A man travels a distance of 4 km every day. One day he increases his speed by 0.8 km/h than usual speed and reaches his destination 15 minutes earlier. What is the normal speed of the man?

Options

A. 3.6 km/h
B. 4.0 km/h
C. 3.2 km/h
D. 3.0 km/h

Answer

To solve the problem step-by-step, we will follow these steps:

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

### Step 2: Calculate the Usual Time Taken
The distance traveled is 4 km. The usual time taken to travel this distance at speed \( x \) is given by the formula:
\[ \text{Usual Time} (t_1) = \frac{\text{Distance}}{\text{Speed}} = \frac{4}{x} \text{ hours} \]

### Step 3: Calculate the New Speed
When the man increases his speed by 0.8 km/h, his new speed becomes:
\[ \text{New Speed} = x + 0.8 \text{ km/h} \]

### Step 4: Calculate the New Time Taken
The new time taken to travel the same distance of 4 km at the new speed is:
\[ \text{New Time} (t_2) = \frac{4}{x + 0.8} \text{ hours} \]

### Step 5: Set Up the Equation for Time Difference
According to the problem, he reaches his destination 15 minutes earlier. We need to convert 15 minutes into hours:
\[ 15 \text{ minutes} = \frac{15}{60} = \frac{1}{4} \text{ hours} \]
Thus, we can set up the equation:
\[ t_1 - t_2 = \frac{1}{4} \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{4}{x} - \frac{4}{x + 0.8} = \frac{1}{4} \]

### Step 6: Simplify the Equation
To simplify the left side, we can find a common denominator:
\[ \frac{4(x + 0.8) - 4x}{x(x + 0.8)} = \frac{1}{4} \]
This simplifies to:
\[ \frac{4 \cdot 0.8}{x(x + 0.8)} = \frac{1}{4} \]
\[ \frac{3.2}{x(x + 0.8)} = \frac{1}{4} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 3.2 \cdot 4 = x(x + 0.8) \]
\[ 12.8 = x^2 + 0.8x \]

### Step 8: Rearrange the Equation
Rearranging the equation gives us a standard quadratic equation:
\[ x^2 + 0.8x - 12.8 = 0 \]

### Step 9: Solve the Quadratic Equation
To solve the quadratic equation, we can use the factorization method. We need to find two numbers that multiply to \(-12.8\) and add to \(0.8\). The factors are:
\[ (x - 3.2)(x + 4) = 0 \]
Setting each factor to zero gives:
\[ x - 3.2 = 0 \quad \text{or} \quad x + 4 = 0 \]
Thus, we find:
\[ x = 3.2 \quad \text{or} \quad x = -4 \]

### Step 10: Determine the Valid Solution
Since speed cannot be negative, we discard \( x = -4 \). Therefore, the normal speed of the man is:
\[ \boxed{3.2} \text{ km/h} \]