3.1 Fractions: Equivalence, Simplification, Comparison, and Operations

Fluency with fractions is essential across both sections of the LANTITE Numeracy test, particularly in Section 2 (Non-Calculator), where you must compare, simplify, and combine fractional quantities mentally or on scratch paper. In school contexts, fractions appear constantly when allocating teaching time, scaling catering recipes, interpreting assessment marks, and tracking student elective enrolments.

Proper Fractions, Improper Fractions, and Mixed Numbers

A fraction $\frac{a}{b}$ (where $a$ and $b$ are integers and $b \neq 0$) represents both $a$ parts of a whole divided into $b$ equal parts and the exact quotient $a \div b$.

• Proper fraction: The numerator is strictly smaller than the denominator ($a < b$ for positive values), so the value is between $0$ and $1$ (for example, $\frac{3}{8}$).
• Improper fraction: The numerator is greater than or equal to the denominator ($a \ge b$), so the value is $1$ or greater (for example, $\frac{17}{5}$).
• Mixed number: Combines a whole-number part and a proper fraction part (for example, $3\frac{2}{5}$).

To convert an improper fraction to a mixed number, divide the numerator by the denominator: the quotient gives the whole number and the remainder becomes the new numerator over the original denominator. Since $17 \div 5 = 3$ with a remainder of $2$, we write:

$$\frac{17}{5} = 3\frac{2}{5}$$

To convert a mixed number back to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the same denominator:

$$4\frac{3}{8} = \frac{(4 \times 8) + 3}{8} = \frac{35}{8}$$

Simplifying Fractions Using the Highest Common Factor (HCF)

Two fractions are equivalent if they represent the same proportion of a whole. Multiplying or dividing both the numerator and denominator by the same non-zero integer preserves the value of the fraction. A fraction is in simplest form (or lowest terms) when the only common factor of the numerator and denominator is $1$.

To simplify a fraction efficiently in a single step, divide both numerator and denominator by their highest common factor (HCF). For example, if $84$ out of $126$ Year 7 students return a camp permission slip, the fraction returned is $\frac{84}{126}$. Because $\text{HCF}(84, 126) = 42$, dividing top and bottom by $42$ gives:

$$\frac{84 \div 42}{126 \div 42} = \frac{2}{3}$$

If you do not spot the HCF immediately in the non-calculator section, divide repeatedly by small prime factors ($2$, $3$, $7$): $\frac{84}{126} = \frac{42}{63} = \frac{14}{21} = \frac{2}{3}$.

Comparing and Ordering Fractions Without a Calculator

When ordering fractions in Section 2, choose the fastest strategy based on the denominators and numerators presented:

When two fractions are each one piece short of a whole—such as $\frac{7}{8}$ and $\frac{5}{6}$—use the complement to $1$: $\frac{7}{8} = 1 - \frac{1}{8}$ and $\frac{5}{6} = 1 - \frac{1}{6}$. Because a missing eighth ($\frac{1}{8}$) is smaller than a missing sixth ($\frac{1}{6}$), $\frac{7}{8} > \frac{5}{6}$.

Operations with Fractions and Mixed Numbers

Addition and Subtraction Using the Lowest Common Denominator

Fractions can only be added or subtracted when their parts are the same size—meaning they share a common denominator. Never add or subtract denominators directly. First find the lowest common denominator (LCD), convert each fraction, and then add or subtract the numerators:

$$\frac{3}{8} + \frac{5}{12} = \frac{9}{24} + \frac{10}{24} = \frac{19}{24}$$

When subtracting mixed numbers where the second fractional part is larger than the first, either convert both to improper fractions or regroup (borrow) $1$ whole from the leading integer. For instance, to find $4\frac{1}{6} - 1\frac{3}{4}$ using $\text{LCD}(6, 4) = 12$:

$$4\frac{1}{6} - 1\frac{3}{4} = 4\frac{2}{12} - 1\frac{9}{12} = 3\frac{14}{12} - 1\frac{9}{12} = 2\frac{5}{12}$$

Multiplication and Division of Fractions

In mathematical word problems, the word "of" signals multiplication. For example, $\frac{3}{4}$ of $120\text{ minutes}$ is $\frac{3}{4} \times 120 = 90\text{ minutes}$. When multiplying two fractions, simplify diagonally by cancelling any common factors between a numerator and a denominator before multiplying across:

$$\frac{14}{15} \times \frac{25}{28} = \frac{14 \div 14}{15 \div 5} \times \frac{25 \div 5}{28 \div 14} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}$$

Always convert mixed numbers to improper fractions before multiplying or dividing. A common error is multiplying whole numbers and fractions separately ($2\frac{1}{2} \times 3\frac{1}{3} \neq 6\frac{1}{6}$); instead:

$$2\frac{1}{2} \times 3\frac{1}{3} = \frac{5}{2} \times \frac{10}{3} = \frac{25}{3} = 8\frac{1}{3}$$

To divide by a fraction, multiply the dividend by the reciprocal (flipped version) of the divisor:

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

For example, if an art teacher divides $4\frac{1}{2}\text{ kg}$ of clay into $\frac{3}{8}\text{ kg}$ blocks, the number of blocks is:

$$4\frac{1}{2} \div \frac{3}{8} = \frac{9}{2} \times \frac{8}{3} = 3 \times 4 = 12\text{ blocks}$$

Multi-Step Complementary Fraction Problems in School Contexts

LANTITE frequently tests multi-step cohort problems where a second fraction applies to the remaining group rather than the original total.

Worked Example: In a Year 9 creative arts cohort, $\frac{2}{5}$ of the students choose drama. Of the remaining students, $\frac{1}{3}$ choose music, and the final $48$ students choose visual art. How many students are in the Year 9 cohort?

1. Find the remainder after drama: $1 - \frac{2}{5} = \frac{3}{5}$ of the total cohort remains.
2. Find the fraction of the total cohort in music: Music takes $\frac{1}{3}$ of the remaining $\frac{3}{5}$, which is $\frac{1}{3} \times \frac{3}{5} = \frac{1}{5}$ of the total cohort.
3. Find the fraction of the total cohort in visual art: Subtract drama and music from the whole ($1 - \frac{2}{5} - \frac{1}{5} = \frac{2}{5}$), or take the remaining $\frac{2}{3}$ of $\frac{3}{5} = \frac{2}{5}$.
4. Scale up to the full cohort: Since $\frac{2}{5}$ of the cohort equals $48$ students, $\frac{1}{5}$ of the cohort is $48 \div 2 = 24$ students, so the full cohort is $24 \times 5 = 120$ students.