The respective ratio between the time taken by a bus to travel a certain distance at x kmph and the time taken by the bus to travel the same distance at (x+20) kmph is 4:3. How much time will the bus take to travel 480 km at a speed of (x+30) kmph?

Options

A. 4 hours 48 minutes
B. 5 hours 10 minutes
C. 4 hours
D. 5 hours 20 minutes

Answer

To solve the problem, we need to find the time taken by the bus to travel 480 km at a speed of (x + 30) km/h, given that the ratio of the time taken at speeds of x km/h and (x + 20) km/h is 4:3.

### Step-by-Step Solution:

1. **Understanding the Relation of Time and Speed**:
The time taken to travel a distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Let's denote the distance as \( D \).

2. **Setting Up the Ratio**:
According to the problem, the ratio of the time taken at speeds \( x \) km/h and \( (x + 20) \) km/h is given as:
\[ \frac{D/x}{D/(x + 20)} = \frac{4}{3} \]

3. **Cross-Multiplying**:
Cross-multiplying the ratio gives:
\[ 3 \cdot \frac{D}{x} = 4 \cdot \frac{D}{x + 20} \]
Since \( D \) is common on both sides, we can cancel it out (assuming \( D \neq 0 \)):
\[ 3/x = 4/(x + 20) \]

4. **Cross-Multiplying Again**:
Now, cross-multiply to eliminate the fractions:
\[ 3(x + 20) = 4x \]

5. **Expanding and Rearranging**:
Expanding the left side:
\[ 3x + 60 = 4x \]
Rearranging gives:
\[ 60 = 4x - 3x \]
Thus:
\[ x = 60 \text{ km/h} \]

6. **Finding the Speed for the Final Calculation**:
Now we need to find the time taken to travel 480 km at a speed of \( (x + 30) \) km/h:
\[ \text{Speed} = x + 30 = 60 + 30 = 90 \text{ km/h} \]

7. **Calculating the Time**:
Using the time formula:
\[ \text{Time} = \frac{480}{90} \]
Simplifying this:
\[ \text{Time} = \frac{480}{90} = \frac{48}{9} = \frac{16}{3} \text{ hours} \]

### Final Answer:
The bus will take \( \frac{16}{3} \) hours or approximately 5 hours and 20 minutes to travel 480 km at a speed of \( (x + 30) \) km/h.

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