10: 500 :: ?

Options

A. `8 : 256`
B. `7 : 343`
C. `9 : 243`
D. `5 : 75`

Explanation

To solve the analogy problem \(10: 500 :: ?\), we need to find the missing value that corresponds to the same relationship established by the first pair.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given the relationship \(10: 500\). This means that there is a specific operation or function applied to the first number (10) to yield the second number (500).

2. **Finding the Operation**:
From the transcript, we see that cubing the first number (10) and then dividing by 2 gives us the second number (500):
\[ 10^3 = 1000 \]
\[ \frac{1000}{2} = 500 \]
This indicates that the operation is to cube the first number and then divide the result by 2.

3. **Applying the Same Logic**:
Now, we need to apply the same operation to find the missing value. Let's denote the missing value as \(x\). We will use the number 8 (as mentioned in the transcript) to find \(x\):
\[ x^3 = 8^3 = 512 \]
\[ \frac{512}{2} = 256 \]
Therefore, the missing value corresponding to \(8\) is \(256\).

4. **Final Answer**:
The complete analogy is:
\[ 10: 500 :: 8: 256 \]