Lesson 19 ~25 min Unit 1 · Financial Maths +85 XP

Multi-Step Financial Problems

Combine percentage, rate, and ratio thinking to solve realistic financial puzzles.

Today's hook: A business buys stock for $\$600$, marks it up $40\%$, then offers a $10\%$ loyalty discount to members. What do members pay?

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Think First

warm-up

A business buys stock for $\$600$, marks it up $40\%$, then offers a $10\%$ loyalty discount to members. What do members pay? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Real-world financial problems often need multiple steps: maybe a markup, then a discount, then GST. Apply each step in turn, often using multipliers, and the total is just the product.

Stock cost $\$600$, mark up $40\%$: $600 \times 1.40 = \$840$. Then a $10\%$ loyalty discount: $840 \times 0.90 = \$756$. Members pay $\$756$. The trick: multipliers stack — $1.40 \times 0.90 = 1.26$, so equivalent to $26\%$ above cost.

Final Price $= $ Cost $\times $ Markup $\times $ Discount $\times $ (1 + GST) $\times \dots$

Step by step

Apply each multiplier in turn.

Or all at once

Multiply all multipliers together for one shortcut.

Read carefully

Each step has a different base — be precise about the order.

What You'll Master

objectives

Know

Multi-step problems use combinations of % and rate methods
Apply each step in turn, then combine
Multipliers can be multiplied together for shortcut
Order can matter when problems include fixed-amount steps

Understand

How real-world prices stack markups, discounts, and taxes
Why multi-step problems test all the skills from this unit
When you can use commutative shortcuts and when you can't

Can Do

Break a multi-step problem into individual steps
Solve combined percentage and ratio scenarios
Identify which technique applies to each step

Words You Need

vocabulary

Multi-stepA problem requiring more than one operation in sequence.

Combined multiplierProduct of all multipliers, captures whole chain.

MarkupIncrease to set a sale price from cost.

Loyalty discountA discount given to repeat customers.

Net priceFinal price after all adjustments.

Working backwardsReverse-engineering each step.

Spot the Trap

heads-up

Wrong: "$40\%$ markup then $10\%$ discount = $30\%$ net markup" — NO. The $10\%$ is on the BIGGER number. Net is $26\%$.

Right: Multipliers: $1.40 \times 0.90 = 1.26$. Net is $26\%$ markup, not $30\%$.

Wrong: "Order doesn't matter for $\$$ amount + $\%$ change" — Depends. With pure $\%$ changes, yes. With fixed amounts mixed in, NO.

Right: For pure $\%$, order doesn't matter. With fixed-amount surcharges, it CAN matter.

Chains of Percentage Changes

+5 XP

Most real prices undergo several changes. Apply each multiplier in turn.

A book has cost $\$30$, marked up $50\%$ to $\$45$ retail. A $20\%$ sale: $\$45 \times 0.80 = \$36$. Then $10\%$ GST: $\$36 \times 1.10 = \$39.60$. Combined multiplier: $1.50 \times 0.80 \times 1.10 = 1.32$. So the customer pays $32\%$ above the original cost.

Final $= $ Cost $\times m_1 \times m_2 \times \ldots$

Step by step

Easier to track.

Combined for shortcut

Multiply all multipliers.

Net effect

Compare combined to 1.

Mixed Operations

+5 XP

Some problems mix percentages with fixed-amount fees, or rates with ratios. Treat each piece distinctly.

A $\$200$ phone bill includes a $\$10$ flat connection fee. The remaining $\$190$ is calls + data, split in ratio $3:2$. Calls: $190 \times \tfrac{3}{5} = \$114$; Data: $190 \times \tfrac{2}{5} = \$76$. This combines ratio with a fixed amount — the connection fee is separate.

Mix of $\%, $ fixed amounts, ratios — apply each piece carefully

Identify pieces

% changes? Fixed fees? Ratio splits?

Order matters

When fixed amounts mix with %, order changes outcome.

Show working

Multi-step problems reward clear layout.

Watch Me Solve It · 3 examples

Watch Me Solve It · Markup then discount

+15 XP per step

PROBLEM

Cost price $\$600$. Markup $40\%$. Then $10\%$ loyalty discount. What do members pay?

1

After markup

$600 \times 1.40 = \$840$

Retail price.
2

After loyalty

$840 \times 0.90 = \$756$

Member price.
3

Verify

Combined multiplier: $1.40 \times 0.90 = 1.26$; $600 \times 1.26 = \$756$ ✓

Same.

AnswerMembers pay $\$756$

Watch Me Solve It · Markup, sale, GST

+15 XP per step

PROBLEM

A retailer buys a jacket at $\$80$ cost. They mark up $75\%$ to set the retail price, then run a $30\%$ sale on it, then add $10\%$ GST. Find the customer's price.

1

After markup

$80 \times 1.75 = \$140$

Retail.
2

After sale

$140 \times 0.70 = \$98$

Sale price exc-GST.
3

After GST

$98 \times 1.10 = \$107.80$

Final inc-GST price.

Answer$\$107.80$

Watch Me Solve It · Bill split

+15 XP per step

PROBLEM

A $\$240$ restaurant bill (inc-GST) is split among 4 friends in the ratio $1:2:2:3$. (a) How much GST is in the bill? (b) What does each friend pay?

