Difficulty: Medium
Correct Answer: 132
Explanation:
Introduction:
This is a number analogy question, where you must identify the relationship between the first pair of numbers and then apply the same pattern to the second pair. Such questions test pattern recognition, factorisation skills, and the ability to see multiplicative relationships rather than just simple addition or subtraction.
Given Data / Assumptions:
- The given analogy is 42 : 56 :: 110 : ?
- The first pair is 42 and 56.
- The second starting number is 110, and we must find the related number from the options 18, 132, 136, and 140.
- We assume a consistent rule links the first number of each pair to the second number of that pair.
Concept / Approach:
A useful approach with such numbers is to factorise them, because many analogies are based on products of consecutive integers. Notice that 42 = 6 * 7 and 56 = 7 * 8. Therefore, the rule appears to be: take a number that is the product of two consecutive integers n and n+1, and map it to the product (n+1) * (n+2). If this rule holds, then we can apply it to 110 by first finding consecutive factors that multiply to 110.
Step-by-Step Solution:
Step 1: Factorise 42. We get 42 = 6 * 7, which are consecutive integers.Step 2: Factorise 56. We get 56 = 7 * 8, again consecutive integers, where 7 is the middle number in both products.Step 3: Observe the pattern: 6 * 7 (which is 42) is mapped to 7 * 8 (which is 56). The first pair uses 6 and 7, and the second pair uses 7 and 8. So the rule is: n * (n+1) → (n+1) * (n+2).Step 4: Factorise 110. We find 110 = 10 * 11, which again are consecutive integers, matching the pattern we suspect.Step 5: Apply the rule: if 110 = 10 * 11, then the related number should be 11 * 12 = 132. This matches option B.
Verification / Alternative check:
To confirm, check that no simpler pattern (like adding a fixed number) works consistently. From 42 to 56, the difference is 14. If we added 14 to 110, we would get 124, which is not in the options. Doubling or other simple operations also do not give 56 from 42. The consecutive product pattern is the only neat and consistent rule matching both pairs and leading to an option present in the list, namely 132.
Why Other Options Are Wrong:
Option A, 18, has no clear relation to 110 in this context and does not arise from 11 * 12 or any simple extension of the pattern. Option C, 136, corresponds to 8 * 17 or other non consecutive factors, so it does not follow the consecutive product rule. Option D, 140, equals 10 * 14 or 7 * 20, again lacking the pattern of (n+1) * (n+2) where 10 * 11 is the starting point. Only 132 can be written as 11 * 12, which fits the analogy perfectly.
Common Pitfalls:
Many students initially look for simple addition, subtraction, or percentage changes between the numbers and miss the importance of factorisation. Others may factorise but not notice that the factors are consecutive or that one factor is carried forward from the first product to the next. A good habit is to immediately check whether numbers like 42, 56, and 110 can be expressed as products of nearby integers, because that often reveals the pattern in analogy questions.
Final Answer:
Using the rule n * (n+1) → (n+1) * (n+2), 110 = 10 * 11 maps to 11 * 12, which is 132.
Discussion & Comments