5.5 Number Comparisons and Equivalents Across Formats

The Fifth Arithmetic Content Area

The Arithmetic placement test has five content areas of equal weight, each carrying 3 to 5 of the 20 questions. Four of them — whole numbers, fractions, decimals, percent — are computation. The fifth is different:

Number comparisons and equivalents: Comparisons of differently formatted values by ordering, using the number line, and using equality/inequality symbol notation; and evaluation of equivalent number statements (to assess mental math strategies).

That parenthetical is the key: this area is designed to assess mental math strategies, not computational grinding. Handheld calculators are not permitted, and the on-screen calculator appears only where computation is not the construct being measured — which is rarely here.

Format Conversion: The Core Routine

Mixed-format comparison is the dominant question type. The reliable approach is to convert everything to decimals, because decimals order by simple left-to-right digit comparison.

Benchmark Values Worth Memorizing

Recalling these is far faster than dividing under test conditions. The eighths in particular appear constantly.

Conversion Rules

• Fraction → decimal: divide numerator by denominator.
• Decimal → percent: multiply by 100 (shift the point two places right).
• Percent → decimal: divide by 100 (shift two places left).
• Decimal → fraction: write over the matching power of ten, then reduce. 0.35 = 35/100 = 7/20.

Ordering Mixed Formats

Order from least to greatest: 0.62, 5/8, 63%, 3/5

Convert all to decimals:

• 0.62 → 0.620
• 5/8 → 0.625
• 63% → 0.630
• 3/5 → 0.600

Ordered: 3/5 (0.600) < 0.62 (0.620) < 5/8 (0.625) < 63% (0.630).

Notice how tightly clustered these are. Test writers choose values within a few thousandths of each other precisely so that estimation fails and conversion is required. Carry three decimal places when values look close.

Comparing Without Converting

Two fraction shortcuts save time.

Common Numerator

When numerators match, the larger denominator gives the smaller fraction — the whole is cut into more pieces:

3/7 vs 3/5 → sevenths are smaller pieces, so 3/7 < 3/5.

Cross Multiplication

To compare a/b and c/d (positive denominators), compare ad against bc:

Compare 7/12 and 9/16: 7 × 16 = 112; 12 × 9 = 108. Since 112 > 108, 7/12 > 9/16.

This is exact and requires no division.

Negative Numbers on the Number Line

The most common error in this content area involves negatives. On the number line, values increase left to right, so the more negative a number, the smaller it is.

Order: −0.8, −0.35, −1.2, −0.5

Distance from zero: 1.2 > 0.8 > 0.5 > 0.35. Reversing for sign:

−1.2 < −0.8 < −0.5 < −0.35

The trap: students compare −0.8 and −0.35 by looking at 8 versus 35 and conclude −0.35 is smaller. It is not. Aligning decimal places first — −0.80 versus −0.35 — makes the magnitudes comparable, and then the sign reverses the order.

Same logic with negative fractions: −1/2 = −0.5 and −1/3 ≈ −0.333, so −1/2 < −1/3.

Inequality Notation

Compound inequalities read from the middle outward:

−2 < x ≤ 5 means x is greater than −2 and less than or equal to 5. So −2 is excluded; 5 is included.

On a number line, an open circle marks a strict inequality (< or >) and a closed circle marks an inclusive one (≤ or ≥).

Equivalent Number Statements

The second half of the content area evaluates whether two expressions are equal — testing mental strategies rather than computation.

Compensation

Is 48 × 25 equal to 12 × 100?

Yes. Divide 48 by 4 and multiply 25 by 4: 48 × 25 = 12 × 100 = 1,200. Halving one factor while doubling the other preserves a product, and the same idea generalizes.

Decomposition

Is 7 × 98 equal to 7 × 100 − 7 × 2?

Yes — the distributive property. 700 − 14 = 686.

Recognizing Non-Equivalence

Is 3/4 + 1/2 equal to 4/6?

No. Adding fractions requires a common denominator: 3/4 + 2/4 = 5/4 = 1.25. Adding numerators and denominators separately is a classic error that produces 4/6 ≈ 0.667.

Is 0.5 × 0.5 equal to 0.25?

Yes — 25/100. Students sometimes expect 0.5 × 0.5 = 0.5 because multiplying by one half of something feels like it should preserve size. Multiplying two numbers between 0 and 1 always produces a smaller result than either factor.

Place-Value Alignment

When comparing decimals of unequal length, pad with zeros:

Compare 0.7 and 0.68. Pad: 0.70 vs 0.68 → 0.7 > 0.68.

Without padding, students see "68 is bigger than 7" and choose wrongly. This is the single most common decimal-comparison error, and padding eliminates it entirely.