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Operations with Positive & Negative Fractions

Maths · Y8 Fractions & Percentages
Year 8
Lesson Guide
Explanation
Year 8 fractions come in two costumes: mixed numbers (2¾) and improper fractions (11/4) — the same value, different notation. Convert mixed → improper with whole × bottom + top: 2 × 4 + 3 = 11, so 11/4. Convert back by dividing: 17/5 = 3 remainder 2 = 3⅖. The working rule for this whole chapter: before multiplying or dividing, ALWAYS convert mixed numbers to improper form — mixed numbers in a product are a famous error factory.
Your Turn
Convert: 3⅖ to improper, and 23/6 to mixed. Check: 17/5; and 3⅚.
Watch Out
2¾ means 2 PLUS ¾, so 2¾ × 2 is NOT 4¾ (that error treats it as 2 AND SEPARATELY ¾). Convert first: 11/4 × 2 = 22/4 = 5½.
Explanation
Adding and subtracting at Y8 level: 2⅓ − 3¾. Convert to improper (7/3 − 15/4), find the LOWEST common denominator — 12 — and rewrite: 28/12 − 45/12. Now your Chapter 1 integer rules run the numerators: 28 − 45 = −17, giving −17/12 = −1 5/12. Sense-check the sign before believing anything: 2⅓ is smaller than 3¾, so negative was inevitable.
Your Turn
Compute: −5/6 + 3/4, and 1½ − 2⅔. Check: −10/12 + 9/12 = −1/12; and 9/6 − 16/6 = −7/6 = −1⅙.
Watch Out
Use the LOWEST common denominator (12 for 3 and 4), not the product when a smaller one exists (6 and 4 → 12, not 24) — big numbers breed errors.
Explanation
Multiplying: −1⅗ × 2½. The routine: sign first (negative × positive → negative), convert to improper (8/5 × 5/2), CANCEL diagonally before multiplying — the 5s kill each other, 8 and 2 reduce to 4 and 1 — then multiply across: 4. Answer: −4. Cancelling first means you multiply tiny numbers and never face a monster simplification at the end. And remember: multiplication needs NO common denominator — that's addition's burden.
Your Turn
Compute: −3/8 × 4/9, and 2⅔ × (−1½). Check: cancel → −1/6; and 8/3 × (−3/2) = −4.
Watch Out
Cancelling is top-with-bottom only, and only across MULTIPLICATION. Cancelling across a + or − sign destroys the value.
Explanation
Dividing: −2¼ ÷ 1½. Improper first (−9/4 ÷ 3/2), then keep-change-flip: keep −9/4, change ÷ to ×, flip the SECOND fraction: −9/4 × 2/3. Sign is negative, cancel (9 with 3, 2 with 4), multiply: −3/2 = −1½. Always estimate as a guard: −2¼ divided by about 1½ should land near −1.5 — if your answer is −6 or −⅜, a flip went wrong somewhere.
Your Turn
Compute: 5/6 ÷ (−10/3), and −3½ ÷ (−7). Check: 5/6 × (−3/10) = −1/4; and −7/2 × (−1/7) = 1/2.
Watch Out
A whole number divisor flips too: ÷ (−7) becomes × (−1/7). Whole numbers are fractions over 1.
Explanation
'Of' means multiply — and Y8 runs it both ways. Forwards: ⅜ of $56 → divide by the bottom (7), multiply by the top ($21). BACKWARDS: two-thirds of a number is 42 — find the number. Unitary thinking: if 2 thirds are 42, then 1 third is 21, so the whole (3 thirds) is 63. Find one part, rebuild the whole. Bank this move — it's the engine behind reverse percentages in the next lessons.
Your Turn
Forwards: ⅚ of 48. Backwards: ¾ of a number is 27 — the number? Check: 40; and one quarter = 9 → 36.
Watch Out
Read which direction you're facing: "⅔ of 42" is 28, but "⅔ of ? is 42" is 63. Same numbers, opposite questions.
