8.2 Fractions, Decimals & Mixed Operations

Fractions, Decimals & Mixed Operations

Core Concept: Fraction and decimal arithmetic is a foundation of the numbers strand that Edutest lists for its Mathematics test. Solving rational number expressions quickly requires abandoning slow, traditional multi-step methods in favor of high-speed cross-cancellation, butterfly additions, recurring decimal algebraic conversion formulas, and benchmark fraction comparisons.

Examiners deliberately construct arithmetic items with awkward denominators, mixed numerals, and repeating decimal tails to test whether a candidate possesses conceptual number sense or remains dependent on slow mechanical algorithms. Candidates who understand structural rational relationships can reduce what appears to be a 90-second calculation into a swift 15-second deduction.

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Core Fraction Operations & Rapid Calculation Shortcuts

1. Addition & Subtraction: The Butterfly Shortcut

When adding or subtracting two fractions with unlike, coprime denominators, finding the Lowest Common Denominator (LCD) through prime factorisation is unnecessarily slow. Use the butterfly cross-multiplication method:

$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$

• Example 1: $\frac{4}{7} + \frac{3}{5} = \frac{(4 \times 5) + (7 \times 3)}{7 \times 5} = \frac{20 + 21}{35} = \frac{41}{35} = 1\frac{6}{35}$
• Example 2: $\frac{7}{8} - \frac{2}{3} = \frac{(7 \times 3) - (8 \times 2)}{8 \times 3} = \frac{21 - 16}{24} = \frac{5}{24}$

Rule of Thumb: Use the butterfly shortcut whenever denominators are coprime or both under 12. When denominators share a large common factor (e.g., $\frac{5}{12} + \frac{7}{18}$), finding the LCD ($36$) directly is faster than simplifying a bloated denominator product ($12 \times 18 = 216$).

2. Mixed Numerals: Partitioning vs. Improper Fractions

When adding mixed numerals, partition wholes and fractions separately:

$3\frac{2}{5} + 4\frac{1}{3} = (3 + 4) + \left(\frac{2}{5} + \frac{1}{3}\right) = 7 + \frac{6 + 5}{15} = 7\frac{11}{15}$

When subtracting mixed numerals where the second fraction exceeds the first, candidates often stumble over borrowing. You have two options:

• Method A (Regrouping / Borrowing): $5\frac{1}{4} - 2\frac{2}{3} = \left(4 + 1\frac{1}{4}\right) - 2\frac{2}{3} = 4\frac{5}{4} - 2\frac{2}{3} = (4 - 2) + \left(\frac{5}{4} - \frac{2}{3}\right) = 2 + \frac{15 - 8}{12} = 2\frac{7}{12}$
• Method B (Direct Improper Conversion): $5\frac{1}{4} - 2\frac{2}{3} = \frac{21}{4} - \frac{8}{3} = \frac{63 - 32}{12} = \frac{31}{12} = 2\frac{7}{12}$

Strategy: For multiplication and division, always convert mixed numerals to improper fractions first. Multiplying whole numbers and fractions separately ($3\frac{1}{2} \times 2\frac{1}{3} \ne 6\frac{1}{6}$) is a fatal error.

3. Aggressive Cross-Cancellation in Multiplication

Never multiply numerators and denominators to form large composite numbers before simplifying. Cancel common factors diagonally and vertically across all terms before computing the final product:

$\text{Evaluate: } \frac{14}{25} \times \frac{15}{49} \times \frac{35}{18}$

• Cancel factors of $7$ between $14$ and $49$: $\frac{2}{25} \times \frac{15}{7} \times \frac{35}{18}$
• Cancel factors of $7$ between $35$ and $7$: $\frac{2}{25} \times \frac{15}{1} \times \frac{5}{18}$
• Cancel factors of $5$ between $25$ and $5$: $\frac{2}{5} \times \frac{15}{1} \times \frac{1}{18}$
• Cancel factors of $5$ between $5$ and $15$: $\frac{2}{1} \times \frac{3}{1} \times \frac{1}{18}$
• Cancel $2 \times 3 = 6$ against $18$: $\frac{6}{18} = \frac{1}{3}$

