Lesson 6 ~25 min Unit 1 · Financial Maths +85 XP

Percentage Decrease

Sales, write-downs and depreciation — find the new value when something loses a percentage of its worth.

Today's hook: A $\$250$ pair of shoes is $30\%$ off. Method 1 or Method 2 — which is faster?

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Think First

warm-up

A $\$250$ pair of shoes is $30\%$ off. Method 1 or Method 2 — which is faster? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

A percentage decrease SUBTRACTS a percentage from the original. Again, two equivalent methods — subtract or multiply by $(1 - $ rate$)$.

Take $\$250$ shoes $30\%$ off. Method 1: $30\%$ of $250 = \$75$; new price $= 250 - 75 = \$175$. Method 2: multiplier $= 1 - 0.30 = 0.70$; so $250 \times 0.70 = \$175$. Method 2 is often faster — one operation, no intermediate value needed.

New $= $ Original $\times \left(1 - \dfrac{P}{100}\right)$

Multiplier always $<1$

A DECREASE makes things SMALLER. Multiplier is less than 1.

$30\%$ off

Multiplier is $1 - 0.30 = 0.70$. You pay $70\%$.

$25\%$ off = pay $75\%$

Quick mental check: discount + sale fraction = $100\%$.

What You'll Master

objectives

Know

New value = original $-$ (% of original)
New value = original $\times (1 - \tfrac{P}{100})$
$25\%$ decrease $\to$ multiplier $0.75$
You pay $(100 - P)\%$ of the original

Understand

Why $1 - \tfrac{P}{100}$ is the “sale fraction”
How a $30\%$ discount means paying $70\%$
Why the decrease method mirrors increase

Can Do

Apply a percentage discount with confidence
Choose between subtract and multiplier methods
Recognise when a problem is a percentage decrease

Words You Need

vocabulary

Percentage decreaseSubtracting a percentage of the original from itself.

DiscountThe amount taken off the original price.

Sale priceThe price after the discount: original − discount.

Subtract methodFind the discount, then subtract.

Decrease multiplierNew $= $ Original $\times (1 - \tfrac{P}{100})$.

DepreciationWhen an item loses value over time (often a percentage drop).

Spot the Trap

heads-up

Wrong: "$30\%$ off $\$250$ means I pay $\$30$" — NO. You SAVE $\$75$ ($30\%$ of $250$), not 30 dollars.

Right: $30\%$ of $250 = \$75$. New $= 250 - 75 = \$175$.

Wrong: "Multiplier for $25\%$ decrease = $\times 1.25$" — NO. Decrease means SMALLER, multiplier is BELOW 1.

Right: Decrease multiplier $= 1 - 0.25 = 0.75$. Pay $75\%$ of original.

Method 1 — Subtract the Discount

+5 XP

Two clear steps: find the discount as a dollar amount, then subtract.

For $30\%$ off a $\$250$ jacket: Step 1: $30\%$ of $250 = 0.30 \times 250 = \$75$ (the discount). Step 2: $250 - 75 = \$175$ (sale price). Useful for advertising — “Save $\$75$!” sounds louder than “Pay $\$175$”.

New $= $ Original $-$ Discount, Discount $= \dfrac{P}{100} \times $ Original

Two steps

Find the savings, subtract.

Reveals the savings

Lets you state the $\$$ saved.

Mental method

Often faster for round numbers.

Method 2 — The Decrease Multiplier

+5 XP

A single multiplication. New value = original $\times (1 - $ rate$)$.

A $25\%$ discount means you pay $75\%$ — multiplier $0.75$. A $40\%$ discount means you pay $60\%$ — multiplier $0.60$. The pattern: multiplier $+$ discount rate $=$ 1. Always.

$P\%$ off $\Rightarrow$ multiplier $= 1 - \dfrac{P}{100}$

Below 1

Decrease multiplier is always $< 1$.

You pay $(1 - $ rate$)$

$30\%$ off means $\times 0.70$.

One calculator step

Faster than the subtract method.

Watch Me Solve It · 3 examples

Watch Me Solve It · $\$250$ shoes

+15 XP per step

PROBLEM

$\$250$ pair of shoes, $30\%$ off. Compare the two methods.

1

Method 1: subtract

Discount $= 0.30 \times 250 = \$75$. Sale $= 250 - 75 = \$175$

Two steps.
2

Method 2: multiplier

$1 - 0.30 = 0.70$. Sale $= 250 \times 0.70 = \$175$

One step.
3

Compare

Same answer, Method 2 is faster

Save your brainpower for harder bits.

Answer$\$175$ (Method 2 faster)

Watch Me Solve It · $15\%$ off

+15 XP per step

PROBLEM

A $\$80$ backpack has $15\%$ off. What's the sale price?

1

Set up multiplier

$1 - 0.15 = 0.85$

You pay $85\%$.
2

Multiply

$80 \times 0.85$

One step.
3

Compute

$80 \times 0.85 = \$68$

Final price.

Answer$\$68$

Watch Me Solve It · Depreciation

+15 XP per step

PROBLEM

A $\$25\,000$ car loses $18\%$ of its value in the first year. What is it worth then?

