Lesson 20 ~25 min Unit 1 · Financial Maths +85 XP

Financial Maths Synthesis and Review

Bringing it all together: percentages, rates, ratios, GST, profit, multi-step problems. Real-world maths in one place.

Today's hook: You're buying a phone, negotiating a pay rise, splitting a bill, and checking a receipt. Every skill from this unit is needed right now.

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Think First

warm-up

You're buying a phone, negotiating a pay rise, splitting a bill, and checking a receipt. Every skill from this unit is needed right now. Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Every situation in this unit boils down to one of: a fraction-decimal-percentage conversion, a percentage of something, a rate, or a ratio. The maths is the same; just be clear about what you're doing each step.

Five powerful ideas: FDP equivalence; percentage of/as quantity; percentage increase/decrease (and successive); rates (unit rates, best buy); ratios (simplify, divide, scale). Mix them in real situations: buying, selling, GST, splitting bills, scaling recipes.

Real-world finance = right tool $\times$ careful working

Identify the tool

Each question fits one of the five categories.

Use multipliers

For % rises and falls.

Always check

Reasonable size? Right units? Question answered?

What You'll Master

objectives

Know

FDP equivalents: $\tfrac{1}{2}=50\%, \tfrac{1}{4}=25\%, \tfrac{1}{5}=20\%$
% as multiplier: $1 \pm \tfrac{P}{100}$
GST in Australia is $10\%$ ($\div 11$ for tax on inc-GST)
Profit/loss on COST price; rate uses different units; ratio uses same

Understand

Why money problems often combine % + ratio + rate
How a real receipt or bill exercises every Year 8 financial skill
When to use mental method vs calculator

Can Do

Pick the right method for any real-world money problem
Combine techniques in multi-step scenarios
Reason about financial choices using percentages and rates

Words You Need

vocabulary

Financial literacyEveryday skills with money: budgeting, comparing, planning.

SynthesisCombining multiple ideas into a single solution.

StrategyChoosing the right method for the right question.

EstimationQuick reasonableness check on an answer.

CheckVerifying answer makes sense: size, sign, units.

Working backwardsReverse-engineering from a given final value.

Spot the Trap

heads-up

Wrong: "Multi-step problems are too hard." — They're just chained simple problems. Solve one piece at a time.

Right: Multi-step = identify each step, apply the right multiplier. Don't panic; just work through.

Wrong: "Percent always means $\div 100$ then add." — Sometimes it's $\div 100$ then SUBTRACT (decrease).

Right: For decreases, use $\times (1 - \tfrac{P}{100})$. For increases, $\times (1 + \tfrac{P}{100})$.

Choosing the Right Method

+5 XP

Every problem fits a pattern. Asking the right first question gets you to the right method.

Ask: What's being compared? Same things → ratio. Different units → rate. What's the verb? "Find $X\%$ of $Y$" → multiplier. "$X$ as $\%$ of $Y$" → fraction $\times 100$. Forwards or backwards? Multiply going forward, divide going back.

Strategy: identify $\to$ choose method $\to$ apply $\to$ check

Same vs different units

Tells ratio vs rate.

Verb of the problem

"Find ... of ..." vs "express as $\%$".

Direction

Forward (× multiplier) or backward (÷ multiplier).

Putting It All Together

+5 XP

A realistic bill or invoice can exercise every skill from this unit.

Imagine a $\$400$ cost, marked up $40\%$ to $\$560$, then $25\%$ off to $\$420$. Add $10\%$ GST: $\$462$. Three friends split in ratio $2:3:5$: 10 parts; 1 part $= 46.20$; shares $\$92.40, \$138.60, \$231.00$. Four skills in one problem.

Complex problems = many simple steps in a chain

Break it up

Each step is just one skill.

Multiplier chains

For % changes.

Sum check

Always verify total adds up.

Watch Me Solve It · 3 examples

Watch Me Solve It · The receipt

+15 XP per step

PROBLEM

A receipt shows: 3 sandwiches at $\$8.50$ each, 2 coffees at $\$4.20$ each, GST included. The friends split the bill equally between 3 people. How much does each pay, and how much GST was in the total?

1

Total before splitting

$3 \times 8.50 + 2 \times 4.20 = 25.50 + 8.40 = \$33.90$

Sum of items.
2

Each share

$33.90 \div 3 = \$11.30$

Equal split.
3

GST

$33.90 \div 11 \approx \$3.08$

Tax on the total.

Answer$\$11.30$ each; GST $\approx \$3.08$

Watch Me Solve It · The pay rise

+15 XP per step

PROBLEM

You currently earn $\$22.50$/hour. Your boss offers two choices: (a) $8\%$ pay rise immediately, or (b) a $\$2$/hour increase. Which gives the higher hourly rate?

1

Option (a)

$22.50 \times 1.08 = \$24.30$/hr

$8\%$ markup.
2

Option (b)

$22.50 + 2 = \$24.50$/hr

Flat increase.
3

Compare

$24.50 > 24.30$

Option (b) is better by $\$0.20$/hour.

AnswerOption (b) — $\$2$/hour flat increase

Watch Me Solve It · Save vs spend

+15 XP per step

PROBLEM

A phone costs $\$880$ inc-GST. You can pay outright, or you can pay $\$95$/month for 10 months. (a) How much extra does the monthly plan cost? (b) What is the extra as a percentage of the outright price?

1

Total monthly cost

$95 \times 10 = \$950$

Cost over 10 months.
2

Extra cost

$950 - 880 = \$70$

Premium for paying monthly.
3

As % of outright

$\tfrac{70}{880} \times 100 \approx 7.95\%$

Effectively a $\approx 8\%$ surcharge.

