Each question contains Quantity I and Quantity II. Read the details and compare the two quantities.\n\nQuantity I: Three years ago, the ratio of the ages of A and B was 3 : 4. Two years from now, the sum of their ages will be 45 years. What is the present age of A?\n\nQuantity II: Five years ago, the ratio of the ages of P and Q was 3 : 4. P’s age after 6 years will be equal to the present age of Q. What is the present age of P?\n\nBased on these values, how does Quantity I compare with Quantity II?

Introduction / Context:
This is a quantitative comparison question involving two different age scenarios. We must compute the present age of A in Quantity I and the present age of P in Quantity II, and then compare these two numerical values. Such questions test your ability to solve age problems and then interpret the relative sizes of derived quantities.

Given Data / Assumptions:

- Quantity I: Three years ago, the ratio of A's age to B's age was 3 : 4. Two years from now, the sum of their ages will be 45 years.
- Quantity II: Five years ago, the ratio of P's age to Q's age was 3 : 4. P's age after 6 years will be equal to Q's present age.
- Ages are positive whole numbers.

Concept / Approach:
We treat each quantity as a separate age problem, form the necessary equations, and solve for the required present ages. Once we have a numerical value for Quantity I and a numerical value for Quantity II, we simply compare them to decide whether Quantity I is greater, smaller or equal to Quantity II. The comparison is then matched with the correct option.

Step-by-Step Solution:
Step 1 (Quantity I): Let the present ages of A and B be A and B years respectively. Step 2 (Quantity I): Three years ago, their ages were A − 3 and B − 3, and the ratio was (A − 3) : (B − 3) = 3 : 4. Step 3 (Quantity I): Two years from now, their ages will be A + 2 and B + 2, and their sum will be 45: (A + 2) + (B + 2) = 45 ⇒ A + B = 41. Step 4 (Quantity I): From the ratio, (A − 3) / (B − 3) = 3 / 4. Step 5 (Quantity I): Also B = 41 − A. Substitute into the ratio: (A − 3) / (41 − A − 3) = 3 / 4 ⇒ (A − 3) / (38 − A) = 3 / 4. Step 6 (Quantity I): Cross-multiply: 4(A − 3) = 3(38 − A) ⇒ 4A − 12 = 114 − 3A ⇒ 7A = 126 ⇒ A = 18. Step 7 (Quantity I): So Quantity I = 18 years. Step 8 (Quantity II): Let the present ages of P and Q be P and Q years respectively. Step 9 (Quantity II): Five years ago, their ages were P − 5 and Q − 5, and (P − 5) : (Q − 5) = 3 : 4. Step 10 (Quantity II): Also, P's age after 6 years equals Q's present age: P + 6 = Q. Step 11 (Quantity II): Use the ratio equation: (P − 5) / (Q − 5) = 3 / 4 and substitute Q = P + 6. Step 12 (Quantity II): Then (P − 5) / (P + 6 − 5) = 3 / 4 ⇒ (P − 5) / (P + 1) = 3 / 4. Step 13 (Quantity II): Cross-multiply: 4(P − 5) = 3(P + 1) ⇒ 4P − 20 = 3P + 3 ⇒ P = 23. Step 14 (Quantity II): So Quantity II = 23 years.

Verification / Alternative check:
We now compare Quantity I and Quantity II numerically: Quantity I = 18 and Quantity II = 23. Clearly, 18 is less than 23. This matches the verbal statement that Quantity I is smaller than Quantity II. There is no ambiguity because each quantity is determined uniquely by its respective conditions.

Why Other Options Are Wrong:
Option a claims Quantity I is greater than Quantity II, which is false since 18 < 23. Option c claims they are equal, which is also false. Option d states both quantities are less than 20, but Quantity II is 23, so this cannot be true. Option e claims both are greater than 25, which is also wrong. Only the statement that Quantity I is smaller than Quantity II matches the computed values.

Common Pitfalls:
Errors can arise from misinterpreting the time phrases like "three years ago" and "after 6 years", or from setting up the ratios incorrectly. Another common mistake is to try to compare the quantities without fully solving them, which can lead to incorrect assumptions. Always write clear equations for each quantity, solve them completely, and only then perform the comparison.

Final Answer:
We find Quantity I = 18 and Quantity II = 23, so Quantity I is smaller than Quantity II.