085i - Percentages in Financial Contexts

Lesson 85 — Solving Applied Percentage Problems in Financial Contexts

Strand: Number | Descriptor: AC9M8N05 | Duration: 45 minutes

Learning Intentions

• To solve multi-step percentage problems in financial contexts, including discounts, GST, and simple interest.
• To use the simple interest relationship to calculate interest, principal, rate, or time.

Success Criteria

I can:

1. Calculate simple interest using , given the principal, rate, and time.
2. Solve multi-step problems combining a discount and GST.
3. Solve percentage problems involving profit, loss, and profit margins.
4. Solve for an unknown quantity (rate, principal, or time) when given a percentage relationship.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. Write the decimal multiplier for a 20% discount.
2. Write the decimal multiplier for adding 10% GST.
3. If an item costs before tax, write an expression for its price after 10% GST.
4. A shop buys a bike for 260. What is the dollar profit?

Answers: 1. ; 2. ; 3. ; 4. $60.

Activities

Activity 1 — Explicit Instruction: Simple Interest (12 min)

Introduce the simple interest formula, where is the principal (starting amount), is the annual interest rate (as a percentage), and is the time in years:

I do: is invested at 4% p.a. for 3 years. Find the interest earned and the final balance.

We do: Find the interest earned on invested at 6% p.a. for 2 years.

You do: (a) Find the interest on at 5% p.a. for 4 years. (b) A loan of accrues interest over 2 years — find the annual rate . (c) invested at 3.5% p.a. earns interest — find the time , in years.

Activity 2 — Explicit Instruction: Discounts, GST and Profit Margins (12 min)

I do — combined discount and GST: A item is discounted by 20%, then 10% GST is added to the sale price.

Note: because multiplication is commutative, gives the same final price regardless of the order the two multipliers are applied — but retailers conventionally apply GST after any discount, on the discounted price.

I do — profit margin: A retailer buys stock for 100 (selling price). Profit . Convention: profit margin is usually calculated on the cost price unless stated otherwise.

We do / You do: Practise further combined discount+GST and profit-margin problems, including one “loss” scenario (selling below cost price, giving a negative profit margin).

Activity 3 — Applied Problem-solving Task (10 min)

Pairs, then whole-class share.

A retailer buys a bike for $240, marks it up 45% to set the retail price, then during a sale offers 20% off the marked price. GST of 10% is then added to the sale price. What does a customer pay, and what is the retailer’s profit margin (as a percentage of cost price) on the transaction, excluding the GST collected (which is passed on to the government, not kept as profit)?

Socratic scaffolding:

Prompt	Purpose
Understand: what are we asked to find?	Two separate things — the customer’s final payment (including GST), and the retailer’s actual profit margin (excluding GST, since that money isn’t kept).
Devise a plan	Work through the chain of changes in order: cost → marked-up retail price → sale price → GST-inclusive price. Then separately find profit sale price (excl. GST) cost.
Carry out the plan	Retail price: . Sale price: . Price incl. GST: . Profit (excl. GST): . Margin: .
Look back — is this sensible?	The 45% markup was substantially eroded by the 20% discount, leaving a much smaller 16% margin — check this is consistent with the size of the discount relative to the markup.
Look back — did we answer both parts?	Customer pays $306.24; retailer’s profit margin is 16%.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the simple interest on at 5% p.a. for 3 years.
2. A $180 item is discounted by 25%, then 10% GST is added. Find the final price.
3. A shop buys an item for 75. Find the profit margin as a percentage of the cost price.
4. Reasoning. Explain why applying a 20% discount and then adding 10% GST gives the same final price as adding the GST first and then applying the discount.

Answers: 1. ; 2. , then ; 3. profit , margin ; 4. Because both operations are multipliers ( and ), and multiplication is commutative — the order in which they are multiplied does not change the final product.

Common Misconceptions

Misconception	How to pre-empt it
The simple interest formula requires the rate as a decimal, not a percentage.	Show is used as the raw percentage number (e.g. , not ) because the formula already divides by .
GST is calculated on the original marked price, even after a discount has been applied.	Model Activity 2 explicitly: GST applies to whatever the customer is actually charged at that point — the discounted price.
Profit margin is always calculated on the selling price.	State the convention explicitly (margin on cost price unless told otherwise) and show how the two conventions give different percentages for the same numbers.
A markup and a discount of the same percentage cancel out.	Show — a 45% markup followed by a 20% discount does not return to the original cost.
GST collected by a retailer counts as part of their profit.	Clarify explicitly, as in Activity 3, that GST is collected on behalf of the government and is not retailer profit.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). An amount of money invested at simple interest doubles in value after 20 years. What is the annual interest rate?

Answer

If the principal is , the interest earned equals (to double the value), over years.

The annual rate is 5%.

E2 (Kangaroo style). A trader buys an item and sells it for 45. What was the cost price?

Answer

The cost price was $150.

E3 (Challenge). A shopkeeper marks all goods up by 50% above cost price, then advertises a “40% off” sale on the marked price. Show, using a general cost price , whether the shopkeeper makes a profit, breaks even, or makes a loss on the sale — and find the percentage.

Answer

Since , the shopkeeper sells at 90% of cost — a 10% loss, despite the seemingly generous 50% markup.

Homework

1. Find the simple interest on: (a) at 4% p.a. for 5 years (b) at 3.5% p.a. for 2 years.
2. A item is discounted by 30%, and 10% GST is then added. Find the final price.
3. A retailer buys stock for 189. Find the profit margin as a percentage of the cost price.
4. A loan of accrues interest over 3 years. Find the annual interest rate.
5. Reasoning. A store manager says, “If I mark up an item by 25% and then discount it by 25% during a sale, I break even.” Show, using a calculation, whether this is true.
6. Challenge. An investment of dollars at simple interest of p.a. grows to exactly after 6 years. Find , and then find how many years it would take the same investment to triple in value at the same rate.

Answers: 1. (a) (b) . 2. ; . 3. profit , margin . 4. . 5. Multiplier — the manager is incorrect; the item ends at of its original price, a net 6.25% loss. 6. . To triple: years.