A motorcyclist covered two thirds of a total journey at his usual speed. He covered the remaining distance at three fourth of his usual speed. As a result, he arrived 30 minutes later than the time he would have taken at usual speed. If the total journey was 180 km, the what is his usual speed?

Options

A. A)40 kmph
B. B)36 kmph
C. C)30 kmph
D. D)32 kmph

Answer

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

### Step 1: Understand the Journey
The total journey is 180 km. The motorcyclist covers two-thirds of this distance at his usual speed and the remaining distance at three-fourths of his usual speed.

### Step 2: Calculate Distances Covered
- Distance covered at usual speed = \( \frac{2}{3} \times 180 \) km = 120 km
- Remaining distance = \( 180 - 120 \) km = 60 km

### Step 3: Define Usual Speed
Let the usual speed of the motorcyclist be \( V \) km/h. Therefore, the speed during the second part of the journey is \( \frac{3}{4}V \) km/h.

### Step 4: Calculate Time Taken for Each Part
- Time taken to cover the first part (120 km) at usual speed \( V \):
\[ \text{Time}_1 = \frac{120}{V} \text{ hours} \]

- Time taken to cover the second part (60 km) at \( \frac{3}{4}V \):
\[ \text{Time}_2 = \frac{60}{\frac{3}{4}V} = \frac{60 \times 4}{3V} = \frac{240}{3V} = \frac{80}{V} \text{ hours} \]

### Step 5: Set Up the Equation for Time Difference
According to the problem, the motorcyclist arrives 30 minutes late. We need to convert 30 minutes into hours:
\[ 30 \text{ minutes} = \frac{30}{60} = \frac{1}{2} \text{ hours} \]

The equation for the time difference is:
\[ \text{Time}_1 + \text{Time}_2 = \text{Time taken at usual speed} + \frac{1}{2} \]
The time taken at usual speed for the entire journey (180 km) is:
\[ \text{Time}_{\text{usual}} = \frac{180}{V} \]

So, we have:
\[ \frac{120}{V} + \frac{80}{V} = \frac{180}{V} + \frac{1}{2} \]

### Step 6: Simplify the Equation
Combine the left side:
\[ \frac{200}{V} = \frac{180}{V} + \frac{1}{2} \]

Subtract \( \frac{180}{V} \) from both sides:
\[ \frac{200 - 180}{V} = \frac{1}{2} \]
\[ \frac{20}{V} = \frac{1}{2} \]

### Step 7: Solve for Usual Speed \( V \)
Cross-multiply to solve for \( V \):
\[ 20 \times 2 = V \]
\[ V = 40 \text{ km/h} \]

### Conclusion
The usual speed of the motorcyclist is **40 km/h**.

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