In the following number pair analogy, 19 is related to 367. Using the same pattern, which of the following number pairs correctly completes the relation ? : ?

Introduction / Context:
This is a number pair analogy where the second number appears to be closely related to the square of the first number. The pair 19 and 367 is given, and we must discover the rule connecting them. Then we have to find another pair in the options that follows exactly the same rule. Such questions often use simple power operations plus or minus a small constant.

Given Data / Assumptions:

• Given pair: 19 : 367
• Option pairs: 21 : 447, 22 : 491, 29 : 850, 31 : 963.
• We assume typical exam level patterns, especially square plus or minus constant.

Concept / Approach:
To find the relation between 19 and 367, start by squaring 19 because 367 is close to 19 squared. If 19 squared plus or minus a constant equals 367, that constant might also be applied to the squares of the numbers in the options. We then check which option pair satisfies the same square plus constant pattern.

Step-by-Step Solution:
Step 1: Compute the square of 19. We have 19^2 = 361. Step 2: Compare 361 with 367. The difference is 367 minus 361 equal to 6. Thus 367 = 19^2 + 6. Step 3: Conclude that the rule is "second number equals square of the first number plus 6". Step 4: Apply this rule to each option. For 21, compute 21^2 = 441, then add 6 to get 447. Step 5: Check if 21 : 447 is present as an option. It is exactly option 21 : 447. Step 6: For completeness, note that 22^2 + 6 = 484 + 6 = 490, not 491. Similarly, 29^2 + 6 = 841 + 6 = 847, not 850, and 31^2 + 6 = 961 + 6 = 967, not 963.

Verification / Alternative check:
The pattern 19^2 + 6 = 367 is straightforward and does not require any extra steps. Applying it to each candidate, only the pair 21 and 447 satisfies second equal to first squared plus 6. Since the rule is simple and consistent, and only one option fits it, we can be confident in the answer.

Why Other Options Are Wrong:
For 22 : 491, the second number would have to be 490 to follow the same pattern, not 491. The pair 29 : 850 deviates because 850 is 9 more than 841, not 6. The pair 31 : 963 also does not match since 31^2 is 961 and 963 differs by 2, not by 6. Thus none of these follow the precise rule that was derived from the original pair.

Common Pitfalls:
Some learners may try to fit more complex formulas, including cubes or combinations of digits, instead of first checking the simplest square based pattern. Others may make calculation mistakes when squaring larger two digit numbers. A disciplined approach is to test n^2, then n^2 plus or minus a small constant. This solves many questions quickly without unnecessary complexity.

Final Answer:
The number pair that correctly completes the analogy 19 : 367 :: ? : ? is 21 : 447.