3.1 Fractions

Fractions appear in every ICAS Mathematics paper: ordering halves, quarters and eighths in Paper A (Year 3), fraction operations and probabilities in Paper D (Year 6), and fractional thinking inside algebra and ratio problems in the senior papers. Because personal calculators are not allowed, ICAS rewards students who can simplify, compare and calculate with fractions quickly by hand. This section builds those skills from the ground up.

What a Fraction Means

A fraction names part of a whole. In 3/4, the denominator (4) says the whole is split into 4 equal parts and the numerator (3) counts how many of those parts we have. Fractions also live on the number line: 3/4 is the point three-quarters of the way from 0 to 1. ICAS often asks you to place fractions on a number line or identify which marked point matches a fraction, so practise picturing the line cut into equal steps.

A fraction also means division: 3/4 is 3 ÷ 4 = 0.75. This link is the key to converting between fractions and decimals later.

Equivalent Fractions and Simplifying

Multiply or divide the numerator and denominator by the same number and the value does not change: 2/3 = 4/6 = 6/9. To simplify (reduce to lowest terms), divide both numbers by their highest common factor (HCF). For 18/24 the HCF is 6, so 18/24 = 3/4.

Common trap: you may only cancel a factor that divides both the top and the bottom. 4/6 simplifies to 2/3, never to 4/3.

Comparing and Ordering

Paper A asks you to order halves, quarters and eighths. Rewrite them all as eighths: 1/2 = 4/8, 1/4 = 2/8, so 1/4 < 3/8 < 1/2 < 5/8. For unlike denominators, two reliable methods:

1. Common denominator: rewrite each fraction using the lowest common multiple of the denominators. Compare 3/5 and 5/8: the LCM of 5 and 8 is 40, so 3/5 = 24/40 and 5/8 = 25/40, hence 5/8 is larger.
2. Cross-multiplication: to compare a/b with c/d, work out a×d and b×c. Here 3×8 = 24 and 5×5 = 25; since 24 < 25, 3/5 < 5/8. This is fast and safe under time pressure.

A handy benchmark: any fraction whose numerator is less than half its denominator is below 1/2. Checking each fraction against 1/2 first often settles an ordering question without any conversion.

Worked Example: Ordering Three Unlike Fractions

Order 2/3, 7/12 and 3/4 from smallest to largest. The LCM of 3, 12 and 4 is 12, so rewrite: 2/3 = 8/12, 7/12 stays, 3/4 = 9/12. Reading off the numerators gives 7/12 < 2/3 < 3/4. Notice the benchmark check agrees: 7/12 is just above 6/12 = 1/2, while the other two are clearly larger than 1/2, so only the close pair needed real comparison.

Mixed Numbers and Improper Fractions

An improper fraction has a numerator greater than or equal to its denominator (7/4); a mixed number combines a whole and a fraction (1 3/4). Convert by dividing: 7 ÷ 4 = 1 remainder 3, so 7/4 = 1 3/4. Backwards, multiply the whole by the denominator and add the numerator: 2 3/5 = (2×5 + 3)/5 = 13/5. Unless told otherwise, give final answers as mixed numbers in simplest form.

The Four Operations with Fractions

• Add / subtract: get a common denominator first. 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
• Multiply: multiply tops, multiply bottoms, and cancel early to keep numbers small. 4/9 × 3/8: cancel 4 with 8 (leaving 1 and 2) and 3 with 9 (leaving 1 and 3), giving 1/6.
• Divide: multiply by the reciprocal (flip the second fraction). 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.

With mixed numbers, convert to improper fractions first — never multiply the whole parts and fraction parts separately.

Fractions of a Quantity

'Of' means multiply, but divide by the denominator first so the numbers stay mental-sized: 3/4 of 28 = 28 ÷ 4 × 3 = 7 × 3 = 21.

Worked problem: a tank holds 36 litres and is 5/6 full. How much water is in it? 36 ÷ 6 × 5 = 30 litres.

ICAS also loves the reverse version. After spending 2/5 of her money, Mia has $24 left. How much did she start with? She has 3/5 left, and 3/5 = $24, so 1/5 = $8 and the whole amount is 5 × $8 = $40.

Common ICAS Traps

• Adding straight across: 2/3 + 1/4 is not 3/7.
• Forgetting to simplify — ICAS answer options almost always use lowest terms.
• Thinking 1/4 > 1/3 because 4 > 3. A larger denominator means smaller pieces.
• Taking a fraction of the wrong whole in word problems: always ask 'of what?'

Equivalents Worth Memorising