A covers `1/4` th of his journey at 20 km/h and `1/3` rd of the rest at 25 km/h and half of the rest at 30 km/h and rest at the speed of 40 km/h. What is the average speed of A?

Options

A. `13(78)/(89)km//h`
B. 12 km/h
C. `26(86)/(89)km//h`
D. 28 km/h

Answer

To solve the problem step by step, we will calculate the average speed of A based on the distances covered at different speeds.

### Step 1: Assume the Total Distance
Let the total distance of the journey be \( D = 48x \) km. We choose 48 because it is a common multiple of the denominators (4, 3, and 2) that will simplify our calculations.

**Hint:** Choose a distance that is a multiple of the denominators in the fractions given in the problem.

### Step 2: Calculate the Distance Covered at Each Speed
1. **First Segment:** A covers \( \frac{1}{4} \) of the journey at 20 km/h.
\[ \text{Distance} = \frac{1}{4} \times 48x = 12x \text{ km} \]

2. **Remaining Distance:**
\[ \text{Remaining Distance} = 48x - 12x = 36x \text{ km} \]

3. **Second Segment:** A covers \( \frac{1}{3} \) of the remaining distance at 25 km/h.
\[ \text{Distance} = \frac{1}{3} \times 36x = 12x \text{ km} \]

4. **Remaining Distance:**
\[ \text{Remaining Distance} = 36x - 12x = 24x \text{ km} \]

5. **Third Segment:** A covers \( \frac{1}{2} \) of the remaining distance at 30 km/h.
\[ \text{Distance} = \frac{1}{2} \times 24x = 12x \text{ km} \]

6. **Remaining Distance:**
\[ \text{Remaining Distance} = 24x - 12x = 12x \text{ km} \]

7. **Fourth Segment:** A covers the remaining distance at 40 km/h.
\[ \text{Distance} = 12x \text{ km} \]

### Step 3: Calculate the Time Taken for Each Segment
1. **Time for First Segment:**
\[ \text{Time} = \frac{12x}{20} = \frac{3x}{5} \text{ hours} \]

2. **Time for Second Segment:**
\[ \text{Time} = \frac{12x}{25} \text{ hours} \]

3. **Time for Third Segment:**
\[ \text{Time} = \frac{12x}{30} = \frac{2x}{5} \text{ hours} \]

4. **Time for Fourth Segment:**
\[ \text{Time} = \frac{12x}{40} = \frac{3x}{10} \text{ hours} \]

### Step 4: Calculate Total Time Taken
Now, we sum up all the times:
\[ \text{Total Time} = \frac{3x}{5} + \frac{12x}{25} + \frac{2x}{5} + \frac{3x}{10} \]

To add these fractions, we need a common denominator. The least common multiple of 5, 25, and 10 is 100.

1. Convert each term:
- \( \frac{3x}{5} = \frac{60x}{100} \)
- \( \frac{12x}{25} = \frac{48x}{100} \)
- \( \frac{2x}{5} = \frac{40x}{100} \)
- \( \frac{3x}{10} = \frac{30x}{100} \)

2. Now add:
\[ \text{Total Time} = \frac{60x + 48x + 40x + 30x}{100} = \frac{178x}{100} = \frac{89x}{50} \text{ hours} \]

### Step 5: Calculate Average Speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{48x}{\frac{89x}{50}} = 48x \times \frac{50}{89x} = \frac{2400}{89} \text{ km/h} \]

### Final Answer
The average speed of A is \( \frac{2400}{89} \) km/h.

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