Mathematics • Year 8 • Unit 1 • Lesson 11
Successive Percentage Changes
Build the “multiply, don't add” habit. One fully-worked example, one guided example with blanks, then eight independent problems from single multipliers up to comparing two chains of changes.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why the answer ISN'T “back to $100”.
Problem. A jacket costs $100. The price goes up 20%, then comes back down 20%. What is the final price?
Step 1 — Turn each percentage change into a multiplier.
+20% means × 1.20. −20% means × 0.80.
Reason: a rise of P% means ×(1 + P/100); a fall of P% means ×(1 − P/100).
Step 2 — Apply the first multiplier.
$100 × 1.20 = $120
Reason: the +20% is on the original $100.
Step 3 — Apply the second multiplier on the NEW value.
$120 × 0.80 = $96
Reason: the −20% is 20% of the BIGGER $120, not the original $100. That's why it doesn't cancel.
Step 4 — Check with the combined multiplier.
1.20 × 0.80 = 0.96 → $100 × 0.96 = $96
Reason: 0.96 is less than 1, so the overall change is a 4% LOSS, not 0%.
Answer: $96 (a 4% overall decrease, not 0%).
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A $200 jacket has its price increased by 25%, then decreased by 10%. Find the final price using multipliers.
Step 1 — Multipliers: +25% is × ______; −10% is × ______
Step 2 — After the rise:
$200 × ______ = $______
Step 3 — After the fall, on the NEW value:
$______ × ______ = $______
Step 4 — Check via combined multiplier:
______ × ______ = ______ → $200 × ______ = $______
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single multiplier or short chains). The middle two are standard (two changes with the second on a different base). The last two are extension (find the equivalent single % or compare two chains).
Foundation — turn each % into a multiplier and apply
3.1 Write the multiplier for each percentage change: (a) +15%, (b) −30%, (c) +8%, (d) −5%. 2 marks
3.2 $100 goes up 10%, then up 10% again. Find the final value using multipliers. 1 mark
3.3 $100 falls 10%, then falls 10% again. Find the final value. 1 mark
3.4 $400 rises 50%, then falls 50%. Find the final value. 1 mark
Standard — two changes on shifting bases
3.5 A $500 phone is discounted 30%, then a further 10% off the new price at the till. Find the final price. 2 marks
3.6 A stock worth $80 rises 15%, then falls 8%. Find the final value. 2 marks
Extension — equivalent single % and comparing chains
3.7 A store offers 25% off, then a further 10% at the till. (a) Find the combined multiplier. (b) Find the equivalent SINGLE percentage discount. (Hint: it is NOT 35%.) 2 marks
3.8 Shop A offers “20% off, then a further 10% off”. Shop B offers a single “28% off”. Both items start at $200. (a) Find the final price at each shop. (b) Which is cheaper, and by how much? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do ($200, +25%, −10%)
Step 1: +25% → 1.25; −10% → 0.90.
Step 2: $200 × 1.25 = $250.
Step 3: $250 × 0.90 = $225.
Step 4: 1.25 × 0.90 = 1.125; $200 × 1.125 = $225. ✓ (Overall +12.5%.)
3.1 — Multipliers
(a) +15% → 1.15; (b) −30% → 0.70; (c) +8% → 1.08; (d) −5% → 0.95.
3.2 — $100 up 10%, then up 10%
$100 × 1.10 × 1.10 = $100 × 1.21 = $121. (Overall +21%, not +20%.)
3.3 — $100 down 10%, then down 10%
$100 × 0.90 × 0.90 = $100 × 0.81 = $81. (Overall −19%, not −20%.)
3.4 — $400 up 50%, then down 50%
$400 × 1.50 × 0.50 = $400 × 0.75 = $300. A 25% overall LOSS.
3.5 — $500, −30%, then −10%
$500 × 0.70 × 0.90 = $500 × 0.63 = $315. (Equivalent single discount = 37%, not 40%.)
3.6 — $80, +15%, then −8%
$80 × 1.15 × 0.92 = $80 × 1.058 = $84.64. (Overall +5.8%.)
3.7 — 25% off then 10% off
(a) Combined multiplier = 0.75 × 0.90 = 0.675.
(b) Equivalent single discount = 1 − 0.675 = 0.325 = 32.5% off (NOT 35%).
3.8 — Shop A vs Shop B
Shop A: $200 × 0.80 × 0.90 = $200 × 0.72 = $144 (28% off the original is the trap answer — it's actually 28% off after combining).
Shop B: $200 × 0.72 = $144.
(b) They are exactly the same: 0.80 × 0.90 = 0.72, which is a 28% combined discount — Shop B's flat 28% matches Shop A's two-stage 20%+10%.