In the given questions, two quantities are given, one as Quantity I and another as Quantity II. You have to determine relationship between two quantities and choose the appropriate option.
The boat takes total time of 4 hours to travel 14 km upstream and 36 km downstream together. The boat takes total time of 5 hours to travel 20 km upstream and 24 km downstream together?
Quantity: I. Speed of the boat in still water (in km/hr).
Quantity: II. 16 km/hr

Options

A. If Quantity I`ge` Quantity II
B. If Quantity I`gt`Quantity II
C. If Quantity I`lt`Quantity II
D. If Quantity I`=`Quantity II or the relationship cannot be established from the information that is given

Answer

To solve the problem, we need to find the speed of the boat in still water based on the given upstream and downstream travel times and distances. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( x \) = speed of the boat in still water (in km/hr)
- \( y \) = speed of the stream (in km/hr)

### Step 2: Set Up Equations for the First Scenario
In the first scenario, the boat travels:
- Upstream: 14 km
- Downstream: 36 km
- Total time = 4 hours

The time taken for upstream travel is given by:
\[ \text{Time}_{\text{upstream}} = \frac{14}{x - y} \]
The time taken for downstream travel is given by:
\[ \text{Time}_{\text{downstream}} = \frac{36}{x + y} \]
Setting up the equation for total time:
\[ \frac{14}{x - y} + \frac{36}{x + y} = 4 \quad \text{(1)} \]

### Step 3: Set Up Equations for the Second Scenario
In the second scenario, the boat travels:
- Upstream: 20 km
- Downstream: 24 km
- Total time = 5 hours

The time taken for upstream travel is:
\[ \text{Time}_{\text{upstream}} = \frac{20}{x - y} \]
The time taken for downstream travel is:
\[ \text{Time}_{\text{downstream}} = \frac{24}{x + y} \]
Setting up the equation for total time:
\[ \frac{20}{x - y} + \frac{24}{x + y} = 5 \quad \text{(2)} \]

### Step 4: Solve the Equations
We have two equations (1) and (2) to solve for \( x \) and \( y \).

**Multiply Equation (1) by \( (x - y)(x + y) \)**:
\[ 14(x + y) + 36(x - y) = 4(x^2 - y^2) \]
This simplifies to:
\[ 14x + 14y + 36x - 36y = 4x^2 - 4y^2 \]
\[ 50x - 22y = 4x^2 - 4y^2 \quad \text{(3)} \]

**Multiply Equation (2) by \( (x - y)(x + y) \)**:
\[ 20(x + y) + 24(x - y) = 5(x^2 - y^2) \]
This simplifies to:
\[ 20x + 20y + 24x - 24y = 5x^2 - 5y^2 \]
\[ 44x - 4y = 5x^2 - 5y^2 \quad \text{(4)} \]

### Step 5: Rearranging and Solving for \( x \) and \( y \)
From equations (3) and (4), we can express \( y \) in terms of \( x \) or vice versa, and then substitute back to find \( x \).

After solving these equations, we find:
\[ x - y = \frac{32}{7} \quad \text{and} \quad x + y = \frac{192}{5} \]

### Step 6: Calculate \( x \)
Adding the two equations:
\[ 2x = \frac{32}{7} + \frac{192}{5} \]
Finding a common denominator (35):
\[ 2x = \frac{32 \cdot 5 + 192 \cdot 7}{35} = \frac{160 + 1344}{35} = \frac{1504}{35} \]
Thus,
\[ x = \frac{752}{35} \approx 21.49 \text{ km/hr} \]

### Step 7: Compare with Quantity II
Quantity I (speed of the boat in still water) is approximately \( 21.49 \) km/hr, and Quantity II is \( 16 \) km/hr.

### Conclusion
Since \( 21.49 \) km/hr is greater than \( 16 \) km/hr, we conclude that:

**Quantity I > Quantity II.**