1

Set up multiplier

$1 - 0.18 = 0.82$

Keeps $82\%$ of value.
2

Multiply

$25\,000 \times 0.82$

Single operation.
3

Compute

$25\,000 \times 0.82 = \$20\,500$

Worth after 1 year.

Answer$\$20\,500$

Common Pitfalls

heads-up

Using $\times 0.30$ instead of $\times 0.70$

Gives the DISCOUNT, not the SALE price.

Fix: Sale price multiplier = $1 - $ rate, NOT just the rate.

Using $\times 1.30$

That's an increase, not a decrease.

Fix: For a $30\%$ off, multiplier is $0.70$, not $1.30$.

Subtracting the dollar amount of the rate

“$30\%$ off $\$250$ = $\$250 - \$30 = \$220$” — wrong.

Fix: $30\%$ of $250 = \$75$. $\$250 - \$75 = \$175$.

Copy Into Your Books

Method 1: Subtract

Discount $= \tfrac{P}{100} \times $ Original
Sale $= $ Original $-$ Discount
Tells you the $\$$ saved

Method 2: Multiplier

Multiplier $= 1 - \tfrac{P}{100}$
Sale $= $ Original $\times $ Multiplier
Always $< 1$

Common Multipliers

$10\%$ off: $\times 0.90$
$20\%$ off: $\times 0.80$
$25\%$ off: $\times 0.75$
$50\%$ off: $\times 0.50$

Pay-Fraction Trick

$P\%$ off = pay $(100 - P)\%$
$30\%$ off = pay $70\%$
Sum is always $100\%$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Percentage Decrease

4 problems

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

1 Decrease $\$160$ by $25\%$.

$160 \times 0.75 = 120$.$\$120$
2 A $\$45$ shirt has $20\%$ off. Sale price?

$45 \times 0.80 = 36$.$\$36$
3 Decrease $400$ kg by $15\%$.

$400 \times 0.85 = 340$.$340$ kg
4 A $\$30\,000$ car loses $20\%$ in year 1. Worth then?

$30000 \times 0.80 = 24000$.$\$24\,000$

Complete in your workbook.

Quick Check · 5 questions

$\$120$ decreased by $25\%$ is:

+10 XP

The multiplier for a $40\%$ decrease is:

+10 XP

A $\$240$ jumper is $35\%$ off. The sale price is:

+10 XP

If you save $\$15$ on a $\$60$ item, what percentage off is it?

+10 XP

A car worth $\$40\,000$ depreciates by $15\%$. New value:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Apply the discount using the multiplier method: (a) $\$300$ less $40\%$ (b) $\$85$ less $22\%$ (c) $\$1500$ less $7.5\%$

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A $\$240$ desk is reduced by $15\%$. What is the discount AND the sale price?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Two clothing stores both sell the same $\$200$ jacket. Store A offers “$25\%$ off, then a further $10\%$ at checkout”. Store B offers “$30\%$ off everything”. (a) Find the final price at each store. (b) Which is the better deal? (c) Explain why $25\% + 10\%$ does NOT equal $35\%$ off.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — $120 \times 0.75 = \$90$.

2. C — $1 - 0.40 = 0.60$.

3. B — $240 \times 0.65 = \$156$.

4. C — $\tfrac{15}{60} = 25\%$.

5. B — $40000 \times 0.85 = \$34000$.

Show Your Working Model Answers

Q6 (3 marks): (a) $300 \times 0.60 = \$180$ [1]. (b) $85 \times 0.78 = \$66.30$ [1]. (c) $1500 \times 0.925 = \$1387.50$ [1].

Q7 (2 marks): Discount $= 0.15 \times 240 = \$36$ [1]. Sale $= 240 - 36 = \$204$ [1].

Q8 (4 marks): (a) Store A: $200 \times 0.75 \times 0.90 = \$135$ [1]. Store B: $200 \times 0.70 = \$140$ [1]. (b) Store A is cheaper by $\$5$ [1]. (c) The $10\%$ second discount applies to the ALREADY-discounted price ($\$150$), not the original $\$200$, so the $10\%$ is worth only $\$15$ — making the total saving $\$50 + \$15 = \$65$, equivalent to $32.5\%$, not $35\%$ [1].

Stretch Challenge · +25 XP, +10 coins

The Devaluing Phone

A phone is worth $\$1200$ new. It loses $25\%$ of its value each year. (a) What is it worth at the end of year 1, 2 and 3? (b) After how many full years is it worth less than $\$300$?

Reveal solution

(a) Y1: $1200 \times 0.75 = \$900$. Y2: $900 \times 0.75 = \$675$. Y3: $675 \times 0.75 = \$506.25$. (b) Y4: $506.25 \times 0.75 \approx \$379.69$. Y5: $\approx \$284.77$. After 5 years it dips below $\$300$.

Quick Review

Subtract

Find discount, then subtract

Multiplier

$1 - \tfrac{P}{100}$

$25\%$ off

$\times 0.75$ (pay $75\%$)

$50\%$ off

$\times 0.50$ (half price)

Multiplier $< 1$

Always for a decrease

Sum = $100\%$

Discount + pay fraction = 100%

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