A man with `(3)/(5)` of his usual speed reaches the destination `2(1)/(2)` hours late. Find his usual time to reach the destination.

Options

A. 4 hours
B. 3 hours
C. `3(3)/(4)` hours
D. `4(1)/(2)` hours

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define Variables
Let the usual speed of the man be \( V \) (in distance per hour) and the usual time to reach the destination be \( T \) (in hours).

### Step 2: Express Distance
The distance \( D \) to the destination can be expressed as:
\[ D = V \times T \]

### Step 3: Determine Reduced Speed
When the man travels at \( \frac{3}{5} \) of his usual speed, his speed becomes:
\[ \text{Reduced Speed} = \frac{3}{5} V \]

### Step 4: Calculate Time Taken at Reduced Speed
The time taken to reach the destination at the reduced speed is:
\[ \text{Time at Reduced Speed} = \frac{D}{\frac{3}{5} V} = \frac{5D}{3V} \]

### Step 5: Set Up the Equation
According to the problem, he arrives \( 2\frac{1}{2} \) hours late, which can be converted to hours:
\[ 2\frac{1}{2} \text{ hours} = \frac{5}{2} \text{ hours} \]
Thus, the equation relating the usual time \( T \) and the time taken at reduced speed is:
\[ \frac{5D}{3V} = T + \frac{5}{2} \]

### Step 6: Substitute Distance
Substituting \( D = VT \) into the equation gives:
\[ \frac{5(VT)}{3V} = T + \frac{5}{2} \]
This simplifies to:
\[ \frac{5T}{3} = T + \frac{5}{2} \]

### Step 7: Rearrange the Equation
Rearranging the equation to isolate \( T \):
\[ \frac{5T}{3} - T = \frac{5}{2} \]
This can be rewritten as:
\[ \frac{5T - 3T}{3} = \frac{5}{2} \]
\[ \frac{2T}{3} = \frac{5}{2} \]

### Step 8: Solve for \( T \)
Multiplying both sides by 3 to eliminate the fraction:
\[ 2T = \frac{15}{2} \]
Now, divide both sides by 2:
\[ T = \frac{15}{4} \]

### Step 9: Convert to Mixed Number
Converting \( \frac{15}{4} \) to a mixed number gives:
\[ T = 3\frac{3}{4} \text{ hours} \]

### Final Answer
The usual time to reach the destination is \( 3\frac{3}{4} \) hours.

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