A minibus takes 6 hours less to cover 1680 km distance, if its speed is increased by 14 km/h.What is the usual time taken by minibus?

Options

A. 15 h
B. 24 h
C. 25 h
D. 30 h

Answer

To solve the problem, we need to find the usual time taken by the minibus to cover a distance of 1680 km. Let's break it down step by step.

### Step 1: Define Variables
Let the usual time taken by the minibus be \( T \) hours.

### Step 2: Calculate Usual Speed
The usual speed of the minibus can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{1680 \text{ km}}{T \text{ hours}} = \frac{1680}{T} \text{ km/h} \]

### Step 3: Calculate New Speed
If the speed is increased by 14 km/h, the new speed becomes:
\[ \text{New Speed} = \frac{1680}{T} + 14 \text{ km/h} \]

### Step 4: Calculate New Time
The new time taken to cover the same distance at the new speed is:
\[ \text{New Time} = \frac{1680 \text{ km}}{\text{New Speed}} = \frac{1680}{\left(\frac{1680}{T} + 14\right)} \]

### Step 5: Set Up the Equation
According to the problem, the new time is 6 hours less than the usual time:
\[ \text{New Time} = T - 6 \]
So we can set up the equation:
\[ \frac{1680}{\left(\frac{1680}{T} + 14\right)} = T - 6 \]

### Step 6: Cross Multiply
Cross multiplying gives:
\[ 1680 = (T - 6)\left(\frac{1680}{T} + 14\right) \]

### Step 7: Expand the Equation
Expanding the right-hand side:
\[ 1680 = (T - 6) \cdot \frac{1680}{T} + 14(T - 6) \]
\[ 1680 = \frac{1680T - 10080}{T} + 14T - 84 \]

### Step 8: Multiply through by T
To eliminate the fraction, multiply through by \( T \):
\[ 1680T = 1680T - 10080 + 14T^2 - 84T \]

### Step 9: Rearrange the Equation
Rearranging gives:
\[ 0 = 14T^2 - 84T - 10080 \]

### Step 10: Simplify the Equation
Dividing the entire equation by 14:
\[ 0 = T^2 - 6T - 720 \]

### Step 11: Factor the Quadratic Equation
Factoring the quadratic:
\[ (T - 30)(T + 24) = 0 \]
Thus, \( T = 30 \) or \( T = -24 \). Since time cannot be negative, we take:
\[ T = 30 \]

### Conclusion
The usual time taken by the minibus is **30 hours**.

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