Anshula walks 35 km partly at the rate 4 km/hour and Partly at 5 km/hour, if she had walked at 5 km/hour. When she walked at 4 km/hour and vice versa. She would have covered 2 km more in the same time. Find the time she was walking.

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = time (in hours) Anshula walks at 4 km/h
- \( y \) = time (in hours) Anshula walks at 5 km/h

### Step 2: Set Up the First Equation
From the problem, we know that the total distance walked is 35 km. Therefore, we can write the equation for distance as:
\[ 4x + 5y = 35 \]

### Step 3: Set Up the Second Equation
The problem states that if she had walked at 5 km/h when she walked at 4 km/h and vice versa, she would have covered 2 km more in the same time. This means:
\[ 5x + 4y = 37 \]

### Step 4: Solve the System of Equations
Now we have the following system of equations:
1. \( 4x + 5y = 35 \) (Equation 1)
2. \( 5x + 4y = 37 \) (Equation 2)

We can solve these equations using the elimination method. First, we will multiply Equation 1 by 5 and Equation 2 by 4 to align the coefficients of \( x \):
\[ (4x + 5y) \times 5 = 35 \times 5 \implies 20x + 25y = 175 \quad (Equation 3) \]
\[ (5x + 4y) \times 4 = 37 \times 4 \implies 20x + 16y = 148 \quad (Equation 4) \]

### Step 5: Subtract the Equations
Now, we will subtract Equation 4 from Equation 3:
\[ (20x + 25y) - (20x + 16y) = 175 - 148 \]
This simplifies to:
\[ 9y = 27 \]
Thus, we can solve for \( y \):
\[ y = \frac{27}{9} = 3 \]

### Step 6: Substitute \( y \) Back to Find \( x \)
Now that we have \( y = 3 \), we can substitute this value back into Equation 1 to find \( x \):
\[ 4x + 5(3) = 35 \]
This simplifies to:
\[ 4x + 15 = 35 \]
Subtracting 15 from both sides gives:
\[ 4x = 20 \]
Now, divide by 4:
\[ x = 5 \]

### Step 7: Calculate Total Time
The total time Anshula was walking is:
\[ x + y = 5 + 3 = 8 \text{ hours} \]

### Final Answer
The total time Anshula was walking is **8 hours**.

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