A man travels the first part of his journey at 20 km/h and the next at 70 km/h, covering the entire journey at an average speed of 50 km/h. What is the ratio of the distance that he covered at 20 km/h to that he covered at 70 km/h?

Options

A. `4:21`
B. `3:22`
C. `1:4`
D. `3:5`

Answer

To solve the problem, we need to find the ratio of the distance covered at 20 km/h to the distance covered at 70 km/h. We will use the concept of average speed and the relationship between speed, time, and distance.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- Speed for the first part of the journey (S1) = 20 km/h
- Speed for the second part of the journey (S2) = 70 km/h
- Average speed for the entire journey (S_avg) = 50 km/h

2. **Use the Allegation Method**:
- The allegation method helps us find the ratio of time taken for each part of the journey based on their speeds.
- The formula for the allegation is:
\[ \text{Time Ratio} = \frac{S2 - S_{avg}}{S_{avg} - S1} \]
- Substituting the values:
\[ \text{Time Ratio} = \frac{70 - 50}{50 - 20} = \frac{20}{30} = \frac{2}{3} \]

3. **Determine the Time Taken**:
- Let the time taken for the first part be 2 units and for the second part be 3 units (based on the ratio we found).

4. **Calculate Distances**:
- Distance is calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- For the first part:
\[ D1 = S1 \times \text{Time}_1 = 20 \times 2 = 40 \text{ km} \]
- For the second part:
\[ D2 = S2 \times \text{Time}_2 = 70 \times 3 = 210 \text{ km} \]

5. **Find the Ratio of Distances**:
- Now, we need to find the ratio of the distances:
\[ \text{Ratio} = \frac{D1}{D2} = \frac{40}{210} \]
- Simplifying this ratio:
\[ \text{Ratio} = \frac{40 \div 10}{210 \div 10} = \frac{4}{21} \]

6. **Final Answer**:
- The ratio of the distance covered at 20 km/h to the distance covered at 70 km/h is:
\[ \text{Ratio} = 4 : 21 \]