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Percentage Increase & Decrease; Unitary Method

Maths · Y8 Fractions & Percentages
Year 8
Lesson Guide
Explanation
Increasing $60 by 15%: you could find 15% ($9) and add it — fine, but two steps. The Y8 upgrade is the MULTIPLIER: after a 15% increase you have 100% + 15% = 115% of the original, and 115% = 1.15, so new = 1.15 × 60 = $69. One multiplication. Build multipliers on sight: +40% → ×1.4, +7% → ×1.07, +100% → ×2. This isn't just elegance — the multiplier is the ONLY method that runs backwards, which reverse problems will demand shortly.
Your Turn
Multipliers for: +25%, +8%, +150%? Then increase 80 by 25%. Check: ×1.25, ×1.08, ×2.5; and 100.
Watch Out
+7% is ×1.07, not ×1.7! Single-digit percentages hide behind a zero — 1.7 would be +70%.
Explanation
Decreasing works the same way, but the multiplier is what you KEEP: a 15% decrease keeps 100% − 15% = 85%, so × 0.85. $60 × 0.85 = $51. On sight: −30% → ×0.7, −5% → ×0.95, −75% → ×0.25. And a built-in error detector: decrease multipliers live below 1, increase multipliers above 1 — if your 'discount' multiplier is 1.15, you just made the price go up!
Your Turn
Multipliers for −20%, −8%, −50%? Then decrease 140 by 20%. Check: ×0.8, ×0.92, ×0.5; and 112.
Watch Out
A 15% decrease is ×0.85, never ×0.15 — ×0.15 keeps only 15%, an 85% crash. The multiplier is the SURVIVOR.
Explanation
The trap that catches adults: up 10% then down 10% does NOT return to the start. $200 → $220 → $198, because the 10% cut acted on the bigger number. Successive changes chain by MULTIPLYING their multipliers: 1.1 × 0.9 = 0.99, a net 1% loss. Likewise +20% then +10% is 1.2 × 1.1 = 1.32 — a 32% rise, not 30%. Never add percentage changes; multiply their multipliers.
Your Turn
A $500 TV rises 20% then falls 25%. Final price and net change? Check: 1.2 × 0.75 = 0.9 → $450, net −10%.
Watch Out
Order doesn't matter (1.1 × 0.9 = 0.9 × 1.1) — but adding the percentages is ALWAYS wrong unless one of them is 0.
Explanation
The reverse skill: prices moved $80 → $92; find the percentage change. Formula: change ÷ ORIGINAL × 100. The change is 12; divide by the ORIGINAL 80 (never the new 92!): 12/80 = 15% increase. Dividing by the new number gives 13% — plausible-looking and wrong, the textbook trap. Verify with the multiplier: 80 × 1.15 = 92 ✓. Decreases identically: $50 → $42 is 8/50 = 16% down.
Your Turn
Find the % change: 60 → 75, and 120 → 90. Check: 15/60 = 25% up; 30/120 = 25% down.
Watch Out
The ORIGINAL is the earlier-in-time number, even when the sentence mentions the new one first. Anchor on "before".
Explanation
The unitary method solves 'backwards' percentage questions: 30% of a number is 45 — find the number. Go down to ONE percent: 45 ÷ 30 = 1.5. Go up to a hundred: 1.5 × 100 = 150. Down to one, up to anything — including sideways jumps: from the same 1% you can get 70% (105) without ever finding the whole. It's exactly the find-one-part move from the fractions lesson, now wearing percentage clothes.
Your Turn
8% of an amount is $6. Find 100%, and also 25%. Check: 1% = 0.75 → $75; and 25% = $18.75.
Watch Out
Divide by the PERCENTAGE NUMBER (÷30 for 30%), not by 100 first. Down to 1%, THEN scale.
