A car covers a distance of 240km with some speed . If the speed is increased by 20km/hr, it will cover the same distance in 2 hours less find the speed of the car .

Answer

To solve the problem, we will follow these steps:

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

### Step 2: Write the equation for the time taken at the original speed
The distance covered is 240 km. The time taken to cover this distance at speed \( x \) is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{240}{x} \]

### Step 3: Write the equation for the time taken at the increased speed
If the speed is increased by 20 km/hr, the new speed becomes \( x + 20 \) km/hr. The time taken to cover the same distance at this new speed is:
\[ \text{Time} = \frac{240}{x + 20} \]

### Step 4: Set up the equation based on the time difference
According to the problem, the time taken at the increased speed is 2 hours less than the time taken at the original speed. Therefore, we can write the equation:
\[ \frac{240}{x} - \frac{240}{x + 20} = 2 \]

### Step 5: Solve the equation
To solve this equation, we first find a common denominator:
\[ \frac{240(x + 20) - 240x}{x(x + 20)} = 2 \]
This simplifies to:
\[ \frac{240 \cdot 20}{x(x + 20)} = 2 \]
\[ \frac{4800}{x(x + 20)} = 2 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 4800 = 2x(x + 20) \]
Expanding the right side:
\[ 4800 = 2x^2 + 40x \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ 2x^2 + 40x - 4800 = 0 \]
Dividing the entire equation by 2 simplifies it to:
\[ x^2 + 20x - 2400 = 0 \]

### Step 8: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-2400\) and add to \(20\). The factors are \(60\) and \(-40\):
\[ (x + 60)(x - 40) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives:
1. \( x + 60 = 0 \) → \( x = -60 \) (not valid since speed cannot be negative)
2. \( x - 40 = 0 \) → \( x = 40 \)

### Conclusion
The speed of the car is \( 40 \) km/hr.

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