Mr. Mohit has decided to walk on a treadmill for a fixed distance. First day, he walks at a certain speed. Next day, he increases the speed of the treadmill by 1 km/h, he takes 6 min less and if he reduces the speed by 1 km/h, then he takes 9 min more. What is the distance that he decided to walk everyday?

Answer

To solve the problem step by step, we can follow this structured approach:

### Step 1: Define Variables
Let:
- \( x \) = speed on the first day (in km/h)
- \( y \) = time taken on the first day (in hours)

### Step 2: Write the Distance Equation
The distance \( d \) that Mr. Mohit walks can be expressed as:
\[ d = x \cdot y \]

### Step 3: Analyze the First Condition (Increased Speed)
When Mr. Mohit increases his speed by 1 km/h, the new speed becomes \( x + 1 \) km/h. He takes 6 minutes less time, which we convert to hours:
\[ 6 \text{ minutes} = \frac{6}{60} \text{ hours} = 0.1 \text{ hours} \]
Thus, the time taken at the increased speed is:
\[ y - 0.1 \]
The distance equation becomes:
\[ d = (x + 1)(y - 0.1) \]

### Step 4: Set Up the First Equation
Since the distance remains the same, we can equate the two expressions for distance:
\[ x \cdot y = (x + 1)(y - 0.1) \]

### Step 5: Expand the Equation
Expanding the right side:
\[ x \cdot y = (x + 1)(y - 0.1) \]
\[ x \cdot y = xy - 0.1x + y - 0.1 \]

### Step 6: Rearrange the First Equation
Rearranging gives:
\[ 0.1x = y - 0.1 \]
Thus, we can express this as:
\[ -0.1x + y = 0.1 \] (Equation 1)

### Step 7: Analyze the Second Condition (Decreased Speed)
When Mr. Mohit decreases his speed by 1 km/h, the new speed becomes \( x - 1 \) km/h. He takes 9 minutes more time, which we convert to hours:
\[ 9 \text{ minutes} = \frac{9}{60} \text{ hours} = 0.15 \text{ hours} \]
Thus, the time taken at the decreased speed is:
\[ y + 0.15 \]
The distance equation becomes:
\[ d = (x - 1)(y + 0.15) \]

### Step 8: Set Up the Second Equation
Again, equating the two expressions for distance gives:
\[ x \cdot y = (x - 1)(y + 0.15) \]

### Step 9: Expand the Second Equation
Expanding the right side:
\[ x \cdot y = (x - 1)(y + 0.15) \]
\[ x \cdot y = xy + 0.15x - y - 0.15 \]

### Step 10: Rearrange the Second Equation
Rearranging gives:
\[ -0.15x - y = -0.15 \]
Thus, we can express this as:
\[ 0.15x - y = 0.15 \] (Equation 2)

### Step 11: Solve the System of Equations
Now we have two equations:
1. \( -0.1x + y = 0.1 \)
2. \( 0.15x - y = 0.15 \)

Adding both equations:
\[ (-0.1x + y) + (0.15x - y) = 0.1 + 0.15 \]
This simplifies to:
\[ 0.05x = 0.25 \]
Thus, solving for \( x \):
\[ x = \frac{0.25}{0.05} = 5 \text{ km/h} \]

### Step 12: Substitute \( x \) Back to Find \( y \)
Substituting \( x \) back into Equation 1:
\[ -0.1(5) + y = 0.1 \]
This simplifies to:
\[ -0.5 + y = 0.1 \implies y = 0.1 + 0.5 = 0.6 \text{ hours} \]

### Step 13: Calculate the Distance
Now, we can find the distance:
\[ d = x \cdot y = 5 \cdot 0.6 = 3 \text{ km} \]

### Final Answer
The distance that Mr. Mohit decided to walk every day is **3 kilometers**.

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