3.4 Comparing, Ordering & Converting Fractions

3.4 Comparing, Ordering & Converting Fractions

Comparing and ordering fractions are core numeracy skills on the HESI A2 exam. In clinical healthcare, nurses must evaluate which dosage is higher or lower, verify whether patient fluid intake exceeds minimum clinical targets, interpret fractional diagnostic metrics, and compare laboratory values. A solid grasp of fraction comparisons, long division conversions, and benchmark values enables you to navigate ordering questions rapidly and with zero ambiguity.

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Comparing Two Fractions: Three Proven Methods

When presented with two fractions, determining which quantity is larger can be accomplished through three reliable methods: finding a common denominator, using decimal conversion, or applying the cross-multiplication shortcut.

Method 1: The Common Denominator Approach

Convert both fractions into equivalent fractions sharing a common denominator (the LCD). With identical denominators, compare the numerators directly: the fraction with the larger numerator is the greater value.

Example: Compare $\frac{5}{6}$ and $\frac{7}{9}$.

1. Find LCD for $6$ and $9$: Multiples of $6$ are $6, 12, \mathbf{18}$; multiples of $9$ are $9, \mathbf{18}$. The LCD is $18$.
2. Convert $\frac{5}{6}$: $\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$.
3. Convert $\frac{7}{9}$: $\frac{7 \times 2}{9 \times 2} = \frac{14}{18}$.
4. Compare: Since $15 > 14$, it follows that $\frac{15}{18} > \frac{14}{18}$, so $\mathbf{\frac{5}{6} > \frac{7}{9}}$.

Method 2: The Cross-Multiplication "Butterfly" Shortcut

When comparing exactly two fractions, the cross-multiplication butterfly method is the fastest technique because it requires no LCD calculation.

To compare $\frac{a}{b}$ and $\frac{c}{d}$:

1. Multiply the numerator of the first fraction by the denominator of the second fraction ($a \times d$). Write this product above the left fraction.
2. Multiply the denominator of the first fraction by the numerator of the second fraction ($b \times c$). Write this product above the right fraction.
3. Compare the two products:
   • If $a \cdot d > b \cdot c$, then $\frac{a}{b} > \frac{c}{d}$.
   • If $a \cdot d < b \cdot c$, then $\frac{a}{b} < \frac{c}{d}$.
   • If $a \cdot d = b \cdot c$, then $\frac{a}{b} = \frac{c}{d}$.

$$\begin{matrix} (a \times d) & & (b \times c) \\ \frac{a}{b} & \longleftrightarrow & \frac{c}{d} \end{matrix}$$

Worked Example: Compare $\frac{7}{11}$ and $\frac{5}{8}$.

1. Compute left product: $7 \times 8 = 56$.
2. Compute right product: $11 \times 5 = 55$.
3. Compare: Since $56 > 55$, it follows immediately that $\mathbf{\frac{7}{11} > \frac{5}{8}}$.

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Converting Fractions to Decimals via Long Division

Every fraction represents an unperformed division problem: $\frac{N}{D} = N \div D$. Converting fractions into decimals is often the easiest way to compare multiple numbers or prepare values for clinical calculators.

Long Division Algorithm

1. Place the numerator ($N$) inside the division bracket (dividend).
2. Place the denominator ($D$) outside the bracket (divisor).
3. Place a decimal point immediately after the numerator and add trailing zeros.
4. Divide through standard long division, carrying the decimal point straight up into the quotient.
5. Continue dividing until the decimal terminates (remainder becomes $0$) or establishes a repeating sequence.

