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 ta 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 distances covered by the man at two different speeds: 20 km/h and 70 km/h, given that his average speed for the entire journey is 50 km/h.

Let's denote:
- Distance covered at 20 km/h = D1
- Distance covered at 70 km/h = D2

### Step 1: Use the concept of average speed
The average speed (V_avg) for the entire journey can be calculated using the formula:
\[ V_{avg} = \frac{Total \ Distance}{Total \ Time} \]

Given:
- \( V_{avg} = 50 \) km/h

### Step 2: Set up the distances and times
Let the time taken to cover distance D1 at 20 km/h be \( t_1 \) and the time taken to cover distance D2 at 70 km/h be \( t_2 \).

Using the formula for time:
\[ t_1 = \frac{D1}{20} \]
\[ t_2 = \frac{D2}{70} \]

### Step 3: Express total distance and total time
The total distance (D) is:
\[ D = D1 + D2 \]

The total time (T) is:
\[ T = t_1 + t_2 = \frac{D1}{20} + \frac{D2}{70} \]

### Step 4: Substitute into the average speed formula
Using the average speed formula:
\[ 50 = \frac{D1 + D2}{\frac{D1}{20} + \frac{D2}{70}} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 50 \left( \frac{D1}{20} + \frac{D2}{70} \right) = D1 + D2 \]

### Step 6: Clear the denominators
Multiply through by 140 (the least common multiple of 20 and 70):
\[ 50 \left( 7D1 + 2D2 \right) = 140(D1 + D2) \]

### Step 7: Expand and simplify
Expanding gives:
\[ 3500D1 + 100D2 = 140D1 + 140D2 \]

Rearranging terms:
\[ 3500D1 - 140D1 = 140D2 - 100D2 \]
\[ 3360D1 = 40D2 \]

### Step 8: Find the ratio of distances
Dividing both sides by 40:
\[ \frac{D1}{D2} = \frac{40}{3360} \]

Simplifying the fraction:
\[ \frac{D1}{D2} = \frac{1}{84} \]

### Step 9: Write the ratio in the required form
Thus, the ratio of the distance covered at 20 km/h to the distance covered at 70 km/h is:
\[ D1 : D2 = 1 : 84 \]

### Final Answer
The ratio of the distance covered at 20 km/h to that covered at 70 km/h is **1:84**.

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