Explanation
The boss level: a jacket costs $68 AFTER a 15% discount — find the original price. Key insight: $68 is the survivor — 85% of the original. Unitary: 68 ÷ 85 = 0.8 per percent, ×100 = $80. (Equivalently, divide by the multiplier: 68 ÷ 0.85.) The famous wrong answer is adding 15% of $68 back on ($78.20) — but the discount was cut from the ORIGINAL, not from the sale price. Name what percentage the given number represents FIRST, then run the machine. Check forwards: 80 × 0.85 = 68 ✓. Quiz and exercises!
Your Turn
A price rose 20% to $96. Original? Check: 96 is 120% → 1% = 0.8 → $80. (Forward: 80 × 1.2 = 96 ✓)
Watch Out
After-increase reverses need ÷1.2 (the given is 120%); after-decrease needs ÷0.85 (the given is 85%). Name the survivor percentage before touching anything.
📌 Key Points — this lesson in a nutshell
  • Increase by p%: multiply by (1 + p/100). Decrease: multiply by (1 − p/100).
  • Successive changes MULTIPLY their multipliers — they never simply add.
  • Percentage change = change ÷ ORIGINAL × 100.
  • Unitary method: find 1% (or one part) first, then scale up.
  • Reverse percentage: divide by the multiplier to recover the original amount.
Step 1: Introduction
Step 1 of 6
Increase $60 by 15% Two-step way: find 15% (0.15 × 60 = $9), then add: 60 + 9 = $69. Works, but two steps and two chances to slip. …and it can't run backwards (next steps!). Multiplier way: new amount = 100% + 15% = 115% of the old. 115% = 1.15 → 1.15 × 60 = $69. ONE multiplication. "Increase by 15%" ⇔ "× 1.15". Burn this equivalence in. Multipliers: +15% → ×1.15 · +40% → ×1.4 · +7% → ×1.07 · +100% → ×2
Decrease $60 by 15% keep 100% − 15% = 85% 0.85 × 60 = $51 The multiplier is what you KEEP, not what you lose: −15% → ×0.85 · −30% → ×0.7 · −5% → ×0.95 · −75% → ×0.25 Sense check: a decrease multiplier is ALWAYS below 1; an increase multiplier is always above 1. Instant error detector.
+10% then −10% ≠ back to start! $200 × 1.1 → $220 × 0.9 → $198 Up 10%, then down 10% lands at $198 — NOT $200! The −10% acted on the BIGGER number ($220), so it took more away. Combined multiplier: 1.1 × 0.9 = 0.99 → a net 1% DECREASE. Chain changes by MULTIPLYING multipliers: +20% then +10% → 1.2 × 1.1 = 1.32 → +32% total (not +30%!).
Price went $80 → $92. What % increase? change = 92 − 80 = 12 % change = change/ORIGINAL × 100 = 12/80 × 100 = 15% ALWAYS divide by the ORIGINAL (the before-number), never the new one. 12/92 gives 13% — wrong and plausible: the perfect trap. Decreases too: $50 → $42 is 8/50 × 100 = 16% decrease. Check: 80 × 1.15 = 92 ✓ — verify with the multiplier you found.
30% of a number is 45. Find the number. 30% = 45 ÷30 → 1% = 1.5 ×100 → 100% = 150 UNITARY = go down to ONE unit (1%), then up to what you want. It also jumps sideways: if 30% = 45, then 70% = 1.5 × 70 = 105. Same move as "⅔ of a number is 42" from the fractions lesson — find one part, rebuild anything.
A jacket costs $68 AFTER a 15% discount. Original? $68 is the SURVIVOR: 85% of the original 85% = 68 → 1% = 0.8 → 100% = $80 (or: 68 ÷ 0.85 = 80 — dividing UNDOES the multiplier) THE trap: taking 15% of $68 and adding it back → $78.20. WRONG — the 15% was cut from the ORIGINAL ($80), not from the sale price. Routine: name what % the given number IS (85%), then unitary or ÷ multiplier. Check forwards: $80 × 0.85 = $68 ✓
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