1

GST in the total

$240 \div 11 = \approx \$21.82$

GST is $\tfrac{1}{11}$ of inc-GST.
2

Total parts

$1+2+2+3 = 8$

Sum.
3

Each share

1 part $= 240/8 = \$30$. Shares: $\$30, \$60, \$60, \$90$

Check: sum = $\$240$ ✓

Answer(a) $\$21.82$ GST; (b) shares $\$30, \$60, \$60, \$90$

Common Pitfalls

heads-up

Wrong order with fixed fees

Adding GST before a $\$$ fee gives a different total than after.

Fix: Decide order based on the real-world process.

Adding percentages instead of multiplying

“$40\%$ markup + $10\%$ off = $30\%$”.

Fix: Multipliers chain: $1.40 \times 0.90 = 1.26$, not 1.30.

Losing track of steps

Mixing up which multiplier goes where.

Fix: Show work step by step. Label each step clearly.

Copy Into Your Books

Multi-Step Method

Identify each step
Use right multiplier for each
Apply in correct order

Combined Shortcut

Multiply all multipliers
One number captures all
Quick check possible

Common Combinations

Markup → Discount → GST
Ratio split + GST extraction
Rate × time × ratio

Show Working

Label each step
Show intermediate values
Verify with check

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Multi-Step Financial Problems

4 problems

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

1 $\$100 \to +20\% \to -20\% \to +10\%$. Final?

$100 \times 1.20 \times 0.80 \times 1.10 = \$105.60$.$\$105.60$
2 Bill $\$220$ inc-GST. GST?

$220 \div 11 = \$20$.$\$20$
3 Cost $\$50$, markup $50\%$, then $20\%$ off. Sale price?

$50 \times 1.50 \times 0.80 = \$60$.$\$60$
4 $\$180$ split in $2:3$. Larger share?

5 parts; 1 part $= 36$; larger $= 3 \times 36 = \$108$.$\$108$

Complete in your workbook.

Quick Check · 5 questions

Cost $\$100$, $25\%$ markup, then $20\%$ off. Sale price:

+10 XP

A $\$132$ inc-GST item gets a $10\%$ discount. Final price:

+10 XP

$\$420$ split in ratio $2:3:1$, second person's share:

+10 XP

A bill of $\$300$ inc-GST is split 3 ways equally. Each pays:

+10 XP

A car worth $\$30\,000$ depreciates $20\%$, then $15\%$. Value:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. A trader buys 50 phones at $\$200$ each. They mark up $30\%$ to set the retail price. They sell 40 at retail and the last 10 at a $20\%$ clearance discount. (a) Total CP. (b) Total revenue. (c) Profit (in $\$$).

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A $\$176$ inc-GST restaurant bill is split between 4 diners. (a) How much GST is in the bill? (b) How much does each diner pay if split equally?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A clothing store buys 100 shirts at $\$25$ each. They mark them up $80\%$ retail. Sale day: $40\%$ off retail, sells 60 shirts. Then a final clearance: $50\%$ off the SALE price, sells 30 shirts. The remaining 10 shirts are wasted. (a) Find retail, sale, and clearance prices. (b) Calculate total revenue. (c) Calculate total profit/loss. (d) What percentage of cost is the profit?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $\$100$.

2. B — $\$118.80$.

3. C — $\$210$.

4. C — $\$100$.

5. B — $\$20\,400$.

Show Your Working Model Answers

Q6 (3 marks): (a) CP $= 50 \times 200 = \$10\,000$ [0.5]. Retail $= 200 \times 1.30 = \$260$; clearance $= 260 \times 0.80 = \$208$ [0.5]. (b) Revenue $= 40 \times 260 + 10 \times 208 = 10400 + 2080 = \$12\,480$ [1]. (c) Profit $= 12480 - 10000 = \$2480$ [1].

Q7 (2 marks): (a) $176 \div 11 = \$16$ GST [1]. (b) $176 \div 4 = \$44$ each [1].

Q8 (4 marks): (a) Retail $= 25 \times 1.80 = \$45$. Sale $= 45 \times 0.60 = \$27$. Clearance $= 27 \times 0.50 = \$13.50$ [1]. (b) Revenue $= 60 \times 27 + 30 \times 13.50 = 1620 + 405 = \$2025$ [1]. (c) CP $= 100 \times 25 = \$2500$. Loss $= 2500 - 2025 = \$475$ loss [1]. (d) $\tfrac{475}{2500} \times 100 = 19\%$ loss on cost [1].

Stretch Challenge · +25 XP, +10 coins

The Cafe Pricing Puzzle

A cafe buys coffee beans at $\$36$/kg. Each cup uses 18 g of beans. They also pay $\$0.40$ per cup for milk and sugar. Wages, electricity, rent: $\$1.20$ per cup. They want a $50\%$ profit on total cost. They charge inc-GST. (a) Cost per cup. (b) Profit per cup. (c) Selling price (exc-GST). (d) Final menu price inc-GST.

Reveal solution

(a) Beans: $0.018 \times 36 = \$0.648$. Milk+sugar: $0.40$. Overheads: $1.20$. Total cost $= \$2.248 \approx \$2.25$. (b) Profit $= 0.50 \times 2.25 = \$1.125$. (c) Selling price exc-GST $= 2.25 + 1.125 = \$3.375$. (d) Inc-GST $= 3.375 \times 1.10 = \$3.71$ ($\approx \$3.75$).

Quick Review

Step by step

Each multiplier in turn

Combine

Multipliers multiply

Order matters

When fixed amounts mix in

Show working

Helps catch errors

Verify

Sum checks, reasonableness checks

Real-world

Markup → Sale → GST is common

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