Quantity I: 'x': Ratio between speed of boat in still water to speed of stream is `3:2.` Total time taken by a man to cover 72km in upstream and come back is 32 hours. 'x' is the downstream speed in kmph
Quantity II: 9 kmph

Options

A. Quantity `Igt` Quantity II
B. Quantity `I lt` Quantity II
C. Quantity `I ge` Quantity II
D. Quantity `I le` Quantity II

Explanation

To solve the problem step by step, we will analyze the given information and derive the required quantities.

### Step 1: Understand the Ratios
The ratio of the speed of the boat in still water (let's denote it as \( b \)) to the speed of the stream (let's denote it as \( s \)) is given as \( 3:2 \). This means:
\[ b:s = 3:2 \]
We can express this in terms of a variable \( x \):
\[ b = 3x \quad \text{and} \quad s = 2x \]

### Step 2: Determine Upstream and Downstream Speeds
When the boat is moving upstream (against the current), its effective speed is:
\[ \text{Upstream speed} = b - s = 3x - 2x = x \]
When the boat is moving downstream (with the current), its effective speed is:
\[ \text{Downstream speed} = b + s = 3x + 2x = 5x \]

### Step 3: Calculate Total Distance and Time
The total distance covered is 72 km upstream and 72 km downstream, which gives a total distance of:
\[ \text{Total distance} = 72 + 72 = 144 \text{ km} \]
The total time taken for this journey is given as 32 hours.

### Step 4: Set Up the Time Equation
We can express the time taken for each part of the journey using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Thus, the total time can be expressed as:
\[ \text{Time upstream} + \text{Time downstream} = 32 \]
Substituting the values, we get:
\[ \frac{72}{x} + \frac{72}{5x} = 32 \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator, which is \( 5x \):
\[ \frac{72 \cdot 5}{5x} + \frac{72}{5x} = 32 \]
This simplifies to:
\[ \frac{360 + 72}{5x} = 32 \]
\[ \frac{432}{5x} = 32 \]
Cross-multiplying gives:
\[ 432 = 32 \cdot 5x \]
\[ 432 = 160x \]
Now, solving for \( x \):
\[ x = \frac{432}{160} = \frac{27}{10} = 2.7 \]

### Step 6: Calculate Downstream Speed
Now that we have \( x \), we can find the downstream speed:
\[ \text{Downstream speed} = 5x = 5 \cdot 2.7 = 13.5 \text{ kmph} \]

### Step 7: Compare Quantities
Now we compare Quantity I (downstream speed \( x = 13.5 \) kmph) with Quantity II (9 kmph):
\[ 13.5 \text{ kmph} > 9 \text{ kmph} \]

### Conclusion
Thus, Quantity I is greater than Quantity II.