Non-Geometric Comparisons: Expressions & Quantitative Relationships

Non-Geometric Comparisons: Expressions & Quantitative Relationships

Non-geometric comparison questions are about 8 of the 52 items on the HSPT Quantitative Skills subtest (52 questions in 30 minutes, roughly 35 seconds each). The item gives three computed numerical quantities labeled (a), (b), and (c) — for example 40% of 80, 25% of 120, and 0.3 x 100 — under the instruction "Examine (a), (b), and (c) and find the best answer."

The four answer choices are complete relationship statements, such as (a) is greater than (b), and (b) and (c) are equal, or (a), (b), and (c) are all equal. Two consequences follow:

• There is no "cannot be determined" option. Every quantity is fully computable, so an item that seems ambiguous has been misread.
• A half-true statement is wrong. Verify every clause of a choice before selecting it.

The key to speed is fast computation, transformation rules that let you compare without evaluating, and benchmark estimation.

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Golden Rules of Comparison Manipulation

To compare any two of the labeled quantities, treat them like the two sides of an inequality sign. Any operation that preserves the direction of an inequality can be applied to both quantities at once to simplify them.

Allowed Operations (Inequality Preserved)

1. Add or Subtract the Same Quantity: You may add or subtract any number or algebraic term from both sides.
   • Example: Compare (a) = x + 14 with (b) = 2x + 9. Subtract x + 9 from both ⇒ 5 versus x.
2. Multiply or Divide by a Known POSITIVE Quantity: You can multiply or divide both sides by any positive number.
   • Example: Compare (a) = y/4 with (b) = y/3 where y > 0. Multiply both by 12 ⇒ 3y versus 4y. Since y > 0, (b) is greater.

Prohibited / Dangerous Operations

1. Never Multiply or Divide by an Unknown Variable: If the sign of x is unknown, multiplying by x may flip the inequality (when x < 0) or preserve it (when x > 0), so the step is invalid. On the HSPT the quantities are always fully determined, so if this situation seems to arise you have overlooked a constraint printed in the item.
2. Never Square Both Quantities Unless Both Are Guaranteed Positive: Squaring negatives flips relative magnitudes (e.g., -5 < -2, but (-5)² = 25 > (-2)² = 4).

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Fast Percent Computation

Convert every percent to a fraction or decimal before comparing.

Worked Example

Which is greatest: (a) 40% of 80, (b) 25% of 120, (c) 0.3 × 100?

• (a) 0.4 × 80 = 32
• (b) ¼ × 120 = 30
• (c) 0.3 × 100 = 30

(a) is greatest—and notice that (b) and (c) tie exactly, a favorite HSPT pattern, so the correct statement must mention both the winner and the tie.

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Estimate First, Compare Without Full Evaluation

• Estimate to eliminate: 49% of 202 ≈ half of 200 = 100; if a rival quantity is 130, you are done in seconds.
• Exploit commutativity: '35% of 60' and '60% of 35' both equal 21—the quantities are equal with no hard computing.
• Compare leftover factors: For '12 × 48' vs '11 × 49', write both as 11 × 48 plus a leftover (48 vs 11); the shared part cancels, so 12 × 48 is greater.

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Common Traps

• Percent of vs. percent increase: 20% of 50 = 10, but '50 increased by 20%' = 60—know whether the item wants the part or the new total.
• Order of operations: 3 + 4 × 5 = 23, not 35; 100 − 40 ÷ 5 = 92, not 12. Apply PEMDAS first.
• Decimal-point slips: 0.3 × 100 = 30, not 3.

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Fraction and Decimal Comparison Tactics

When comparing fractional quantities, avoid long division with these high-speed shortcuts:

1. The Cross-Multiplication Method

To compare a/b and c/d where the denominators b and d are positive:

Compare a × d against b × c

Worked Example: Compare 7/13 with 11/20.

• Cross-product over the first fraction: 7 × 20 = 140
• Cross-product over the second fraction: 13 × 11 = 143
• Since 143 > 140, 11/20 is greater.

2. Benchmark Comparisons (1/2 or 1)

Compare fractions to standard benchmarks like 1/2:

• 9/19 is less than 1/2 (since half of 19 is 9.5).
• 13/24 is greater than 1/2 (since half of 24 is 12).
• Therefore, 13/24 > 9/19.

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Exponent, Root, and Absolute Value Dynamics

The behavior of powers (x^n) and roots (√x) changes dramatically depending on whether the base is greater than 1, between 0 and 1, or negative.

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The Strategic Five-Number Test Method

HSPT items always constrain their quantities enough to be decided, but the same behavior — how expressions change across the number line — drives many of the traps. Testing these five representative values is the fastest way to verify your own algebra and to see why a tempting choice fails:

          Negative Region         Zero      Fractional      Integer
  <───[ -2 ]───────[ -1/2 ]───────[ 0 ]───────[ 1/2 ]───────[ 2 ]───>
     Neg Integer  Neg Fraction   Zero     Pos Fraction   Pos Integer

1. x = 2 (Positive Integer Test)
2. x = 1/2 (Positive Fraction Test — crucial for exponent/fraction inversions)
3. x = 0 (Zero Test — collapses product terms to zero)
4. x = -1/2 (Negative Fraction Test)
5. x = -2 (Negative Integer Test — exposes even vs odd exponent differences)

Rule: If two different test values produce two different orderings, the comparison genuinely depends on the value of x. On the HSPT that means you have missed a stated restriction — reread the item for a condition such as x is a positive integer or x is greater than 1, which will pin the ordering down.