A person covers a distance of 61 km in 9 hours. Some part of this journey is covered on foot at the speed of 4 km/h and the rest by cycle at the speed of 9km/h. Find the distance of both cases .

Answer

To solve the problem, we need to find the distances covered on foot and by cycle during a journey of 61 km in 9 hours, where the person walks at a speed of 4 km/h and cycles at a speed of 9 km/h.

### Step-by-Step Solution:

1. **Define Variables**:
Let:
- \( d_1 \) = distance covered on foot (in km)
- \( d_2 \) = distance covered by cycle (in km)

We know that:
\[ d_1 + d_2 = 61 \quad \text{(1)} \]

2. **Time Calculation**:
The time taken to cover each distance can be expressed as:
- Time on foot = \( \frac{d_1}{4} \) hours (since speed = 4 km/h)
- Time by cycle = \( \frac{d_2}{9} \) hours (since speed = 9 km/h)

The total time for the journey is given as 9 hours:
\[ \frac{d_1}{4} + \frac{d_2}{9} = 9 \quad \text{(2)} \]

3. **Substituting Equation (1) into Equation (2)**:
From equation (1), we can express \( d_2 \) in terms of \( d_1 \):
\[ d_2 = 61 - d_1 \]
Substitute this into equation (2):
\[ \frac{d_1}{4} + \frac{61 - d_1}{9} = 9 \]

4. **Finding a Common Denominator**:
The common denominator for 4 and 9 is 36. Rewrite the equation:
\[ \frac{9d_1}{36} + \frac{4(61 - d_1)}{36} = 9 \]
Simplifying gives:
\[ \frac{9d_1 + 244 - 4d_1}{36} = 9 \]
Combine like terms:
\[ \frac{5d_1 + 244}{36} = 9 \]

5. **Clearing the Fraction**:
Multiply both sides by 36:
\[ 5d_1 + 244 = 324 \]
Now, isolate \( d_1 \):
\[ 5d_1 = 324 - 244 \]
\[ 5d_1 = 80 \]
\[ d_1 = \frac{80}{5} = 16 \text{ km} \]

6. **Finding \( d_2 \)**:
Substitute \( d_1 \) back into equation (1):
\[ d_2 = 61 - d_1 = 61 - 16 = 45 \text{ km} \]

### Final Distances:
- Distance covered on foot: **16 km**
- Distance covered by cycle: **45 km**