Analogy Reasoning: Decoding the Number Pattern

This question involves a number analogy problem. We need to identify the relationship between the pairs of numbers and apply that same relationship to find the missing term. The given analogy is:

512 : 10 :: ? : 11 :: 1331 : 13

Identifying the Relationship

Let's examine the relationship between the numbers in the known pairs:

• Pair 1: 512 and 10
• Pair 3: 1331 and 13

We look for a mathematical pattern connecting the first number to the second number in each pair.

Analyzing Pair 1 (512 : 10)

The first number, 512, is a perfect cube. We can find its cube root:

$$ \sqrt[3]{512} = 8 $$

So, 512 is $8^3$. Now let's see how 8 relates to the second number, 10.

We observe that $10 = 8 + 2$.

Thus, the pattern seems to be: If the first number is $N^3$, the second number is $N+2$.

Analyzing Pair 3 (1331 : 13)

Let's check if this pattern holds for the third pair.

The first number, 1331, is also a perfect cube.

$$ \sqrt[3]{1331} = 11 $$

So, 1331 is $11^3$. Now let's relate 11 to the second number, 13.

We see that $13 = 11 + 2$.

The pattern $N^3 : N+2$ holds true for this pair as well, where $N=11$.

Applying the Pattern to Find the Missing Term

Now, we need to apply this confirmed pattern ($N^3 : N+2$) to the middle pair:

? : 11

In this pair, the second number is 11. According to our pattern, this number should be equal to $N+2$.

$$ N + 2 = 11 $$

Solving for $N$:

$$ N = 11 - 2 $$

$$ N = 9 $$

The first number in the pair should be $N^3$. Since we found $N=9$, the missing term is:

$$ 9^3 $$

Calculating the value:

$$ 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729 $$

Conclusion

The missing term in the analogy is 729. The complete analogy is:

512 : 10 :: 729 : 11 :: 1331 : 13

This fits the pattern where the first number is the cube of $N$ ($N^3$) and the second number is $N+2$.

For 512 : 10: $N = \sqrt[3]{512} = 8$. Then $N+2 = 8+2 = 10$.

For ? : 11: $N+2 = 11$, so $N = 9$. Then the term is $N^3 = 9^3 = 729$.

For 1331 : 13: $N = \sqrt[3]{1331} = 11$. Then $N+2 = 11+2 = 13$.

Therefore, the correct option is the one containing 729.