Solving the Number Analogy: 27:65 :: 64:?

This question asks us to find the missing number in a numerical analogy. We are given the pair 27 : 65 and asked to find the number that relates to 64 in the same way.

To solve this, we first need to identify the relationship between the numbers in the given pair, 27 and 65.

Analyzing the Relationship Between 27 and 65

Let's look for a pattern involving these numbers. Often, analogy questions involve operations like squaring, cubing, adding, subtracting, multiplying, or dividing, or combinations of these.

• We notice that 27 is a perfect cube: $3^3 = 3 \times 3 \times 3 = 27$.
• Now let's look at 65. How might it relate to 27 or 3?
• Consider the number immediately following 3, which is 4. Let's cube 4: $4^3 = 4 \times 4 \times 4 = 64$.
• We can see that 65 is one more than 64: $64 + 1 = 65$.

This suggests a potential pattern: If the first number is $n^3$, the second number is $(n+1)^3 + 1$. Let's verify this with the given pair:

• For 27: $27 = 3^3$. Here, $n=3$.
• According to the pattern, the second number should be $(3+1)^3 + 1 = 4^3 + 1 = 64 + 1 = 65$.

The pattern $n^3 : (n+1)^3 + 1$ holds true for the given pair 27 : 65.

Applying the Pattern to Find the Missing Number

Now we apply the same pattern to the second part of the analogy, 64 : ?

• The first number is 64. We need to express 64 as a cube. We know that $64 = 4^3$.
• So, for this pair, $n=4$.
• Following the identified pattern $n^3 : (n+1)^3 + 1$, the missing number should be $(n+1)^3 + 1$.
• Substituting $n=4$, the missing number is $(4+1)^3 + 1 = 5^3 + 1$.
• Calculating $5^3$: $5^3 = 5 \times 5 \times 5 = 125$.
• Now, add 1: $125 + 1 = 126$.

Thus, the missing number is 126.

Comparing with the Options

Let's check if 126 is among the given options:

1. 126
2. 127
3. 125
4. 124

The calculated missing number, 126, matches option 1.

Step-by-Step Solution

1. Analyze the first pair (27 : 65) to find the relationship.
2. Identify that $27 = 3^3$.
3. Identify that $65 = 64 + 1 = 4^3 + 1$.
4. Conclude the pattern is $n^3 : (n+1)^3 + 1$.
5. Apply the pattern to the second pair (64 : ?).
6. Identify that $64 = 4^3$. So, here $n=4$.
7. Calculate the missing number using $(n+1)^3 + 1$ with $n=4$.
8. Missing number = $(4+1)^3 + 1 = 5^3 + 1 = 125 + 1 = 126$.
9. Confirm that 126 is one of the options.

The missing number is 126.

Revision Table: Key Concepts in Number Analogies

Additional Information: Strategies for Solving Number Analogy Problems

Number analogy questions test your ability to identify patterns and relationships between numbers. Here are some strategies:

• Look for common patterns: Check for squares, cubes, prime numbers, consecutive numbers, multiples, factors, etc.
• Check for arithmetic relationships: See if there's a constant difference, ratio, or a linear relationship ($ax+b$).
• Look for combined operations: The pattern might involve more than one step, like squaring and adding, or cubing and subtracting.
• Analyze the structure: Sometimes the relationship involves the digits of the numbers or their position in a sequence.
• Test your hypothesis: Once you find a potential pattern in the first pair, apply it to the first number of the second pair and see if you get one of the options.
• Practice: Familiarity with common number patterns and types of relationships comes with practice.

Understanding perfect cubes and how they relate to consecutive numbers, as seen in this problem ($n^3$ and $(n+1)^3$), is often key to solving such analogies.