Answer$\$70$ extra, $\approx 8\%$ premium

Common Pitfalls

heads-up

Mixing up percentage of vs as

“$\$30$ of $\$120$” vs “$\$30$ as $\%$ of $\$120$”.

Fix: “of” = $\times$; “as $\%$” = $\div$ then $\times 100$.

Forgetting GST is $\tfrac{1}{11}$

Treating $10\%$ of inc-GST as the tax.

Fix: GST is $10\%$ of exc-GST, so $\tfrac{1}{11}$ of inc-GST.

Skipping sanity check

Submitting an answer like a $300\%$ profit without questioning it.

Fix: Always ask “does this make sense?”

Copy Into Your Books

Key Formulas

$P\%$ of $Q = \tfrac{P}{100} \times Q$
$A$ as % of $B = \tfrac{A}{B} \times 100$
% change = $\tfrac{\Delta}{\text{old}} \times 100$

Multipliers

Increase by $P\%$: $\times (1 + P/100)$
Decrease by $P\%$: $\times (1 - P/100)$
GST: $\times 1.10$ or $\div 1.10$

Rates and Ratios

Rate = different units (per kg, per h)
Ratio = same units (recipe parts)
Unit price = total ÷ quantity

Strategy

Identify the question type
Pick the method
Apply, check, reason

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Financial Maths Synthesis and Review

4 problems

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

1 Find $15\%$ of $\$240$.

$0.15 \times 240 = \$36$.$\$36$
2 GST on a $\$77$ inc-GST item.

$77/11 = \$7$.$\$7$
3 Split $\$480$ in $3:5$. Bigger share?

8 parts; 1 part $= 60$; bigger $= 5 \times 60 = \$300$.$\$300$
4 CP $\$200$, SP $\$260$. % profit?

$\tfrac{60}{200} \times 100 = 30\%$.$30\%$

Complete in your workbook.

Quick Check · 5 questions

Convert $\tfrac{3}{4}$ to a percentage.

+10 XP

A car bought for $\$15\,000$ is sold for $\$13\,500$. % loss:

+10 XP

A $\$300$ jacket has $30\%$ off, then $10\%$ off the sale price. Final:

+10 XP

Split $\$420$ in ratio $1:2:4$. Middle share:

+10 XP

A 750 g bag of rice costs $\$3$. The price per kg is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. (a) Convert $\tfrac{7}{8}$ to a percentage. (b) Find $18\%$ of $\$240$. (c) Split $\$315$ in ratio $4:5$.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A jacket has CP $\$80$, marked up $40\%$. Then a $15\%$ discount is offered at the till. (a) What's the final selling price? (b) What's the profit on this sale?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A family's monthly grocery bill is $\$840$. They spend in the ratio Food:Toiletries:Cleaning = $7:2:1$. Food bills go up $10\%$ this month due to inflation, but cleaning items go down $15\%$ due to a sale. Toiletries stay the same. (a) Calculate each original cost. (b) Calculate each new cost. (c) Calculate the new total. (d) Express the new total as a percentage change from the original.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. D — $75\%$.

2. A — $10\%$ loss.

3. B — $\$189$.

4. B — $\$120$.

5. D — $\$4/$kg.

Show Your Working Model Answers

Q6 (3 marks): (a) $7 \div 8 \times 100 = 87.5\%$ [1]. (b) $0.18 \times 240 = \$43.20$ [1]. (c) 9 parts; 1 part = $\$35$; shares $\$140, \$175$ [1].

Q7 (2 marks): (a) $80 \times 1.40 \times 0.85 = \$95.20$ [1]. (b) Profit $= 95.20 - 80 = \$15.20$ [1].

Q8 (4 marks): (a) 10 parts; 1 part $= \$84$. Food $= 7 \times 84 = \$588$; Toiletries $= 2 \times 84 = \$168$; Cleaning $= 1 \times 84 = \$84$ [1]. (b) Food $= 588 \times 1.10 = \$646.80$; Toiletries $= \$168$; Cleaning $= 84 \times 0.85 = \$71.40$ [1]. (c) New total $= 646.80 + 168 + 71.40 = \$886.20$ [1]. (d) $\tfrac{886.20 - 840}{840} \times 100 \approx 5.5\%$ increase [1].

Stretch Challenge · +25 XP, +10 coins

The Big Synthesis

You start an honest small business. You buy 100 t-shirts at $\$8$/each. You sell 70 at $\$25$ each (full retail), 20 at $\$18$ each (sale), and 10 are damaged and unsold. All prices inc-GST. (a) Total CP. (b) Total revenue (inc-GST). (c) Total revenue (exc-GST). (d) Total profit on the exc-GST revenue. (e) % profit on cost. (f) What's the total GST you owe?

Reveal solution

(a) CP $= 100 \times 8 = \$800$. (b) Revenue $= 70 \times 25 + 20 \times 18 + 10 \times 0 = 1750 + 360 = \$2110$ inc-GST. (c) Revenue exc-GST $= 2110 \div 1.10 = \approx \$1918.18$. (d) Profit (exc-GST) $= 1918.18 - 800 = \$1118.18$. (e) % profit = $\tfrac{1118.18}{800} \times 100 \approx 139.8\%$. (f) GST $= 2110 - 1918.18 \approx \$191.82$ ($\approx 2110/11$).

Quick Review

Identify the type

% / rate / ratio / FDP

Use multipliers

For % changes

GST = $\times 1.10$ or $\div 11$

Add or strip

Unit rate

Lower = cheaper

Ratio split

Sum, 1 part, multiply

Sanity check

Reasonable size, right units

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