A cyclist covering a distance of 40 km would have reached 1 hour earlier, if he had run at an increased speed of 2 km/hr. His speed in (km/hr) was

Options

A. 6
B. 8
C. 10
D. 12

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the initial speed of the cyclist be \( x \) km/hr.

### Step 2: Calculate the time taken at the initial speed
The distance covered is 40 km. The time taken at the initial speed can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{40}{x} \text{ hours} \]

### Step 3: Calculate the new speed
If the cyclist increases his speed by 2 km/hr, the new speed becomes:
\[ \text{New Speed} = x + 2 \text{ km/hr} \]

### Step 4: Calculate the time taken at the new speed
The time taken to cover the same distance at the new speed is:
\[ \text{New Time} = \frac{40}{x + 2} \text{ hours} \]

### Step 5: Set up the equation based on the time difference
According to the problem, the cyclist would have reached 1 hour earlier if he had traveled at the increased speed. Therefore, we can set up the equation:
\[ \frac{40}{x} - \frac{40}{x + 2} = 1 \]

### Step 6: Solve the equation
To solve the equation, we first find a common denominator:
\[ \frac{40(x + 2) - 40x}{x(x + 2)} = 1 \]
This simplifies to:
\[ \frac{80}{x(x + 2)} = 1 \]
Cross-multiplying gives:
\[ 80 = x(x + 2) \]
Expanding this, we get:
\[ x^2 + 2x - 80 = 0 \]

### Step 7: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ x^2 + 2x - 80 = 0 \]
We need two numbers that multiply to -80 and add to 2. These numbers are 10 and -8. Thus, we can factor it as:
\[ (x - 8)(x + 10) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 8 = 0 \quad \text{or} \quad x + 10 = 0 \]
This results in:
\[ x = 8 \quad \text{or} \quad x = -10 \]
Since speed cannot be negative, we discard \( x = -10 \).

### Step 9: Conclusion
Thus, the initial speed of the cyclist is:
\[ \boxed{8 \text{ km/hr}} \]

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