In the number analogy "97 : 63 :: 67 : ____", which number correctly completes the pattern using a product of digits?

Introduction / Context:
This is a numerical analogy where the second number in each pair is formed by using the digits of the first number in a particular way. The pair "97 : 63" suggests that 63 has been derived from 97 by applying some digit based operation. You must uncover that operation and then apply it to 67 in order to find the correct answer from the options. Such questions test your ability to experiment with sums, differences, and products of digits.

Given Data / Assumptions:

• First pair: 97 corresponds to 63.
• Second pair: 67 corresponds to an unknown value.
• Options: 38, 56, 42, 45, 36.
• We assume the relationship uses simple operations on individual digits.

Concept / Approach:
One common approach is to multiply the digits of the first number. For 97, the digits are 9 and 7, and their product is 9 * 7 = 63, which matches the second number in the pair. Therefore, the rule is that the second number is the product of the two digits of the first number. To complete the analogy, we apply the same pattern to 67. The digits 6 and 7 give 6 * 7 = 42, so 42 must be the missing term.

Step-by-Step Solution:

Step 1: Take the number 97 and separate its digits: 9 and 7. Step 2: Compute the product of the digits: 9 * 7 = 63. Step 3: Confirm that the result 63 matches the given pair, so the rule is consistent. Step 4: Now take the number 67 and separate its digits: 6 and 7. Step 5: Compute the product of these digits: 6 * 7 = 42. Step 6: Select 42 from the options as the number that fits the pattern.

Verification / Alternative check:
Check the other options quickly. 38 would require 6 and 7 to combine in a more complex way, such as 6 * 7 plus or minus something, which is not supported by the first pair. 56, 45, and 36 are also not equal to 6 * 7. Because the pattern 9 * 7 = 63 is clean and directly mirrored by 6 * 7 = 42, no other candidate fits as well as 42. This confirms that we have identified the intended rule.

Why Other Options Are Wrong:
The values 38, 56, 45, and 36 do not equal the product of the digits 6 and 7. Any rule that would generate these numbers from 67 would fail to produce 63 from 97, so it would not be a valid analogy rule. Since analogies require a single consistent relationship across pairs, these alternatives must be rejected.

Common Pitfalls:
Students sometimes attempt to use the sum of digits, difference of digits, or even reverse the digits without checking whether these operations reproduce the second number in the first pair. The simplest habit is always to start by trying sums and products of digits and see whether they match. Here, the product of digits instantly gives the correct mapping, making the problem straightforward once that is tried.

Final Answer:
Using the product of digits rule, 97 maps to 9 * 7 = 63 and 67 maps to 6 * 7 = 42. So the completed analogy is "97 : 63 :: 67 : 42".