A person has to cover a distance of 48 km . If the increases his speed by 4 km/h he reaches 1 hour early . Find the his initial speed .

Answer

To solve the problem step by step, we will use the relationship between distance, speed, and time.

### Step-by-Step Solution:

1. **Define Variables**:
Let the initial speed of the person be \( S \) km/h.

2. **Calculate Time at Initial Speed**:
The time taken to cover 48 km at the initial speed \( S \) is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{48}{S} \text{ hours} \]

3. **Calculate Time at Increased Speed**:
If the speed is increased by 4 km/h, the new speed becomes \( S + 4 \) km/h. The time taken to cover the same distance at this new speed is:
\[ \text{Time} = \frac{48}{S + 4} \text{ hours} \]

4. **Set Up the Equation**:
According to the problem, the person reaches 1 hour earlier when he increases his speed. Therefore, we can set up the equation:
\[ \frac{48}{S} - \frac{48}{S + 4} = 1 \]

5. **Solve the Equation**:
To solve the equation, we first find a common denominator:
\[ \frac{48(S + 4) - 48S}{S(S + 4)} = 1 \]
Simplifying the numerator:
\[ \frac{48S + 192 - 48S}{S(S + 4)} = 1 \]
This simplifies to:
\[ \frac{192}{S(S + 4)} = 1 \]

6. **Cross Multiply**:
Cross multiplying gives us:
\[ 192 = S(S + 4) \]

7. **Rearrange the Equation**:
Rearranging the equation gives:
\[ S^2 + 4S - 192 = 0 \]

8. **Factor the Quadratic Equation**:
We can factor the quadratic equation:
\[ (S + 16)(S - 12) = 0 \]

9. **Find the Values of S**:
Setting each factor to zero gives us:
\[ S + 16 = 0 \quad \Rightarrow \quad S = -16 \quad (\text{not valid since speed cannot be negative}) \]
\[ S - 12 = 0 \quad \Rightarrow \quad S = 12 \]

10. **Conclusion**:
The initial speed of the person is \( \boxed{12} \) km/h.