4. Division: Keep-Change-Flip & Compound Fractions

Division by a fraction is equivalent to multiplication by its reciprocal:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

For compound (complex) fractions, multiply the outer terms (extremes) for the numerator and the inner terms (means) for the denominator:

$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}$

Example: $\frac{\frac{5}{12}}{\frac{15}{16}} = \frac{5 \times 16}{12 \times 15} = \frac{1 \times 4}{3 \times 3} = \frac{4}{9}$

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Decimal Arithmetic Without a Calculator

1. Addition & Subtraction

Always align decimal points vertically when you write a calculation. Fill empty trailing decimal places with placeholder zeroes to prevent alignment drift:

$$\begin{array}{rl} 42.700 & \\ + \;\; 3.085 & \\ - \;\; 0.940 & \\ \hline 44.845 & \end{array}$$

2. Decimal Multiplication: The Digit Count Rule

1. Temporarily ignore all decimal points and multiply the numbers as integers.
2. Count the total number of digits to the right of the decimal point across both original factors.
3. Place the decimal point in the product so that it has that exact total number of decimal places, prepending leading zeroes if necessary.

Example: $0.035 \times 0.04$

• Integer product: $35 \times 4 = 140$
• Total decimal places: $3$ (in $0.035$) $+ 2$ (in $0.04$) $= 5$ decimal places
• Shift decimal left $5$ positions: $0.00140 = 0.0014$

3. Decimal Division: Clearing the Divisor

Never attempt long division with a decimal divisor. Multiply both the dividend (numerator) and divisor (denominator) by powers of 10 ($10, 100, 1000$) to convert the divisor into an integer:

$\frac{0.072}{0.0015} = \frac{0.072 \times 10,000}{0.0015 \times 10,000} = \frac{720}{15} = 48$

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Terminating vs. Recurring Decimals

A critical number theory theorem frequently tested on scholarship exams governs whether a fraction terminates or repeats indefinitely:

The Rational Decimal Invariant: In a fully simplified vulgar fraction $\frac{a}{b}$ (where $\gcd(a,b) = 1$):

• The decimal expansion terminates if and only if the prime factorisation of the denominator $b$ contains no prime factors other than 2 and 5 ($b = 2^m \cdot 5^n$).
• If the denominator contains any other prime factor (such as $3, 7, 11, 13$), the decimal expansion is infinitely recurring.

Examples:

• $\frac{7}{40}$: $40 = 2^3 \times 5^1 \implies$ Terminating ($0.175$)
• $\frac{9}{24}$: Simplify first $\implies \frac{3}{8}$. $8 = 2^3 \implies$ Terminating ($0.375$)
• $\frac{5}{14}$: $14 = 2 \times 7 \implies$ Recurring (contains prime $7$)

Converting Recurring Decimals to Vulgar Fractions

1. Pure Recurring Decimals (Repeating starts immediately after decimal point)

Every block of repeating digits corresponds to an equal number of $9$s in the denominator:

$0.\dot{a} = \frac{a}{9}, \quad 0.\dot{a}\dot{b} = \frac{ab}{99}, \quad 0.\dot{a}\dot{b}\dot{c} = \frac{abc}{999}$

• $0.\dot{7} = \frac{7}{9}$
• $0.\dot{1}\dot{8} = \frac{18}{99} = \frac{2}{11}$
• $0.\dot{1}\dot{4}\dot{8} = \frac{148}{999} = \frac{4}{27}$

2. Mixed Recurring Decimals (Non-repeating prefix before recurring digits)

To convert a mixed recurring decimal like $0.41\dot{6}$ or $0.3\dot{8}$ under exam pressure, use the direct algebraic shortcut:

$\text{Fraction} = \frac{\text{Entire Number (non-repeating + repeating)} - \text{Non-repeating Prefix}}{\text{A } 9 \text{ for each repeating digit, followed by a } 0 \text{ for each non-repeating digit}}$

