27: 65 :: 125 : ?

Options

A. 217
B. 216
C. 215
D. 227

Explanation

To solve the analogy problem \(27 : 65 :: 125 : ?\), we need to identify the relationship between the numbers in the first pair and apply the same relationship to the second pair.

### Step-by-Step Solution:

1. **Identify the first number in the first pair:**
- The first number is \(27\).
- We can express \(27\) as \(3^3\) (since \(3 \times 3 \times 3 = 27\)).

2. **Identify the second number in the first pair:**
- The second number is \(65\).
- Notice that \(65\) can be expressed as \((3 + 1)^3 + 1\).
- Calculating this gives us:
\[ (3 + 1)^3 = 4^3 = 64 \]
\[ 64 + 1 = 65 \]

3. **Establish the relationship:**
- From the first pair, we see that:
\[ 27 \rightarrow 65 \text{ can be expressed as } 3^3 \rightarrow (3 + 1)^3 + 1 \]

4. **Apply the same relationship to the second pair:**
- The first number in the second pair is \(125\).
- We can express \(125\) as \(5^3\) (since \(5 \times 5 \times 5 = 125\)).

5. **Calculate the second number in the second pair:**
- Following the established pattern, we need to calculate:
\[ (5 + 1)^3 + 1 \]
- Calculating this gives us:
\[ (5 + 1)^3 = 6^3 = 216 \]
\[ 216 + 1 = 217 \]

6. **Conclusion:**
- Therefore, the value of the question mark is \(217\).

### Final Answer:
The answer to the analogy \(27 : 65 :: 125 : ?\) is \(217\).

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