Quantitative Comparison Strategy • Topic 2 of 4

Simplify Both Columns Equally

Because you only need the relationship, you may treat a QC like a balance scale and perform the same operation on both quantities without changing which side is heavier — provided the operation is legal. You can add or subtract the same amount from both, and you can multiply or divide both by the same positive number, all without touching the comparison. This lets you strip away shared clutter: cancel a common term, clear a denominator, or subtract a value that appears on both sides, until the comparison becomes obvious. The one rule you must never break: multiplying or dividing both quantities by a negative number flips the inequality, and because a variable might be negative, you generally avoid multiplying both sides by an unknown whose sign you do not know. Squaring both sides is only safe when you know both quantities are non-negative. Used carefully, simplification turns a scary-looking comparison into a one-line answer.

✅ Solved examples

1. Quantity A: 7x + 5. Quantity B: 7x + 9. Which choice?
Subtract 7x from both (legal, and it removes the unknown): Quantity A becomes 5, Quantity B becomes 9. 5 < 9 regardless of x, so Choice B.
2. Quantity A: (3/4) × 88. Quantity B: (2/3) × 88. Which choice?
Divide both by 88 (a positive number, legal): compare 3/4 with 2/3. Cross-multiply: 3×3 = 9 versus 2×4 = 8, so 3/4 > 2/3. Choice A.
3. a > 0. Quantity A: a + 12. Quantity B: a + 15. Which choice?
Subtract a from both: 12 versus 15. 12 < 15 no matter what positive a is, so Choice B. (The value of a is a red herring.)
4. Quantity A: 5⁴. Quantity B: 4⁵. Which choice?
No shared factor to cancel, so compute: 5⁴ = 625, 4⁵ = 1024. Both fixed, and 625 < 1024, so Choice B. (When nothing cancels, the basic calculator or direct computation settles it.)

✏️ Practice — try these, take hints as needed

1. Quantity A: 4x + 7. Quantity B: 4x + 2. Which choice?
Subtract 4x from both.
Compare 7 and 2.
The x drops out.
A
2. Quantity A: (5/6) × 120. Quantity B: (4/5) × 120. Which choice?
Divide both by 120 (positive, legal).
Compare 5/6 and 4/5.
Cross-multiply: 25 vs 24.
A (5/6 > 4/5)
3. y > 0. Quantity A: 3y. Quantity B: 3y − 4. Which choice?
Subtract 3y from both.
0 versus −4.
Zero exceeds a negative.
A
4. Quantity A: 2⁶. Quantity B: 6². Which choice?
Nothing cancels — compute both.
64 versus 36.
Both fixed, so not D.
A (64 > 36)
5. Quantity A: n/5 + 3. Quantity B: n/5 − 1. Which choice?
Subtract n/5 from both.
Compare 3 and −1.
Independent of n.
A

📝 Topic test — 8 questions

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