Worked Walkthrough: Convert $\frac{3}{8}$ to a Decimal

$$\begin{array}{r} 0.375 \\ 8 {\overline{\smash{\big)}\,3.000}} \\ \underline{-2.4\phantom{00}} \\ 0.60\phantom{0} \\ \underline{-0.56\phantom{0}} \\ 0.040 \\ \underline{-0.040} \\ 0 \end{array}$$

1. $8$ goes into $30$ three times ($3 \times 8 = 24$), leaving a remainder of $6$.
2. Bring down a $0$ to form $60$. $8$ goes into $60$ seven times ($7 \times 8 = 56$), remainder $4$.
3. Bring down another $0$ to form $40$. $8$ goes into $40$ exactly five times ($5 \times 8 = 40$), remainder $0$.
4. Result: $\mathbf{0.375}$.

Terminating vs. Repeating Decimals

• Terminating Decimals: The division reaches a remainder of zero. Denominators whose prime factors consist solely of $2$s and/or $5$s always terminate (e.g., $\frac{1}{2} = 0.5$, $\frac{1}{4} = 0.25$, $\frac{3}{8} = 0.375$, $\frac{7}{20} = 0.35$).
• Repeating Decimals: The division enters an infinite recurring pattern. Denominators with prime factors other than $2$ or $5$ (such as $3, 7, 11$) produce repeating decimals (e.g., $\frac{1}{3} = 0.333\dots = 0.\overline{3}$, $\frac{2}{3} = 0.666\dots = 0.\overline{6}$, $\frac{1}{6} = 0.1666\dots = 0.1\overline{6}$). When a question does not say how to round, carry repeating decimals to at least the hundredth ($0.33$ or $0.67$) or the thousandth, and follow any rounding instruction that is given.

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High-Frequency Clinical Benchmark Fractions

In nursing school and on the HESI A2, certain fraction-to-decimal conversions occur repeatedly. Memorizing these benchmark equivalents saves valuable calculation time on exam day.

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Ordering Sets of Fractions: Step-by-Step Strategies

HESI questions often present a set of 3 or 4 fractions and mixed numbers, instructing you to arrange them from least to greatest or greatest to least.

Recommended Strategy: Decimal Conversion

When ordering three or more fractions with diverse denominators (such as $5, 8, 3,$ and $12$), converting each number into a decimal value is faster and far less prone to arithmetic error than calculating a massive LCD.

Worked Example 1: Arrange the following fractions in order from least to greatest:

$\frac{5}{8}, \quad \frac{2}{3}, \quad \frac{7}{12}, \quad \frac{3}{5}$

Step-by-Step Solution:

1. Convert each fraction to its decimal equivalent (carry to three decimal places):
   • $\frac{5}{8} = 5 \div 8 = \mathbf{0.625}$
   • $\frac{2}{3} = 2 \div 3 = 0.6666\dots \approx \mathbf{0.667}$
   • $\frac{7}{12} = 7 \div 12 = 0.5833\dots \approx \mathbf{0.583}$
   • $\frac{3}{5} = 3 \div 5 = \mathbf{0.600}$
2. Compare the decimal values: $0.583 < 0.600 < 0.625 < 0.667$
3. Replace each decimal with its original fraction: $\mathbf{\frac{7}{12} < \frac{3}{5} < \frac{5}{8} < \frac{2}{3}}$

Ordering Sets with Mixed Numbers

When mixed numbers are included in the set, compare whole numbers first. Only fractions sharing the same whole number need direct fractional comparison.

Worked Example 2: Arrange from greatest to least:

$2\frac{3}{4}, \quad 2\frac{5}{8}, \quad 2\frac{7}{10}, \quad 2\frac{4}{5}$

1. All four values share the same whole integer ($2$). Focus on the fractional parts.
2. Convert each fraction to a decimal:
   • $\frac{3}{4} = 0.750$
   • $\frac{5}{8} = 0.625$
   • $\frac{7}{10} = 0.700$
   • $\frac{4}{5} = 0.800$
3. Rank from greatest to least: $0.800 > 0.750 > 0.700 > 0.625$
4. Write in original mixed number form: $\mathbf{2\frac{4}{5} > 2\frac{3}{4} > 2\frac{7}{10} > 2\frac{5}{8}}$