In this number analogy, “12 is to 20 as 30 is to ______”. Identify the number that should replace the question mark so that the same numerical pattern or rule holds for both pairs.

Introduction / Context:
This is a number analogy question where we must discover the numerical pattern connecting 12 and 20, and then apply the same pattern to 30 to find the missing term. These questions test recognition of hidden relationships such as products, sums, or patterns in factors, rather than simple addition or subtraction alone.

Given Data / Assumptions:

• First pair: 12 : 20. • Second pair: 30 : ? (unknown). • Options: 15, 35, 42, 48. • We assume a single consistent numerical rule applies to both pairs. • The rule may involve factorisation, patterns with consecutive integers, or simple arithmetic operations.

Concept / Approach:
Rather than guessing, a good strategy is to look at factor patterns. Notice that 12 can be written as 3 * 4 and 20 can be written as 4 * 5. Both numbers are products of consecutive integers, and the second number in the pair uses the next higher consecutive pair. We then check if 30 can be written similarly and see which option matches the product of the next pair of integers.

Step-by-Step Solution:
Step 1: Factorise 12. We can write 12 as 3 * 4. Step 2: Factorise 20. We can write 20 as 4 * 5. Step 3: Observe the pattern. The first number 12 uses the product 3 * 4. The second number 20 uses 4 * 5. So, the right hand number is formed by shifting the pair of consecutive integers up by one: (3,4) → (4,5). Step 4: Express 30 similarly. We can write 30 as 5 * 6. Step 5: Apply the pattern again by increasing each factor by one: (5,6) → (6,7). The product 6 * 7 = 42. Step 6: Therefore, 30 should be related to 42 to preserve the same “product of consecutive integers shifting up” rule.

Verification / Alternative check:
If we tried a simple addition rule like “add 8” (12 + 8 = 20), we would get 30 + 8 = 38, which is not among the options. If we tried doubling 12 (24) and adding some constant, it would not match 20 either. The only neat and consistent pattern that simultaneously explains both 12 and 20 and gives a valid option for 30 is the factor pattern using consecutive integers: 3 * 4 → 4 * 5 and 5 * 6 → 6 * 7.

Why Other Options Are Wrong:
• 15: Not obtained by shifting consecutive factors of 30; 30 is not 3 * 5 in the same sense as the earlier pattern. • 35: Equals 5 * 7, which does not follow directly from 5 * 6 by shifting both factors up by one. • 48: Equals 6 * 8 or 12 * 4, but there is no matching shift rule here comparable to the 12 : 20 pair.

Common Pitfalls:
Candidates often look only at addition or subtraction and miss patterns involving multiplication and factorisation. Another mistake is to force a pattern that only works for the first pair but fails for the second. Always check your hypothesised rule against both given pairs before committing to an answer. Here, the factor pattern uniquely points to 42.

Final Answer:
The missing number that completes the analogy is 42.