In each of questions two quantities I and II are given compare both quantities choose correct option and give your answer accordingly
Quantity I: Y:the ratio of speed of boat in still water to speed of stream is 2:1 total time taken by man to cover 72km upstream and comes back in 32 hours, Y is the downstream speed in kmph
QuantityII:9kmph

Options

A. `quantityIltquantityII`
B. `quantity I gtquantityII`
C. `quantityIlequantityII`
D. `quantityI = quantityII`

Answer

To solve the problem, we need to find the downstream speed of the boat, denoted as Y, given the ratio of the speed of the boat in still water to the speed of the stream is 2:1. We also know that the total time taken to cover 72 km upstream and then return downstream is 32 hours.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the speed of the boat in still water be \( 2x \) km/h.
- Let the speed of the stream be \( x \) km/h.

2. **Calculate Upstream Speed:**
- The speed of the boat when going upstream is \( 2x - x = x \) km/h.

3. **Calculate Downstream Speed:**
- The speed of the boat when going downstream is \( 2x + x = 3x \) km/h.

4. **Calculate Time Taken for Upstream and Downstream:**
- Time taken to go upstream for 72 km:
\[ \text{Time}_{\text{upstream}} = \frac{72}{x} \]
- Time taken to come back downstream for 72 km:
\[ \text{Time}_{\text{downstream}} = \frac{72}{3x} \]

5. **Set Up the Equation for Total Time:**
- The total time for the journey is given as 32 hours:
\[ \frac{72}{x} + \frac{72}{3x} = 32 \]

6. **Simplify the Equation:**
- Find a common denominator (which is \( 3x \)):
\[ \frac{72 \cdot 3}{3x} + \frac{72}{3x} = 32 \]
\[ \frac{216 + 72}{3x} = 32 \]
\[ \frac{288}{3x} = 32 \]

7. **Cross-Multiply to Solve for x:**
- Cross-multiplying gives:
\[ 288 = 32 \cdot 3x \]
\[ 288 = 96x \]
\[ x = \frac{288}{96} = 3 \]

8. **Calculate Downstream Speed (Y):**
- Now, substitute \( x \) back to find \( Y \):
\[ Y = 3x = 3 \cdot 3 = 9 \text{ km/h} \]

### Conclusion:
- **Quantity I (Y)**: 9 km/h
- **Quantity II**: 9 km/h

Since both quantities are equal, we conclude that:

**Answer: Quantity I is equal to Quantity II.**