Lesson 11 ~25 min Unit 1 · Financial Maths +85 XP

Successive Percentage Changes

Why a $20\%$ rise followed by a $20\%$ drop doesn't bring you back to where you started.

Today's hook: A price goes up $20\%$ then comes down $20\%$. Are you back to the original? Most people get this wrong.

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Think First

warm-up

A price goes up $20\%$ then comes down $20\%$. Are you back to the original? Most people get this wrong. Jot down your first reaction, then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Two percentage changes applied in sequence DO NOT add. They multiply. A $20\%$ rise then $20\%$ fall ends with $96\%$ of the original, a $4\%$ overall LOSS.

Start with $\$100$. Rise $20\%$: $100 \times 1.20 = \$120$. Now fall $20\%$: $120 \times 0.80 = \$96$. NOT back to $\$100$. The percent rise was on $\$100$, but the percent fall was on the BIGGER $\$120$. Combined multiplier: $1.20 \times 0.80 = 0.96$, a $4\%$ overall decrease.

New $= $ Original $\times m_1 \times m_2 \times \dots$

Multiply, don't add

$20\%$ up + $20\%$ down $\neq 0\%$.

Different bases

The second $\%$ is on the NEW amount, not the original.

Order doesn't matter

$1.20 \times 0.80 = 0.80 \times 1.20$. Same result.

What You'll Master

objectives

Know

Successive changes multiply, never add
Combined multiplier = product of individual multipliers
Overall % change = $(m_1 m_2 - 1) \times 100$
Order of multiplication doesn't affect the final value

Understand

Why the second percentage is on a different base than the first
How equal-and-opposite percentage moves don't cancel
Why this matters for sales, investments, and depreciation

Can Do

Compute final value after multiple successive percentage changes
Find the equivalent single percentage change
Recognise common traps in everyday percentage thinking

Words You Need

vocabulary

Successive changeOne percentage change after another, each on the new value.

Combined multiplierProduct of all multipliers; gives the overall change in one number.

Equivalent single %The single change that gives the same final result.

Base shiftThe reason percentages don't add, each acts on a different base.

CommutativeOrder doesn't matter when multiplying.

Net changeFinal minus original, as a percentage of original.

Spot the Trap

heads-up

Wrong: "$20\%$ up then $20\%$ down = $0\%$ change", NO. It's a $4\%$ DECREASE.

Right: Multipliers: $1.20 \times 0.80 = 0.96$. That's $4\%$ down.

Wrong: "$10\%$ off then $5\%$ off = $15\%$ off", NO. It's about $14.5\%$ off (multipliers $0.90 \times 0.95 = 0.855$).

Right: $0.90 \times 0.95 = 0.855$, you pay $85.5\%$, save $14.5\%$, not $15\%$.

Why The Second % is on the New Value

+5 XP

Each percentage change uses the CURRENT value as the base. So $20\%$ of $\$120$ is bigger than $20\%$ of $\$100$, that's where the asymmetry comes from.

Start at $\$100$, rise $20\%$ to $\$120$. The next $20\%$ DECREASE is on the new $\$120$, not the original $\$100$. So the drop is $\$24$, not $\$20$. The price falls to $\$96$, $\$4$ below where we started.

$\$100 \xrightarrow{+20\%} \$120 \xrightarrow{-20\%} \$96$

Drop is bigger

$20\%$ of $\$120 > 20\%$ of $\$100$.

Final $< $ original

When rise and fall are equal $\%$ size.

The gap

Always a small loss for equal-up-then-down.

Combined Multiplier Method

+5 XP

Multiply all the individual multipliers together to get one combined multiplier. This represents the entire chain of changes as a single multiplication.

Three changes: $+10\%, -20\%, +5\%$. Multipliers: $1.10, 0.80, 1.05$. Combined: $1.10 \times 0.80 \times 1.05 = 0.924$. So the overall change is $0.924 - 1 = -0.076 = -7.6\%$. The starting value ends up at $92.4\%$ of its original sizea $7.6\%$ overall decrease.

Combined multiplier $= m_1 \times m_2 \times \dots \times m_n$

Multiply them all

Don't add the percentages.

One number

Combined multiplier replaces the whole chain.

Compare to 1

$< 1$ = overall decrease; $> 1$ = overall increase.

Watch Me Solve It · 3 examples

Watch Me Solve It · $20\%$ up, $20\%$ down

+15 XP per step

PROBLEM

Start at $\$100$. Increase by $20\%$, then decrease by $20\%$. Final value?

1

After $+20\%$

$100 \times 1.20 = \$120$

Multiplier 1.20.
2

After $-20\%$

$120 \times 0.80 = \$96$

New multiplier on the $\$120$.
3

Combined

$1.20 \times 0.80 = 0.96 \Rightarrow$ $\$100 \times 0.96 = \$96$

Same answer via one multiplier.

Answer$\$96$, a $4\%$ overall loss

Watch Me Solve It · Two discounts

+15 XP per step

PROBLEM

A store offers $25\%$ off, then a further $10\%$ at the till. What's the equivalent single discount on a $\$200$ item?

1

Multipliers

$0.75 \times 0.90 = 0.675$

Combined.
2

Final price

$200 \times 0.675 = \$135$

Sale price.
3

Equivalent single discount

$1 - 0.675 = 0.325 = 32.5\%$

Not $35\%$!

