Quantity I: value of
("Sum of single digit posotive odd intergers" /"Sum of single digit posotive even intergers")`
Quantity II: Value of
`("Sum of cubes of single digit posotive odd intergers" /"Sum of cubes of single digit posotive even intergers")`

Options

A. Quantity `IgtQ` uantity II
B. Quantity `IltQ` uantity II
C. Quantity `IleQ` uantity II
D. Quantity `IgeQ` uantity II

Answer

To solve the problem, we will compute the values for both Quantity I and Quantity II step by step.

### Step 1: Calculate Quantity I

**Quantity I** is defined as:
\[ \text{Quantity I} = \frac{\text{Sum of single digit positive odd integers}}{\text{Sum of single digit positive even integers}} \]

**Step 1.1: Identify single digit positive odd integers**
The single digit positive odd integers are: 1, 3, 5, 7, 9.

**Step 1.2: Calculate the sum of odd integers**
\[ 1 + 3 + 5 + 7 + 9 = 25 \]

**Step 1.3: Identify single digit positive even integers**
The single digit positive even integers are: 2, 4, 6, 8.

**Step 1.4: Calculate the sum of even integers**
\[ 2 + 4 + 6 + 8 = 20 \]

**Step 1.5: Calculate Quantity I**
Now, we can substitute the sums into the formula:
\[ \text{Quantity I} = \frac{25}{20} = \frac{5}{4} = 1.25 \]

### Step 2: Calculate Quantity II

**Quantity II** is defined as:
\[ \text{Quantity II} = \frac{\text{Sum of cubes of single digit positive odd integers}}{\text{Sum of cubes of single digit positive even integers}} \]

**Step 2.1: Calculate the cubes of single digit positive odd integers**
The cubes of the odd integers are:
- \(1^3 = 1\)
- \(3^3 = 27\)
- \(5^3 = 125\)
- \(7^3 = 343\)
- \(9^3 = 729\)

**Step 2.2: Calculate the sum of cubes of odd integers**
\[ 1 + 27 + 125 + 343 + 729 = 1225 \]

**Step 2.3: Calculate the cubes of single digit positive even integers**
The cubes of the even integers are:
- \(2^3 = 8\)
- \(4^3 = 64\)
- \(6^3 = 216\)
- \(8^3 = 512\)

**Step 2.4: Calculate the sum of cubes of even integers**
\[ 8 + 64 + 216 + 512 = 800 \]

**Step 2.5: Calculate Quantity II**
Now, we can substitute the sums into the formula:
\[ \text{Quantity II} = \frac{1225}{800} \]

**Step 2.6: Simplify Quantity II**
To simplify:
\[ \frac{1225}{800} = \frac{49}{32} \approx 1.53125 \]

### Step 3: Compare Quantity I and Quantity II

Now we have:
- Quantity I = 1.25
- Quantity II ≈ 1.53125

Since \(1.25 < 1.53125\), we conclude:
\[ \text{Quantity I} < \text{Quantity II} \]

### Final Answer
Thus, the answer is:
**Quantity I is less than Quantity II.**

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