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Quantity Comparison (Quantity I vs Quantity II): How to Decide the Relation

Compute both quantities • Two-root cases • ≥ and ≤ • When the relation cannot be established

Banking 📅 Last updated: 2026-10-24

In a quantity comparison question, bank exams give two expressions, Quantity I and Quantity II, and ask which relation holds between them. The standard options are: I > II, I < II, I ≥ II, I ≤ II, I = II, and "relation cannot be established". The topic tests ordinary arithmetic, from percentages and interest to quadratics and probability, but the scoring trap is choosing the relation.

The method has two steps. First, compute each quantity on its own, writing every possible value if there is more than one. Then compare all the possible values before choosing an option.

When each quantity is one fixed number, the answer is simply >, < or =.

Worked example 1: equal values

Quantity I: 20% of 450 = 90. Quantity II: 3/5 of 150.

Quantity II is 3/5 × 150 = 90, so Quantity I = Quantity II. Equal pairs appear often precisely because careless rounding turns them into > or <.

Worked example 2: interest

Quantity I: simple interest on ₹5,000 at 8% for 3 years = 5,000 × 8 × 3 ÷ 100 = ₹1,200. Quantity II: compound interest on ₹5,000 at 8% for 2 years.

Quantity II is 5,000 × (1.08² − 1) = ₹832, so Quantity I > Quantity II. Comparing the time periods first would have hinted at this: three years of simple interest beats two years of compound interest at the same rate.

Worked example 3: a word problem

Quantity I: A's age, if A is three times as old as B and their ages add up to 48. A = 48 × 3/4 = 36. Quantity II: 35. So Quantity I > Quantity II.

Worked example 4: probability

Quantity I: the probability of two heads in two tosses of a fair coin, 1/2 × 1/2 = 0.25. Quantity II: the probability that two dice show a sum of 7.

Six of the 36 outcomes give a sum of 7, so Quantity II = 6/36 ≈ 0.1667, and Quantity I > Quantity II.

Quadratic equations and square roots give two values. Compare every value of one quantity with every value of the other.

Worked example 5: roots against a number

Quantity I: x, where x² − 7x + 12 = 0. Quantity II: 4. The larger root is (7 + 1) ÷ 2 = 4.

The smaller root is (7 − 1) ÷ 2 = 3. So x is 3 or 4: never more than 4 and equal to it in one case. The relation is Quantity I ≤ Quantity II.

Worked example 6: a square that hides a sign

Quantity I: x, where x² = 25. Quantity II: 2. Here √25 = 5, but x can be 5 or −5. One value is above 2 and one below, so the relation cannot be established.

Contrast this with "Quantity I: √25", which means only the positive root, 5. The wording decides whether the negative value counts.

Possible valuesRelation
Every value of I is above every value of III > II
Every value of I is below every value of III < II
I is never below II, and they can be equalI ≥ II
I is never above II, and they can be equalI ≤ II
Both are the same single valueI = II
I can be above or below IICannot be established

When you have two roots for each quantity, a quick way to compare is to write them on a number line. If one pair ends exactly where the other begins, so that they share only that boundary value, the answer is ≥ or ≤. If the values interleave in any other way, for example x = 3 or 7 against y = 5 or 7, the relation cannot be established.

  • ✘ Forgetting the negative root when a quantity is defined by x² = k.
  • ✘ Choosing > when the quantities can also be equal; that case is ≥.
  • ✘ Rounding during calculation and turning an equal pair into a strict inequality.
  • ✘ Comparing only the larger root of each quadratic.

Quadratic quantity questions use the same factorising as quadratic equations for bank exams. The other prelims sets are number series, approximation and simplification and data sufficiency; see also compound interest formulas and the Quant cheat code.

Q1. Quantity I: 15% of 640. Quantity II: 4/5 of 120. Which relation holds? easy

✅ Answer: (c) Quantity I = Quantity II

💡 Explanation: 15% of 640 is 96, and 4/5 of 120 is also 96. Both quantities are single fixed values and they are equal.

Q2. Quantity I: x, where x² − 9x + 20 = 0. Quantity II: y, where y² − 11y + 30 = 0. Which relation holds? medium

✅ Answer: (b) Quantity I ≤ Quantity II

💡 Explanation: x is 4 or 5, and y is 5 or 6. Every value of x is less than or equal to every value of y, and they can both be 5, so the relation is Quantity I ≤ Quantity II.

Q3. Quantity I: x, where x² = 49. Quantity II: 6. Which relation holds? medium

✅ Answer: (a) Relation cannot be established

💡 Explanation: x can be 7 or −7. If x = 7, Quantity I is larger; if x = −7, it is smaller. Since the relation changes with the root, it cannot be established.

Q4. Quantity I: compound interest on ₹10,000 at 10% a year for 2 years. Quantity II: simple interest on ₹10,000 at 10.5% a year for 2 years. Which relation holds? medium

✅ Answer: (d) Quantity I = Quantity II

💡 Explanation: CI = 10,000 × (1.1² − 1) = ₹2,100. SI = 10,000 × 10.5 × 2 ÷ 100 = ₹2,100. The two interests are exactly equal.

Q5. Quantity I: the speed in km/h of a 180 m train that crosses a pole in 12 seconds. Quantity II: 50 km/h. Which relation holds? easy

✅ Answer: (b) Quantity I > Quantity II

💡 Explanation: The train covers its own length, 180 m, in 12 s, which is 15 m/s. Converting, 15 × 18/5 = 54 km/h, which is more than 50 km/h.

Q6. Quantity I: the average of the first 10 natural numbers. Quantity II: the median of 2, 3, 7, 9, 10. Which relation holds? easy

✅ Answer: (c) Quantity I < Quantity II

💡 Explanation: The average of 1 to 10 is 55 ÷ 10 = 5.5. The median of the five sorted values is the middle one, 7. So Quantity I is less than Quantity II.

When should I choose 'relation cannot be established'?

Choose it when one quantity can take values on both sides of the other, as with a square root that can be positive or negative, or roots that straddle the other value. If every possible value of one quantity stays on one side, a definite relation exists.

What does Quantity I ≥ Quantity II mean in these questions?

It means that in every possible case Quantity I is either greater than or equal to Quantity II, and the two are equal in at least one case. It is the right answer when a shared root makes them equal and all other values put Quantity I above.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 18 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-24 against the sources listed above.

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