Solving the Number Analogy: Finding the Relationship Pattern

The question asks us to find the number that completes the analogy 132 : 10 :: 42 : 5 :: 272 : ?. This is a number analogy problem where we need to identify the relationship between the first two numbers in each pair and apply that same relationship to the third pair to find the missing term. Let's analyze the given pairs to discover the underlying pattern.

Analyzing the First Pair: 132 and 10

We need to find a connection between 132 and 10. Let's consider common mathematical operations or properties related to these numbers. Large numbers are often related to their square roots or squares of numbers close to them.

Let's look at perfect squares near 132:

• $11^2 = 121$
• $12^2 = 144$

The number 132 is between these two perfect squares. Let's see which one is closer:

• Difference from $11^2$: $132 - 121 = 11$
• Difference from $12^2$: $144 - 132 = 12$

132 is closer to 121, which is $11^2$. The square root is 11. How can we get 10 from 11? By subtracting 1.

So, a possible relationship is: Find the closest perfect square to the first number, take its square root, and subtract 1.

Let's test this potential pattern on the second pair.

Analyzing the Second Pair: 42 and 5

Now, let's apply the proposed pattern to the second pair, 42 and 5. We look for perfect squares near 42:

• $6^2 = 36$
• $7^2 = 49$

The number 42 is between these two perfect squares. Let's see which one is closer:

• Difference from $6^2$: $42 - 36 = 6$
• Difference from $7^2$: $49 - 42 = 7$

42 is closer to 36, which is $6^2$. The square root is 6. How can we get 5 from 6? By subtracting 1.

Applying the rule: Find the closest perfect square to 42 ($36$), take its square root ($6$), and subtract 1 ($6 - 1 = 5$). This matches the given second term (5) in the pair (42 : 5).

The pattern holds for both given pairs. The relationship seems to be: Take the first number, identify the square root of the closest perfect square to it, and subtract 1 to get the second number.

Applying the Pattern to the Third Pair: 272 and ?

Now, we apply the established relationship pattern to the third term, 272, to find the missing fifth term. We need to find the perfect squares nearest to 272:

• $16^2 = 256$
• $17^2 = 289$

The number 272 is between these two perfect squares. Let's determine which one is closer:

• Difference from $16^2$: $272 - 256 = 16$
• Difference from $17^2$: $289 - 272 = 17$

272 is closer to 256, which is $16^2$. The square root is 16.

Now, apply the final step of the pattern: subtract 1 from the square root.

Missing term = $16 - 1 = 15$.

Conclusion

The number related to 272 by the same pattern is 15.

The analogy follows the pattern:

First number : ($\sqrt{\text{Closest Perfect Square to First Number}}$ - 1)

Let's verify:

• 132: Closest perfect square is 121 ($11^2$). $\sqrt{121} - 1 = 11 - 1 = 10$. Correct.
• 42: Closest perfect square is 36 ($6^2$). $\sqrt{36} - 1 = 6 - 1 = 5$. Correct.
• 272: Closest perfect square is 256 ($16^2$). $\sqrt{256} - 1 = 16 - 1 = 15$.

The missing term is 15.

Revision Table: Key Concepts for Number Analogies

Additional Information: Strategies for Solving Number Reasoning Problems

Number analogy and series problems are common in logical and quantitative reasoning tests. Here are some general strategies to approach them:

• Look for basic arithmetic operations: Check for addition, subtraction, multiplication, or division between the numbers or their digits.
• Consider squares, cubes, and roots: Numbers might be related to perfect squares, cubes, or their roots.
• Analyze differences or ratios: Look at the difference or ratio between consecutive terms. These might follow a pattern.
• Examine digit properties: The relationship might involve the sum of digits, product of digits, or other manipulations of the digits themselves.
• Prime numbers: Numbers in the series or analogy might be prime numbers or related to them.
• Combine operations: Sometimes the pattern involves a combination of operations (e.g., multiply by 2 then add 1).
• Position/Index: In series, the position of the number can be part of the pattern.
• Look for multiple patterns: Sometimes there might be alternating patterns or a combination of rules.

For analogy problems like this one, the key is consistency. The rule you find must work for all the given pairs before you apply it to find the missing term.