In each of the following questions, two quantities I and II are given. Compare both the quantities and choose the correct option and give answer accordingly.
Quantity I Value of `(2^30-2^29)/2`
Quantity II Value of `2^28`

Options

A. Quantity I = Quantity II or No relation can be established
B. `Quantity I gt Quantity II`
C. `Quantity I lt Quantity II`
D. `Quantity I lt Quantity II`

Answer

To solve the problem, we need to compare the two quantities given:

**Quantity I:** \((2^{30} - 2^{29}) / 2\)
**Quantity II:** \(2^{28}\)

Let's break down the calculations step by step.

### Step 1: Simplify Quantity I

We start with the expression for Quantity I:

\[ \text{Quantity I} = \frac{2^{30} - 2^{29}}{2} \]

### Step 2: Factor out common terms

Notice that both terms in the numerator have a common factor of \(2^{29}\):

\[ 2^{30} - 2^{29} = 2^{29}(2 - 1) = 2^{29} \cdot 1 = 2^{29} \]

### Step 3: Substitute back into Quantity I

Now we substitute back into the expression for Quantity I:

\[ \text{Quantity I} = \frac{2^{29}}{2} \]

### Step 4: Simplify further

Dividing \(2^{29}\) by \(2\) gives us:

\[ \text{Quantity I} = 2^{29 - 1} = 2^{28} \]

### Step 5: Compare with Quantity II

Now we can compare Quantity I with Quantity II:

- **Quantity I:** \(2^{28}\)
- **Quantity II:** \(2^{28}\)

### Conclusion

Since both quantities are equal:

\[ \text{Quantity I} = \text{Quantity II} \]

Thus, the correct option is that Quantity I is equal to Quantity II.

### Final Answer

**Option 1:** Quantity I is equal to Quantity II.

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