Quantitative comparison
Two quantities to compare. The four choices never change, in the same order, so learn them and don’t waste time reading them.
Two quantities to compare. The four choices never change, in the same order, so learn them and don’t waste time reading them.
The four choices
| Choice | Meaning |
|---|---|
| A، Quantity A is greater | A is larger in every case |
| B، Quantity B is greater | B is larger in every case |
| C، The two are equal | Equal always |
| D، Cannot be determined | The relationship changes with the values |
Method
- Simplify or work out both quantities.
- If both are plain numbers, the answer is never D.
- With variables, try zero, one, a negative, a fraction, a big number as the condition allows.
- If the result changes between tries → D. If it never changes → choose what stayed true.
Comparison patterns
Numbers and fractions
Compare fractions by cross-multiplying.
Variables with a condition
Test special values; the next table sums up the classic cases.
Percentages
Trick: a% of b = b% of a.
Two rises in a row
Two rises in a row are more than one rise of their total.
Roots and powers
Square both sides when positive: √50 vs 7 → 50 vs 49.
Geometry comparisons
In a triangle, the longest side faces the largest angle.
Root rules
√a + √b is always bigger than √(a + b), and √a × √b = √(a × b). For a fraction between 0 and 1: its root is bigger and its square is smaller.
Different units
Convert both quantities to the same unit before comparing.
Powers of numbers
Write both sides with the same base and compare the powers, or work them out if they are small.
Areas of different shapes
Work out each area and use π ≈ 3.14 when comparing with a plain number.
Two means
If both lists have the same count, just compare the totals, no dividing needed.
Sum of consecutive numbers
1 + 2 + … + n = n(n + 1) 2
A negative or fractional base
Negative base: even power positive, odd power negative. A fraction between 0 and 1: the bigger the power, the smaller it gets.
Two close fractions
Cross-multiply: a × d against c × b; the bigger product marks the bigger fraction.
A shaded figure against a fraction
Turn the figure into a fraction first (shaded ÷ total), then cross-multiply.
Mean vs median
Work out both; extreme values pull the mean but don’t move the median.
Classic cases, memorise this table
| Condition | Comparison | Result |
|---|---|---|
| 0 < x < 1 | x vs x² | x is greater |
| x > 1 | x vs x² | x² is greater |
| x < 0 | x² vs x³ | x² is greater |
| any real x | 3x vs 2x | cannot tell |
| x² = 9 | x vs 3 | cannot tell (±3) |
| x³ = 8 | x vs 2 | equal |
| x > 0 | x/2 vs x/3 | A is greater |
| x < 0 | x/2 vs x/3 | B is greater |
| any real x | x² vs 0 | cannot tell (may be equal) |
- The “0-1-neg-frac-big” rule: zero, one, a negative, a fraction, a big number, try them in order in every comparison with a variable.
Advanced comparisons
Two percent changes against one
Work out the combined change a + b + ab÷100, then compare.
A figure with an exterior angle, triangles between parallels, a decimal area, and “3x = 4y”
Get the interior angle from the exterior one (180° − exterior), then the longer side faces the larger angle. Triangles between parallels: compare the bases. “3x = 4y” alone: not enough information (they may be negative or zero); with “positive”, x > y.