Complete the proportion 16 : 56 :: 32 : ?. By applying the same numerical relationship that relates 16 to 56, determine the missing number.

Introduction / Context:
This is another analogy question of the form a : b :: c : ?. We must determine what number stands in the same relation to 32 as 56 stands to 16. Such problems test pattern recognition and the ability to identify simple arithmetic relations such as multiplication by a constant factor.

Given Data / Assumptions:
• First pair: 16 and 56. • Second pair: 32 and an unknown number. • We assume a consistent numerical rule connects each pair.

Concept / Approach:
We look for a simple relationship such as multiplication, addition or a combination that converts 16 to 56. The simplest pattern is multiplication by a constant. Observing 16 * 3.5 = 56 suggests that the rule might be "multiply by 3.5". If that is the consistent rule, then the missing number is 32 * 3.5. We check if that value appears among the options.

Step-by-Step Solution:
Step 1: Examine 16 to 56: compute 56 / 16. Step 2: 56 / 16 = 3.5, so 16 * 3.5 = 56. Step 3: Assume the same multiplier applies to 32. Step 4: Compute 32 * 3.5 = 32 * (7 / 2). Step 5: 32 * 7 = 224; now divide by 2: 224 / 2 = 112. Step 6: Therefore, the missing number should be 112.

Verification / Alternative check:
Check whether any other simple pattern exists. For instance, if we tried addition, 16 + 40 = 56, but 32 + 40 = 72, which does not match any given option. If we tried squaring or other operations, they would not consistently match both pairs. The constant multiplier 3.5 works neatly with both 16 and 32 and directly yields one of the provided options, 112.

Why Other Options Are Wrong:
• 196, 148 and 132 do not follow from multiplying 32 by any simple factor that also converts 16 to 56 in the same way. • Choosing any of these would break the required proportional relationship between the pairs.

Common Pitfalls:
Candidates sometimes overcomplicate the pattern or assume only integer multipliers are allowed. Another common error is to guess without checking whether the same rule works consistently for both pairs. Always compute the transformation on the first pair and verify that it applies cleanly to the second.

Final Answer:
The number that correctly completes the proportion is 112.