A person has to cover 360 km distance. If he increases his speed by 10 km/h he reaches 3 hours early. Find his initial speed.

Answer

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

### Step 1: Define the Variables
Let the initial speed of the person be \( x \) km/h.

### Step 2: Calculate the Time Taken at Initial Speed
The time taken to cover 360 km at the initial speed \( x \) is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{360}{x} \text{ hours} \]

### Step 3: Calculate the Time Taken at Increased Speed
If the person increases his speed by 10 km/h, his new speed becomes \( x + 10 \) km/h. The time taken to cover the same distance at this new speed is:
\[ \text{Time} = \frac{360}{x + 10} \text{ hours} \]

### Step 4: Set Up the Equation
According to the problem, if he increases his speed by 10 km/h, he reaches 3 hours earlier. Therefore, we can set up the equation:
\[ \frac{360}{x} - \frac{360}{x + 10} = 3 \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator:
\[ \frac{360(x + 10) - 360x}{x(x + 10)} = 3 \]
This simplifies to:
\[ \frac{3600}{x(x + 10)} = 3 \]

Now, cross-multiply to eliminate the fraction:
\[ 3600 = 3x(x + 10) \]
Expanding the right side gives:
\[ 3600 = 3x^2 + 30x \]

Rearranging the equation yields:
\[ 3x^2 + 30x - 3600 = 0 \]

### Step 6: Simplify the Equation
Dividing the entire equation by 3 simplifies it to:
\[ x^2 + 10x - 1200 = 0 \]

### Step 7: Factor the Quadratic Equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to -1200 and add to 10. The factors are 40 and -30:
\[ (x + 40)(x - 30) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
\[ x + 40 = 0 \quad \Rightarrow \quad x = -40 \quad (\text{not valid, as speed cannot be negative}) \]
\[ x - 30 = 0 \quad \Rightarrow \quad x = 30 \]

### Conclusion
The initial speed of the person is \( 30 \) km/h.

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