A person covers a distance of 80 km in 8 hours. some part of this journey is completed at the speed of 8 km/h and the rest at 16 km/h. Find distance covered at both speeds respectively.

Answer

To solve the problem, we need to find the distances covered at two different speeds: 8 km/h and 16 km/h. Let's denote:

- \( d_1 \) = distance covered at 8 km/h
- \( d_2 \) = distance covered at 16 km/h

From the problem, we know:

1. The total distance covered is 80 km:
\[ d_1 + d_2 = 80 \quad \text{(1)} \]

2. The total time taken is 8 hours. We can express the time taken for each part of the journey using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

Therefore, the time taken for each part of the journey is:
- Time taken at 8 km/h:
\[ \text{Time}_1 = \frac{d_1}{8} \quad \text{(2)} \]

- Time taken at 16 km/h:
\[ \text{Time}_2 = \frac{d_2}{16} \quad \text{(3)} \]

The total time taken is given as 8 hours:
\[ \frac{d_1}{8} + \frac{d_2}{16} = 8 \quad \text{(4)} \]

Now we have two equations (1) and (4) to solve for \( d_1 \) and \( d_2 \).

### Step 1: Solve for \( d_2 \) in terms of \( d_1 \)
From equation (1):
\[ d_2 = 80 - d_1 \quad \text{(5)} \]

### Step 2: Substitute \( d_2 \) in equation (4)
Substituting equation (5) into equation (4):
\[ \frac{d_1}{8} + \frac{80 - d_1}{16} = 8 \]

### Step 3: Find a common denominator and simplify
The common denominator for 8 and 16 is 16. Rewriting the equation:
\[ \frac{2d_1}{16} + \frac{80 - d_1}{16} = 8 \]

Combining the fractions:
\[ \frac{2d_1 + 80 - d_1}{16} = 8 \]
\[ \frac{d_1 + 80}{16} = 8 \]

### Step 4: Multiply both sides by 16
\[ d_1 + 80 = 128 \]

### Step 5: Solve for \( d_1 \)
\[ d_1 = 128 - 80 \]
\[ d_1 = 48 \text{ km} \]

### Step 6: Use equation (5) to find \( d_2 \)
Substituting \( d_1 \) back into equation (5):
\[ d_2 = 80 - 48 \]
\[ d_2 = 32 \text{ km} \]

### Final Answer
The distance covered at 8 km/h is 48 km, and the distance covered at 16 km/h is 32 km.