Worked Example: Converting Mixed Recurring Decimals

• Convert $0.3\dot{8}$ ($0.3888\dots$): $\text{Fraction} = \frac{38 - 3}{90} = \frac{35}{90} = \frac{7}{18}$
• Convert $0.41\dot{6}$ ($0.41666\dots$): $\text{Fraction} = \frac{416 - 41}{900} = \frac{375}{900} = \frac{15}{36} = \frac{5}{12}$
• Convert $0.1\dot{2}\dot{7}$ ($0.1272727\dots$): $\text{Fraction} = \frac{127 - 1}{990} = \frac{126}{990} = \frac{14}{110} = \frac{7}{55}$

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High-Frequency Fraction-Decimal Benchmark Table

Memorizing these common equivalences eliminates conversion time entirely:

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Rapid Fraction Comparison & Ordering Techniques

When a question asks you to order four fractions or identify the largest/smallest, calculating a massive common denominator wastes valuable minutes. Deploy these four rapid diagnostic techniques:

1. The Cross-Multiplication Bowtie (Comparing 2 Fractions)

To compare $\frac{a}{b}$ and $\frac{c}{d}$, cross-multiply upward:

$\frac{a}{b} \gtrless \frac{c}{d} \iff a \cdot d \gtrless b \cdot c$

Compare $\frac{7}{11}$ and $\frac{9}{14}$:

• Upward cross-products: $7 \times 14 = 98$ vs $11 \times 9 = 99$
• Since $98 < 99$, it follows that $\frac{7}{11} < \frac{9}{14}$. Solved in 4 seconds.

2. Residual Comparison (Distance to 1)

When comparing proper fractions where numerators and denominators are both large and close to 1, compare their deficits from 1 ($1 - \frac{a}{b}$):

$\text{The fraction with the smaller deficit from 1 is the LARGER fraction.}$

Which is greater: $\frac{17}{18}$ or $\frac{24}{25}$?

• Deficit for $\frac{17}{18} = 1 - \frac{17}{18} = \frac{1}{18}$
• Deficit for $\frac{24}{25} = 1 - \frac{24}{25} = \frac{1}{25}$
• Because $\frac{1}{25} < \frac{1}{18}$, the deficit is smaller for $\frac{24}{25}$. Therefore, $\frac{24}{25} > \frac{17}{18}$.

3. Benchmarking Against 1/2

Instantly partition fractions into two groups by comparing each numerator to half of its denominator:

• In $\frac{13}{28}$, half of $28$ is $14$. Since $13 < 14$, $\frac{13}{28} < \frac{1}{2}$.
• In $\frac{19}{36}$, half of $36$ is $18$. Since $19 > 18$, $\frac{19}{36} > \frac{1}{2}$.
• Thus, $\frac{19}{36} > \frac{13}{28}$ without any calculation.

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High-Yield Traps in Fractions & Decimals

1. Inverting the Wrong Factor in Division

In expressions with multiple divisions like $12 \div \frac{3}{4} \div \frac{2}{3}$, evaluate strictly from left to right: $12 \times \frac{4}{3} = 16, \quad 16 \div \frac{2}{3} = 16 \times \frac{3}{2} = 24$ Do not multiply the two divisors together first ($12 \div (\frac{3}{4} \div \frac{2}{3})$), which produces the incorrect value $10\frac{2}{3}$.

2. The Direct Numerator/Denominator Addition Fallacy

Under acute time stress, untrained candidates occasionally add numerators and denominators directly: $\frac{2}{5} + \frac{3}{7} \to \frac{5}{12}$. This is fundamentally false. Fractions represent parts of a whole and cannot be combined additively without a common scale.

3. Decimal Shift Drift in Multi-Zero Divisions

In calculations such as $0.0048 \div 0.06$, students frequently miscount zeroes. By clearing the divisor ($0.06 \times 100 = 6$), the dividend shifts by the exact same power ($0.0048 \times 100 = 0.48$). Then $0.48 \div 6 = 0.08$. Writing out the shifted numbers explicitly prevents misplaced decimal places.