In this numerical analogy question, 56 is related to 72, so you must select the number that stands in the same relation to 90 from the given alternatives.

Introduction / Context:

This is a numerical analogy where the relationship between the first pair of numbers needs to be understood and then applied to the second number to find the correct partner. The pair "56 : 72" suggests a pattern involving products of consecutive integers. Once that pattern is uncovered, it can be applied to 90 to obtain the required number.

Given Data / Assumptions:

- First pair: 56 and 72.
- Second pair to complete: 90 and ?.
- Options: 96, 97, 100, 110 and 120.
- We assume standard arithmetic operations and knowledge of basic multiplication tables.

Concept / Approach:

We look for a factorisation pattern in 56 and 72. Notice that 56 can be written as 7 * 8, the product of two consecutive integers. Similarly, 72 is 8 * 9, again the product of two consecutive integers, with 8 being the common factor. Thus, the transformation from 56 to 72 can be seen as shifting from 7 * 8 to 8 * 9. We will check if 90 can also be expressed as a product of consecutive integers and then advance that pattern by one step.

Step-by-Step Solution:

Verification / Alternative check:

We can verify by thinking of the sequence of such products: 7 * 8 = 56, 8 * 9 = 72, 9 * 10 = 90, 10 * 11 = 110. Clearly, the numbers 56, 72, 90 and 110 form a simple pattern where each is the product of two successive integers, and the second element in each pair uses the next consecutive product. None of the other options (96, 97, 100 or 120) can be expressed as the immediate next product of consecutive integers following 9 * 10.

Why Other Options Are Wrong:

96 is 12 * 8 or 16 * 6, but neither factor pair is a pair of consecutive integers directly following 9 and 10.

97 is a prime number and cannot be expressed as a product of two natural numbers greater than 1, so it clearly breaks the pattern.

100 is 10 * 10, which uses equal factors, not consecutive integers like 9 and 10 or 10 and 11.

120, although equal to 10 * 12, has factors that differ by 2, so it does not fit the simple consecutive product rule.

Common Pitfalls:

Some candidates attempt to find patterns in differences or ratios, such as adding a fixed number or multiplying by a constant, but those do not work neatly here. Others may notice that 72 is 56 plus 16 and then try to add the same difference to 90, which yields 106, not an option. The clean factor pattern is much more consistent and leads directly to the correct answer.

Final Answer:

Following the product-of-consecutive-integers pattern, 90 is correctly paired with 110.