Explanation
The graduation exercise: (1½ − 2⅓) × 6/5 ÷ (−2). Brackets first — LCD 6 gives −5/6. Multiply: −5/6 × 6/5 = −1 (glorious total cancellation). Divide: −1 ÷ (−2) = ½. Every tool fired in order: BODMAS, improper conversion, LCD, cancel-first, keep-change-flip, sign rules — and one operation per line, always. This discipline is exactly what algebra with fractions will demand next term. Quiz and exercises now!
Your Turn
Evaluate: (2/3 + 1/6) × (−4/5). Check: brackets → 5/6; then 5/6 × (−4/5) = −2/3.
Watch Out
Under pressure, brackets get skipped. Circle them when you read the expression — they always fire first.
📌 Key Points — this lesson in a nutshell
  • Convert mixed numbers to improper fractions before multiplying or dividing.
  • Add and subtract with the LCD — and the sign travels with its fraction.
  • Multiply: cancel common factors first, then multiply across.
  • Divide: keep, change, flip.
  • A fraction OF a quantity = multiply; the reverse ("2/3 of it is 18") = divide by the fraction.
Step 1: Introduction
Step 1 of 6
Two names for the same number 2¾ = 11/4 mixed number improper fraction Mixed → improper: whole × bottom + top, over the same bottom. 2¾ → 2 × 4 + 3 = 11 → 11/4 Improper → mixed: divide, remainder stays as the fraction. 17/5 → 17 ÷ 5 = 3 r 2 → 3⅖ RULE for this chapter: convert mixed → improper BEFORE any × or ÷.
2⅓ − 3¾ 1. To improper: 7/3 − 15/4 2. LCD of 3 and 4 is 12: 28/12 − 45/12 3. Integer rules on top: (28 − 45)/12 = −17/12 4. Back to mixed: −1 5/12 LCD = lowest common denominator (the smallest shared bottom). Sense check: 2⅓ is smaller than 3¾, so the answer HAD to be negative. The Ch 1 sign rules run the numerators: 28 − 45 = −17.
−1⅗ × 2½ 1. Sign first: (−) × (+) → negative 2. To improper: 8/5 × 5/2 3. Cancel diagonally: 8̸4/5̸ × 5̸/2̸ = 4/1 × 1/1 4. Multiply across: = −4 Cancel any top with any bottom (5 with 5, 8 with 2) BEFORE multiplying — small numbers, no end-simplification, fewer errors. No common denominator needed for × — that is addition's burden only.
−2¼ ÷ 1½ 1. To improper: −9/4 ÷ 3/2 2. Keep · change · flip: −9/4 × 2/3 3. Sign, cancel, multiply: −(9̸3 × 2̸)/(4̸2 × 3̸) = −3/2 4. As a mixed number: −1½ Why flip? ÷ 3/2 asks "how many 1½s fit?" — same as × 2/3. Sense check: −2¼ ÷ 1½ should be about −1.5 ✓ — estimate to catch flips gone wrong.
"of" means × — both directions Forwards: ⅜ of $56 56 ÷ 8 = 7, then × 3 = $21. (Or 3/8 × 56/1, cancel the 8.) Backwards: ⅔ of a number is 42. What is the number? If 2 thirds = 42, then 1 third = 21, so 3 thirds (the whole) = 63. ✓ (⅔ of 63 = 42) Backwards = the unitary method: find ONE part, then rebuild the whole. This exact move powers reverse percentages next lesson!
(1½ − 2⅓) × 6/5 ÷ (−2) Brackets: 3/2 − 7/3 = 9/6 − 14/6 = −5/6 Then × 6/5: −5/6 × 6/5 = −1 (everything cancels!) Then ÷ (−2): −1 × (−1/2) = 1/2 The full stack, in firing order: BODMAS decides the order · mixed → improper before × ÷ LCD for + − · cancel-first for × · keep-change-flip for ÷ sign rules throughout · sense-check the size and sign at the end One operation per line. Every line. No exceptions at Y8.
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