In the number analogy “40 : 100 :: 56 : ____”, which number should replace the blank to follow the same rule?

Introduction / Context:
This numerical analogy compares two number pairs and asks you to identify the pattern that links the first number to the second in each pair. You are given “40 : 100” as the first pair and “56 : ____” as the second, with several options. Your task is to determine how 100 is derived from 40 and then apply the same rule to 56 to find the correct corresponding number. This type of question tests your ability to identify simple multiplicative relationships.

Given Data / Assumptions:

• First pair: 40 and 100.
• Second starting number: 56.
• Options: 140, 112, 118, 148, 104.
• We assume simple arithmetic operations such as multiplication or division.

Concept / Approach:
Start by looking for a straightforward operation that converts 40 into 100. One obvious approach is multiplication. If we multiply 40 by 2, the result is 80, which is too small. If we multiply 40 by 2.5, we get 40 * 2.5 = 100 exactly. Thus, the rule appears to be “multiply the first number by 2.5”. To keep consistency, we must apply this same rule to 56. So, 56 * 2.5 = 140. Therefore, the correct answer is 140.

Step-by-Step Solution:

Step 1: Examine the first pair. Try simple operations: 40 + 60 = 100, but adding 60 is less useful as a general rule. Step 2: Try multiplication. 40 * 2 = 80, 40 * 3 = 120, but neither equals 100. Step 3: Consider multiplying by a fraction. 40 * 2.5 = 40 * (5 / 2) = (40 * 5) / 2 = 200 / 2 = 100, which matches the second number. Step 4: Accept the rule: second number = first number * 2.5. Step 5: Apply the rule to 56. Compute 56 * 2.5 = (56 * 5) / 2 = 280 / 2 = 140. Step 6: Choose 140 from the options as the number that correctly fits the pattern.

Verification / Alternative check:
Check if any alternative rule could be consistent for both pairs. For example, adding 60 to 40 gives 100, but then adding 60 to 56 would produce 116, which is not among the options. Squaring or cubing 40 would give very large numbers that do not match 100. The factor 2.5 is the only simple, exact multiplier that works for the first pair and produces one of the options for the second pair.

Why Other Options Are Wrong:
112 is 56 * 2, 118 is a nearby but arbitrary value, 148 would be 56 * 2.64 approximately, and 104 would be 56 * 1.857. None of these multipliers match the factor of 2.5 discovered from the first pair. Since the pattern must be consistent, no option other than 140, which equals 56 * 2.5, can be correct.

Common Pitfalls:
A common mistake is to assume that the relationship must use an integer multiplier, so students try to force a rule like “multiply by 2 and add something.” However, analogies often use fractional multipliers when they yield clean results. Another pitfall is to guess based solely on closeness to 100 or 150 rather than determining the actual operation. Always derive the rule from the first pair and confirm it with the second.

Final Answer:
Following the rule “multiply by 2.5,” we get 40 → 100 and 56 → 140, so the analogy is “40 : 100 :: 56 : 140”.