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Comparison Type (Greater / Smaller)

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Introduction

In this pattern, two quantities are compared to determine which is greater, smaller, or equal. You are not asked to compute exact values - only to decide whether the given data is sufficient to establish the comparison.

Pattern: Comparison Type (Greater / Smaller)

Pattern: Comparison Type (Greater / Smaller)

Key concept - Determine whether the given statements can conclusively show if A > B, A < B, or A = B.

Each statement may present equations, ratios, or relations. You must test whether these are enough to establish a clear comparison between A and B.

Step-by-Step Example

Question

Which is greater - A or B?
(I) A = B + 3
(II) A² = B² + 9

Solution

  1. Step 1: Analyze Statement (I)

    From (I): A = B + 3 ⇒ A - B = 3 → a positive constant.
    Hence, A is always greater than B regardless of sign or value of B.
    ✅ (I) alone is sufficient.

  2. Step 2: Analyze Statement (II)

    From (II): A² = B² + 9 ⇒ A² - B² = 9 ⇒ (A - B)(A + B) = 9.
    The result depends on the specific values of A and B (for example, A - B could be 9 or 3 or even -9, depending on A + B).
    So, (II) alone is ambiguous. ❌ Insufficient.

  3. Step 3: Combine

    Even when both are combined, (I) alone already gives a definite result; (II) adds nothing new.

  4. Final Answer:

    Only (I) is sufficient

  5. Quick Check:

    (I) → A - B = 3 → A > B always ✅
    (II) → multiple possible cases ❌

Quick Variations

1. Comparison using differences (A - B = constant).

2. Comparison using ratios (A/B = k), which may depend on sign of B.

3. Comparison using squares or absolute values often leads to ambiguity unless sign information is provided.

Trick to Always Use

  • Step 1: Express A - B if possible - check whether it’s always positive or negative.
  • Step 2: Watch for sign ambiguity when equations involve A², |A|, or ratios.
  • Step 3: If sign of one variable affects the result, that statement is not sufficient.

Summary

  • A statement is sufficient only if it always establishes A > B, A < B, or A = B.
  • Any case-dependent result (depending on sign or value) means insufficiency.
  • Check both statements independently, then together if needed.
  • Equations with squares or absolute values frequently lose direction - handle carefully.

Example to remember:
(I) A = B + 3 → A > B always (sufficient).
(II) A² = B² + 9 → ambiguous (insufficient).

Practice

(1/5)
1. Which is greater - A or B?
(I) A = B + 4
(II) A² = B² + 16
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    A = B + 4 ⇒ A - B = 4 → A is always 4 greater than B. Hence, A > B always.

  2. Step 2: Analyze (II)

    A² = B² + 16 ⇒ A² - B² = 16 ⇒ (A - B)(A + B) = 16. The result depends on values of A and B, so it’s ambiguous.

  3. Final Answer:

    Only (I) is sufficient → Option A

  4. Quick Check:

    (I) → A - B = 4 → A > B always ✅
Hint: If A - B is a positive constant, it’s always sufficient.
Common Mistakes: Treating squared difference as clear directional information.
2. Which is greater - A or B?
(I) A/B = 2
(II) A + B = 0
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Both statements together are necessary
D. Each statement alone is sufficient

Solution

  1. Step 1: From (I)

    A/B = 2 ⇒ A = 2B. If B > 0, A > B; if B < 0, A < B → depends on sign of B → insufficient.

  2. Step 2: From (II)

    A + B = 0 ⇒ A = -B. Alone, cannot compare without specific sign values.

  3. Step 3: Combine

    From (I) and (II): 2B = -B ⇒ 3B = 0 ⇒ B = 0 ⇒ A = 0 → A = B. Combined gives equality conclusively.

  4. Final Answer:

    Both statements together are necessary → Option C

  5. Quick Check:

    Only when combined we get A = B ✅
Hint: If ratio and sum are given, combine to resolve sign ambiguity.
Common Mistakes: Assuming ratio alone is sufficient without considering sign cases.
3. Compare values of X and Y.
(I) X - Y = 5
(II) X + Y = 20
medium
A. Each statement alone is sufficient
B. Only (I) is sufficient
C. Only (II) is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: From (I)

    X - Y = 5 → X is always greater than Y by 5 units → sufficient.

  2. Step 2: From (II)

    X + Y = 20 → can’t decide which is greater without relative info → insufficient.

  3. Final Answer:

    Only (I) is sufficient → Option B

  4. Quick Check:

    (I) → clear X > Y ✅
Hint: Difference (X-Y) directly determines direction of comparison.
Common Mistakes: Misreading sum (X+Y) as comparison indicator.
4. Which is greater - P or Q?
(I) P + Q = 20
(II) P - Q = 0
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    P + Q = 20 → gives total, not comparison → insufficient.

  2. Step 2: Analyze (II)

    P - Q = 0 ⇒ P = Q → comparison clear (equal).

  3. Final Answer:

    Only (II) is sufficient → Option B

  4. Quick Check:

    (II) alone → P = Q ✅
Hint: When difference equals zero, both are equal → sufficient.
Common Mistakes: Assuming totals (sums) reveal comparisons.
5. Compare M and N.
(I) M² = N²
(II) M = N
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Both statements together are necessary
D. Each statement alone is sufficient

Solution

  1. Step 1: Analyze (I)

    M² = N² ⇒ M = ±N → could be equal or opposite → insufficient.

  2. Step 2: Analyze (II)

    M = N → clearly M and N are equal → sufficient.

  3. Final Answer:

    Only (II) is sufficient → Option B

  4. Quick Check:

    (I) ambiguous sign; (II) clear equality ✅
Hint: Square equations lose sign direction - insufficient alone.
Common Mistakes: Assuming M² = N² implies M = N always.