4.2 Fractions, Decimals, and Value Comparison Shortcuts

4.2 Fractions, Decimals, and Value Comparison Shortcuts

Wonderlic SLE Fast Fact: Fraction and decimal comparison items frequently induce decision paralysis among test-takers who attempt to calculate lowest common denominators (LCDs) or execute multi-step long division. Number comparisons are one of the item types Wonderlic lists for the SLE. Fraction-order, mixed-number, and decimal-magnitude items reward candidates who use benchmark conversions and cross-multiplication instead of common denominators.

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The Cognitive Burden of Traditional Fraction Arithmetic

When an examinee encounters a question such as "Which of the following is greatest: 5/8, 7/11, or 9/14?", the instinctive grade-school response is to search for a common denominator. For 8, 11, and 14, the least common multiple is 616. Calculating $\frac{5 \times 77}{616} = \frac{385}{616}$, $\frac{7 \times 56}{616} = \frac{392}{616}$, and $\frac{9 \times 44}{616} = \frac{396}{616}$ requires nearly a full minute of scratch-paper calculation. Under a 14.4-second pacing constraint, spending 60 seconds on one question severely damages your overall exam score.

Mastering quantitative speed requires adopting two tactical alternatives:

1. Instant Benchmark Conversion: Replacing fractions with known decimal equivalents.
2. The Cross-Multiplication Shortcut: Comparing numerators across denominators pairwise in under 4 seconds.

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Fraction-Decimal-Percent Benchmarks to Memorize

No-calculator items tend to use a small family of benchmark fractions. Memorizing these values turns fraction arithmetic into quick decimal estimation.

The Power of the Eighths Family

The eighths family (1/8, 3/8, 5/8, 7/8) is worth special attention because the decimal expansions terminate in three places (0.125, 0.375, 0.625, 0.875). When questions ask you to compare fractions like 5/8 to 0.62 or 3/8 to 0.38, recalling that 5/8 = 0.625 and 3/8 = 0.375 resolves the inequality instantly without a single calculation (0.625 > 0.62 and 0.375 < 0.38).

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The Cross-Multiplication Comparison Shortcut

When comparing two fractions $\frac{a}{b}$ and $\frac{c}{d}$ that are not familiar benchmarks, never find a common denominator. Instead, execute cross-multiplication:

$$\frac{a}{b} \quad \text{vs} \quad \frac{c}{d}$$ $$\text{Compute: } (a \times d) \quad \text{vs} \quad (b \times c)$$

1. Multiply the numerator of the left fraction by the denominator of the right fraction ($a \times d$), and note this product above the left fraction.
2. Multiply the numerator of the right fraction by the denominator of the left fraction ($b \times c$), and note this product above the right fraction.
3. Compare the two products. Whichever product is larger designates the larger fraction.

Mathematical Proof

Multiplying both sides of the inequality $\frac{a}{b} > \frac{c}{d}$ by the positive product of the denominators ($b \times d$) yields:

$$\frac{a \times (b \times d)}{b} > \frac{c \times (b \times d)}{d} \implies a \times d > b \times c$$

Because b and d are positive integers, this transformation strictly preserves the directional order of the inequality without changing the relative values.

Worked Comparison Examples

• Compare 5/8 and 7/11:

  • Left product: $5 \times 11 = 55$.
  • Right product: $7 \times 8 = 56$.
  • Since 56 > 55, it follows that $\frac{7}{11} > \frac{5}{8}$. Elapsed time: 2 seconds.

• Compare 9/14 and 11/17:

  • Left product: $9 \times 17 = 9 \times (10 + 7) = 90 + 63 = 153$.
  • Right product: $11 \times 14 = 11 \text{ rule on } 14 = 154$.
  • Since 154 > 153, it follows that $\frac{11}{17} > \frac{9}{14}$. Elapsed time: 4 seconds.

Tournament Elimination for Four Fractions

When an item asks to identify the greatest among four fractions [A, B, C, D]:

1. Compare A and B; discard the smaller.
2. Compare the winner against C; discard the smaller.
3. Compare that winner against D. Two or three rapid cross-multiplications identify the maximum fraction in under 12 seconds, completely bypassing common denominators.

