A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

Answer

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

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

### Step 2: Calculate the Time Taken at Original Speed
The time taken to cover 400 km at the original speed \( x \) is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{400}{x} \text{ hours} \]

### Step 3: Calculate the New Speed
If the speed is increased by 12 km/h, the new speed becomes:
\[ \text{New Speed} = x + 12 \text{ km/h} \]

### Step 4: Calculate the Time Taken at New Speed
The time taken to cover the same distance (400 km) at the new speed is:
\[ \text{New Time} = \frac{400}{x + 12} \text{ hours} \]

### Step 5: Set Up the Equation Based on Time Difference
According to the problem, the time taken at the original speed minus the time taken at the new speed is equal to 1 hour and 40 minutes. We need to convert 1 hour and 40 minutes into hours:
\[ 1 \text{ hour} + \frac{40}{60} \text{ hours} = \frac{5}{3} \text{ hours} \]

Thus, we can set up the equation:
\[ \frac{400}{x} - \frac{400}{x + 12} = \frac{5}{3} \]

### Step 6: Solve the Equation
To solve the equation, we will first find a common denominator, which is \( x(x + 12) \):
\[ \frac{400(x + 12) - 400x}{x(x + 12)} = \frac{5}{3} \]
This simplifies to:
\[ \frac{4800}{x(x + 12)} = \frac{5}{3} \]

### Step 7: Cross Multiply
Cross-multiplying gives:
\[ 4800 \cdot 3 = 5 \cdot x(x + 12) \]
\[ 14400 = 5x^2 + 60x \]

### Step 8: Rearrange the Equation
Rearranging the equation gives:
\[ 5x^2 + 60x - 14400 = 0 \]

### Step 9: Simplify the Equation
Dividing the entire equation by 5:
\[ x^2 + 12x - 2880 = 0 \]

### Step 10: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ x^2 + 60x - 48x - 2880 = 0 \]
Factoring gives:
\[ (x - 48)(x + 60) = 0 \]

### Step 11: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 48 = 0 \quad \Rightarrow \quad x = 48 \]
\[ x + 60 = 0 \quad \Rightarrow \quad x = -60 \quad (\text{not valid since speed cannot be negative}) \]

### Conclusion
Thus, the original speed of the car is:
\[ \boxed{48 \text{ km/h}} \]