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

Options

A. 4 hours
B. 7 hours
C. `3(3)/(4)` hours
D. `4(1)/(2)` hours

Answer

To solve the problem step by step, we need to find the usual time taken by the man to reach the destination. Let's break it down:

### Step 1: Define Variables
Let the usual speed of the man be \( S \) and the usual time taken to reach the destination be \( T \).

### Step 2: Calculate New Speed
The man travels at \( \frac{3}{5} \) of his usual speed. Therefore, his new speed is:
\[ \text{New Speed} = \frac{3}{5} S \]

### Step 3: Calculate Distance
Using the formula for distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
The distance to the destination can be expressed using the usual speed and time:
\[ \text{Distance} = S \times T \]

### Step 4: Calculate Time Taken with New Speed
When he travels at the new speed, the time taken to reach the destination can be calculated as:
\[ \text{Time with New Speed} = \frac{\text{Distance}}{\text{New Speed}} = \frac{S \times T}{\frac{3}{5} S} = \frac{5T}{3} \]

### Step 5: Relate the Late Time
According to the problem, he is \( 2\frac{1}{3} \) hours late, which can be converted to an improper fraction:
\[ 2\frac{1}{3} = \frac{7}{3} \text{ hours} \]
Thus, we can set up the equation:
\[ \text{Time with New Speed} = \text{Usual Time} + \text{Late Time} \]
Substituting the values:
\[ \frac{5T}{3} = T + \frac{7}{3} \]

### Step 6: Solve for T
Now, we will solve for \( T \):
1. Rearranging the equation:
\[ \frac{5T}{3} - T = \frac{7}{3} \]
2. Express \( T \) with a common denominator:
\[ \frac{5T}{3} - \frac{3T}{3} = \frac{7}{3} \]
3. Simplifying gives:
\[ \frac{2T}{3} = \frac{7}{3} \]
4. Multiplying both sides by 3:
\[ 2T = 7 \]
5. Dividing by 2:
\[ T = \frac{7}{2} \text{ hours} \]

### Final Answer
Thus, the usual time taken by the man to reach the destination is:
\[ T = 3.5 \text{ hours} \text{ or } 3\frac{1}{2} \text{ hours} \]