know that speed and time are inversely proportional when the distance is constant. This means that if speed decreases, time increases.

### Step 2: Define the variables
Let the usual speed of the man be \( S \) (in hours). When he walks at \( \frac{3}{4} \) of his usual speed, his new speed becomes \( \frac{3}{4}S \).

### Step 3: Set up the time equations
Let the usual time taken to cover the distance be \( T \) (in hours). The time taken at the reduced speed is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Since the distance remains constant, we can express the time taken at the reduced speed as:
\[ \text{Time at reduced speed} = \frac{D}{\frac{3}{4}S} = \frac{4D}{3S} \]
And the usual time is:
\[ \text{Usual time} = \frac{D}{S} = \frac{D}{S} \]

### Step 4: Relate the times
According to the problem, the man is \( 1 \frac{1}{2} \) hours late when walking at \( \frac{3}{4} \) of his speed. This means:
\[ \text{Time at reduced speed} - \text{Usual time} = 1 \frac{1}{2} \text{ hours} \]
In terms of \( T \):
\[ \frac{4D}{3S} - T = \frac{3}{2} \]

### Step 5: Substitute the usual time
Since \( T = \frac{D}{S} \), we can substitute this into the equation:
\[ \frac{4D}{3S} - \frac{D}{S} = \frac{3}{2} \]

### Step 6: Simplify the equation
To simplify, find a common denominator:
\[ \frac{4D - 3D}{3S} = \frac{3}{2} \]
This simplifies to:
\[ \frac{D}{3S} = \frac{3}{2} \]

### Step 7: Cross-multiply to solve for D
Cross-multiplying gives:
\[ 2D = 9S \]
Thus, we can express \( D \) in terms of \( S \):
\[ D = \frac{9S}{2} \]

### Step 8: Find the usual time
Now, substitute \( D \) back into the equation for usual time:
\[ T = \frac{D}{S} = \frac{\frac{9S}{2}}{S} = \frac{9}{2} \text{ hours} \]

### Step 9: Convert to mixed fraction
To convert \( \frac{9}{2} \) into a mixed fraction:
\[ \frac{9}{2} = 4 \frac{1}{2} \text{ hours} \]

### Final Answer
The usual time to cover the same distance is \( 4 \frac{1}{2} \) hours.

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