Each question below contains two quantities, Quantity I and Quantity II. Read the information and decide the relationship between the two quantities. Quantity I: Three numbers are in the ratio 5 : 6 : 10. The sum of the largest and the smallest numbers is 126 more than the middle number. Find the largest number. Quantity II: Twelve percent of the first number is equal to twenty five percent of the second number. The difference between these two numbers is 78. Find the larger number.

Introduction / Context:
This is a comparison type question where two different mathematical situations are described as Quantity I and Quantity II. We need to compute the numerical value in each case and then compare them. Questions like this appear in competitive exams to check a candidate ability to handle multiple small problems and then make a logical comparison rather than only finding individual answers.

Given Data / Assumptions:

• Quantity I: Three numbers are in the ratio 5 : 6 : 10. The sum of the largest and smallest numbers is 126 more than the middle number.
• Quantity II: Twelve percent of the first number equals twenty five percent of the second number. The difference between these two numbers is 78.
• All numbers are real and positive.

Concept / Approach:
For Quantity I we use the idea that numbers in a ratio can be written as multiples of a common variable. We then translate the given condition into an equation in that variable and solve. For Quantity II we interpret the percentage statements as algebraic equations linking two unknown numbers. After solving each situation, we compare the final numerical values of the required largest numbers in both quantities.

Step-by-Step Solution:
For Quantity I, let the three numbers be 5k, 6k and 10k. Smallest = 5k, middle = 6k, largest = 10k. Given: largest + smallest is 126 more than the middle number. So 10k + 5k = 6k + 126. This gives 15k = 6k + 126, so 9k = 126 and k = 14. Largest number in Quantity I = 10k = 10 * 14 = 140. For Quantity II, let the two numbers be A and B, with A larger than B. Given 12% of A = 25% of B. So 0.12 * A = 0.25 * B. Hence A / B = 0.25 / 0.12 = 25 / 12. Let B = 12x and A = 25x. Difference A − B = 25x − 12x = 13x = 78. So x = 78 / 13 = 6. Therefore A = 25x = 25 * 6 = 150. Largest number in Quantity II = 150.

Verification / Alternative check:
Check the conditions for Quantity II: 12% of A = 12% of 150 = 18 and 25% of B = 25% of 72 = 18, so the percentage relation holds. The difference A − B = 150 − 72 = 78, matching the statement. For Quantity I, with k = 14 we have numbers 70, 84 and 140. Largest plus smallest = 210 and middle + 126 = 84 + 126 = 210, so that condition also holds. Thus both values 140 and 150 are correct, and the comparison is valid.

Why Other Options Are Wrong:
Quantity I > Quantity II is wrong because 140 is less than 150. Quantity I = Quantity II is incorrect since the values are not equal. Relationship cannot be determined is incorrect because both values can be uniquely computed. Both quantities are zero is clearly not true given the positive values obtained.

Common Pitfalls:
Typical errors include misidentifying which number is largest in Quantity I or mishandling percentages in Quantity II. Students sometimes invert the ratio when moving from 12% of A equals 25% of B to A : B, which leads to wrong values. Some also forget that the numbers in a ratio must be multiplied by the same factor and instead treat them as final values. A careful step-by-step algebraic approach avoids these mistakes.

Final Answer:
Quantity I is less than Quantity II, so Quantity I < Quantity II.