5.5 Fractions, Rational Expressions, and Non-Calculator Computation
Key Takeaways
- Adding or subtracting fractions requires a common denominator; multiplying and dividing never does, which is the single most common source of fraction errors.
- Dividing by a fraction is multiplying by its reciprocal, and this rule extends unchanged to algebraic rational expressions.
- A rational expression is simplified by factoring the numerator and denominator completely and cancelling common factors, never common terms.
- Any value that makes a denominator zero is excluded from the domain, and the exam tests this as an explicit restriction.
- Complex fractions collapse fastest by multiplying every term in the numerator and denominator by the least common denominator of the inner fractions.
5.5 Fractions, Rational Expressions, and Non-Calculator Computation
Core Principle: No calculator is permitted anywhere on the ASVAB. Mathematics Knowledge gives you about 124 seconds per scored item, which is generous — but only if fraction arithmetic is automatic. Candidates rarely lose MK points to hard concepts; they lose them to a common denominator applied where it did not belong, or a factor cancelled that was really a term.
1. The Four Fraction Operations
| Operation | Rule | Worked example |
|---|---|---|
| Addition | Convert to a common denominator, then add numerators only | $\frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20}$ |
| Subtraction | Same requirement | $\frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$ |
| Multiplication | No common denominator. Multiply straight across, cancelling first | $\frac{3}{8} \times \frac{4}{9} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ |
| Division | Multiply by the reciprocal of the divisor | $\frac{5}{6} \div \frac{10}{3} = \frac{5}{6} \times \frac{3}{10} = \frac{1}{4}$ |
The number one fraction error on the ASVAB: finding a common denominator before multiplying. It is unnecessary, it wastes 30 seconds, and it usually produces an unreduced answer that does not appear among the options.
Cancel before you multiply. In $\frac{3}{8} \times \frac{4}{9}$, the 4 and 8 share a factor of 4, and the 3 and 9 share a factor of 3. Reducing first turns a three-digit multiplication into $\frac{1}{2} \times \frac{1}{3}$.
Mixed numbers must be converted before any operation. $3\frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{17}{5}$. Attempting to multiply mixed numbers directly is a reliable way to generate a wrong answer that still looks plausible.
2. Fraction, Decimal, and Percent Fluency
These conversions appear constantly in both MK and Arithmetic Reasoning, and knowing them cold saves whole minutes.
| Fraction | Decimal | Percent | Fraction | Decimal | Percent | |
|---|---|---|---|---|---|---|
| $1/2$ | 0.5 | 50% | $1/8$ | 0.125 | 12.5% | |
| $1/3$ | $0.\overline{3}$ | 33⅓% | $3/8$ | 0.375 | 37.5% | |
| $2/3$ | $0.\overline{6}$ | 66⅔% | $5/8$ | 0.625 | 62.5% | |
| $1/4$ | 0.25 | 25% | $7/8$ | 0.875 | 87.5% | |
| $3/4$ | 0.75 | 75% | $1/6$ | $0.1\overline{6}$ | 16⅔% | |
| $1/5$ | 0.2 | 20% | $1/9$ | $0.\overline{1}$ | 11⅑% |
Comparing fractions without decimals — cross multiplication. To decide whether $\frac{7}{12}$ or $\frac{5}{9}$ is larger, cross multiply: $7 \times 9 = 63$ against $5 \times 12 = 60$. The larger product sits above the larger fraction, so $\frac{7}{12} > \frac{5}{9}$. This takes about four seconds and is exact.
3. Rational Expressions: Fractions with Variables
A rational expression is a quotient of two polynomials, such as $\dfrac{x^2 - 9}{x^2 + 5x + 6}$. Every fraction rule above carries over unchanged; the only new requirement is that you must factor first.
Simplifying
- Factor the numerator completely.
- Factor the denominator completely.
- Cancel factors that appear in both.
Cancel factors, never terms. In $\dfrac{x + 4}{4}$ you cannot cancel the fours, because the 4 in the numerator is a term joined by addition, not a factor joined by multiplication. Only quantities multiplied across the entire numerator and denominator may be cancelled. This is the most heavily tested trap in the whole topic.
Domain restrictions
Division by zero is undefined, so any value of the variable that makes an original denominator zero is excluded. In the example above, $x^2 + 5x + 6 = (x+2)(x+3)$, so $x \neq -2$ and $x \neq -3$ — and the restriction $x \neq -3$ survives even though the $(x+3)$ factor cancelled. Exclusions are read from the expression before simplification.
Multiplying and dividing
Factor everything, invert the divisor if you are dividing, then cancel across the whole product:
Adding and subtracting
Here a common denominator is required, exactly as with numeric fractions:
When subtracting, distribute the minus sign across the entire second numerator. Writing $\frac{a - b + c}{d}$ where the intended value was $\frac{a - (b + c)}{d}$ is the second most common error in this topic.
4. Complex Fractions
A complex fraction has fractions inside its numerator, its denominator, or both. Two reliable methods exist; the second is faster.
Method 1 — treat the main bar as division:
Method 2 — multiply top and bottom by the LCD of the inner fractions. For
the inner LCD is $x^2$. Multiplying every term above and below by $x^2$ clears all inner fractions at once:
Method 2 scales to expressions with three or four inner fractions where Method 1 becomes unmanageable.
5. Non-Calculator Arithmetic Shortcuts
| Shortcut | Rule | Example |
|---|---|---|
| Multiply by 5 | Multiply by 10, halve it | $48 \times 5 = 480 \div 2 = 240$ |
| Divide by 5 | Double it, divide by 10 | $ 340 \div 5 = 680 \div 10 = 68$ |
| Multiply by 25 | Multiply by 100, divide by 4 | $36 \times 25 = 3600 \div 4 = 900$ |
| Squares near 50 | $(50 \pm n)^2 = 2500 \pm 100n + n^2$ | $53^2 = 2500 + 300 + 9 = 2809$ |
| Difference of squares | $a^2 - b^2 = (a-b)(a+b)$ | $41^2 - 39^2 = 2 \times 80 = 160$ |
| Percent swap | $a%$ of $b$ equals $b%$ of $a$ | 16% of 25 = 25% of 16 = 4 |
Estimate before you compute. If an item asks for $\frac{7}{8}$ of 512 and the options are 56, 448, 512, and 576, you know instantly that the answer is slightly below 512 — only 448 qualifies, and you never had to multiply. On a no-calculator test, checking the magnitude of your answer against the option spread is the cheapest error-detection tool you have.
Simplify completely and state the excluded values: (x² − 16) / (x² − x − 12).
A candidate simplifies (x + 6)/6 to x by cancelling the sixes. Why is this incorrect?
Evaluate (3/8) × (4/9) ÷ (1/6).
Simplify the complex fraction (1 + 1/x) / (1 − 1/x²), assuming x is neither 0, 1, nor −1.