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Simplifying Fractions: The Difference Rule for GCF

Simplifying large fractions by testing small primes (dividing by 2, then by 2 again) is painfully slow. To reduce fractions like $\frac{72}{108}$ or $\frac{51}{85}$ instantly, use the Difference Rule for finding the Greatest Common Factor (GCF):

The Difference Rule Theorem: The greatest common factor of two positive integers A and B (where A < B) must divide their positive difference (B - A).

$$\text{GCF}(A, B) \mid (B - A)$$

How to Apply the Difference Rule

1. Calculate the difference: D = B - A.
2. Check if D divides both A and B. If it does, D is the GCF!
3. If D does not divide both numbers, test the largest factors of D.

• Example 1: Simplify 48/64

  • Difference: 64 - 48 = 16.
  • Does 16 divide 48? Yes ($48 \div 16 = 3$).
  • Does 16 divide 64? Yes ($64 \div 16 = 4$).
  • The GCF is 16. The reduced fraction is 3/4.

• Example 2: Simplify 51/85

  • Difference: 85 - 51 = 34.
  • Does 34 divide 51? No (51 is odd).
  • Factors of 34: 1, 2, 17, 34. The largest odd factor is 17.
  • Test 17: $51 \div 17 = 3$; $85 \div 17 = 5$.
  • The GCF is 17. The reduced fraction is 3/5.

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Mixed Numbers and Improper Fractions Speed Rules

Converting mixed numbers into improper fractions before performing operations often inflates numbers unnecessarily, increasing the risk of arithmetic error.

Additive Splitting

For addition, always split the whole numbers from the fractions:

$$4 \frac{2}{3} + 3 \frac{1}{2} = (4 + 3) + \left(\frac{2}{3} + \frac{1}{2}\right) = 7 + \left(\frac{4 + 3}{6}\right) = 7 + \frac{7}{6} = 7 + 1 \frac{1}{6} = 8 \frac{1}{6}$$

Subtraction via Regrouping

When the fractional part of the minuend is smaller than the subtrahend, borrow 1 from the whole number without converting the entire expression to an improper fraction:

• Compute $8 \frac{1}{5} - 3 \frac{4}{5}$:
  1. Recognize that 1/5 < 4/5.
  2. Borrow 1 from 8: convert $8 \frac{1}{5}$ to $7 + \left(1 + \frac{1}{5}\right) = 7 \frac{6}{5}$.
  3. Subtract whole numbers: 7 - 3 = 4.
  4. Subtract fractions: $\frac{6}{5} - \frac{4}{5} = \frac{2}{5}$.
  5. Result: $4 \frac{2}{5}$.

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Decimal Alignment & Place-Value Traps

Decimal comparison questions assess whether candidates understand that decimal value is governed by place value, not the count of digits after the point.

The "Longer Decimal" Intuition Trap

Under time pressure, human intuition falsely associates length with magnitude. In whole numbers, 1,245 is greater than 89. In decimals, however, 0.0894 is significantly smaller than 0.7, despite having four digits versus one. Test writers deliberately construct answer sets where the smallest value contains the most digits.

The Zero-Padding Alignment Method

To eliminate perceptual confusion, equalize all numbers to the same number of decimal places by adding trailing zeroes:

• Question: Arrange from least to greatest: 0.7, 0.075, 0.705, 0.07, 0.699.
• Step 1: Identify the maximum number of decimal places (three places).
• Step 2: Pad all values to three digits:
  • 0.700
  • 0.075
  • 0.705
  • 0.070
  • 0.699
• Step 3: Read the values as whole integers: 700, 75, 705, 70, 699.
• Step 4: Sort effortlessly: 70 < 75 < 699 < 700 < 705.
• Result: 0.07 < 0.075 < 0.699 < 0.7 < 0.705.

Rapid Conversion Reference: Benchmark Fractions vs Decimals