Answer$32.5\%$ off (not $35\%$)

Watch Me Solve It · Three changes

+15 XP per step

PROBLEM

A stock rises $15\%$, then falls $10\%$, then rises $8\%$. What's the equivalent single change?

1

Set up multipliers

$1.15, 0.90, 1.08$

Convert each.
2

Combine

$1.15 \times 0.90 \times 1.08 = 1.1178$

Multiply all three.
3

Equivalent change

$1.1178 - 1 = 0.1178 = 11.78\%$ increase

Net up, not $13\%$.

Answer$\approx 11.8\%$ overall increase

Common Pitfalls

heads-up

Adding instead of multiplying

$+20\%, -20\%$ gives $0$ if you add. Wrong.

Fix: Always MULTIPLY multipliers.

Forgetting the changed base

Treating the second $\%$ as if it were of the original.

Fix: Each $\%$ is of the CURRENT value, not the start.

Skipping the conversion

Working with raw percentages instead of multipliers.

Fix: Convert each $\%$ to a decimal multiplier ($1+r$ or $1-r$), THEN multiply.

Copy Into Your Books

Combined Multiplier

Multiply all multipliers together
$1.20 \times 0.80 = 0.96$
$96\%$ = $4\%$ loss

Asymmetry Rule

$+P\%$ then $-P\%$ $\neq 0$
Always slight LOSS
Different bases cause it

Equivalent Single %

Combined $-$ 1, then $\times 100$
$0.96 \to -4\%$
$1.10 \to +10\%$

Order Doesn't Matter

$0.80 \times 1.20 = 1.20 \times 0.80$
Multiplication is commutative
Apply in any order

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Successive Percentage Changes

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 $\$100 \to +10\% \to -10\%$. Final?

$100 \times 1.10 \times 0.90 = \$99$.$\$99$
2 $\$200, -25\%$ then $-20\%$. Final?

$200 \times 0.75 \times 0.80 = \$120$.$\$120$
3 Combined multiplier for $+30\%$ and $+20\%$:

$1.30 \times 1.20 = 1.56 \Rightarrow 56\%$ rise.$56\%$
4 $\$400 \to +50\%, -50\%$. Final?

$400 \times 1.50 \times 0.50 = \$300$.$\$300$

Complete in your workbook.

Quick Check · 5 questions

$\$100$ rises $10\%$ then falls $10\%$. Final value:

+10 XP

Two successive $20\%$ discounts give an equivalent single discount of:

+10 XP

$\$500, +30\%$ then $-10\%$. Final value:

+10 XP

A combined multiplier of $0.945$ means an overall:

+10 XP

A stock falls $50\%$, then rises $50\%$. The overall change is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. A jacket has its price modified twice: $+20\%$, then $-15\%$, on an original of $\$80$. (a) Find the price after each change. (b) Compute the combined multiplier. (c) What is the equivalent single percentage change?

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A shop offers $30\%$ off, then a $10\%$ student discount on top. What does a $\$200$ item cost the student?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Asha thinks a $40\%$ pay rise followed by a $40\%$ pay cut puts her back at her original salary. (a) Starting from $\$50\,000$, show with calculations whether she is right or wrong. (b) Calculate the actual percentage change. (c) Explain in plain words WHY two equal-and-opposite percentages don't cancel.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B$\$99$.

2. B$36\%$ off.

3. C$\$585$.

4. A$5.5\%$ decrease.

5. C$-25\%$.

Show Your Working Model Answers

Q6 (3 marks): (a) After $+20\%$: $80 \times 1.20 = \$96$. After $-15\%$: $96 \times 0.85 = \$81.60$ [1]. (b) Combined: $1.20 \times 0.85 = 1.02$ [1]. (c) $2\%$ overall increase [1].

Q7 (2 marks): $200 \times 0.70 \times 0.90 = \$126$ [2].

Q8 (4 marks): (a) After $+40\%$: $50000 \times 1.40 = \$70\,000$. After $-40\%$: $70000 \times 0.60 = \$42\,000$ [2]. (b) She finishes at $\$42\,000$, an overall $16\%$ decrease ($1.40 \times 0.60 = 0.84$) [1]. (c) The $40\%$ rise was on her original $\$50k$ (a $\$20k$ gain), but the $40\%$ cut was on the larger $\$70k$ (a $\$28k$ loss). The second percentage acts on a DIFFERENT base, so the changes don't cancel [1].

Stretch Challenge · +25 XP, +10 coins

The Compound Equivalent

A house price changes in 3 consecutive years: $+15\%$, $-8\%$, $+5\%$. (a) Find the combined multiplier. (b) Find the equivalent single percentage change. (c) If we replaced the three changes with three equal $+P\%$ changes (compounded), what single yearly $P\%$ would give the same total? (Hint: cube root.)

Reveal solution

(a) Combined $= 1.15 \times 0.92 \times 1.05 = 1.1109$. (b) Overall change $\approx +11.09\%$. (c) $1.1109^{1/3} \approx 1.0357$, so $P \approx 3.57\%$ per year, compounded.

Quick Review

Multiply

Multipliers multiply, percentages don't add

Different bases

Second % is on new value

Combined $= m_1 \times m_2$

One number captures the chain

$+P\%, -P\%$

Always slight loss

Order doesn't matter

Commutative multiplication

Equivalent single %

Combined $- 1$, then $\times 100$

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