079i - Percentages in Financial Contexts

Lesson 79 — Solving Problems Involving Percentages in Financial Contexts

Strand: Number | Descriptor: AC9M7N09 | Duration: 45 minutes

Learning Intentions

• To solve financial problems involving discounts, mark-ups, GST and simple interest.
• To interpret percentage answers in terms of the situation.

Success Criteria

I can:

1. Calculate discounts, mark-ups and GST.
2. Calculate simple interest on savings.
3. Compare offers using percentages.
4. Work backwards from a final price to an original price.

Warmup

(6 minutes — retrieval from Lesson 66, mini whiteboards)

1. of 260$.
2. 8025%$.
3. 4510%$.
4. After a discount, the price is what fraction of the original?
5. A price doubles. What percentage increase is that?

Answers: 1. 26$60$49.50\tfrac45100%$.

Activities

Activity 1 — The Financial Percentage Toolkit (14 min)

Mark-up — how shops price goods. A store buys jackets at 4560%$:

GST — Australia’s goods and services tax. A service is quoted at 180$ before GST:

The reverse — GST already included. A shelf price of 6666 - 6.60$:

The GST component is 6$ — one eleventh of the shelf price, not one tenth. This trips up adults; treat it slowly.

Simple interest — savings that grow. Interest each year rate principal; simple interest pays on the original amount only.

I do. 8004%3$ years:

We do:

1. A bike bought wholesale for 22045%$. Selling price?
2. A quote of 450$ plus GST. Total?
3. A shelf price of 88$ includes GST. Find the pre-GST price and the GST.
4. 12003.5%2$ years. Interest and balance?

(Answers: 319$495$80$8$84$1284$.)

Activity 2 — Multi-step Financial Problems (14 min)

Pairs. Estimate → solve → sentence.

Problem 1 — The sale rack, reprised. A store buys shirts at 1880%25%$. (a) Find the marked and sale prices. (b) Does the store still profit on each shirt? By how much and what percentage of cost?

Problem 2 — Compare the offers. The same laptop: Store A 124915%$118010%$999$ plus GST. Rank the final prices.

Problem 3 — The savings goal. Mia has 650$7005%18$ months without depositing more?

Problem 4 — Backwards from the docket. A restaurant bill totals 154$ including GST. Find the pre-GST amount and the GST.

Socratic scaffolding for Problem 1(b):

Prompt	Purpose
Track the money. Cost, marked, sale?	.
Profit per shirt?	6.30$.
As a percentage — of what?	Of cost: .
Why did an up and down not cancel to ?	Percentages act multiplicatively on different bases: .
Looking back	The multiplier chain gives the whole story in one line: .

Answers:

1. (a) 32.40$24.30$6.3035%$ margin on cost.
2. A: 1061.651180 \times 0.9 = $1062999 \times 1.1 = $1098.9035$ cents over B — worth noting how close.
3. (a) 32.5018= 1.5I = 48.75$698.75$1.25$.
4. 140= $14$.

Activity 3 — Inquiry: the Misleading Discount (7 min)

Pairs.

A store raises a 20025%25%$ OFF!” on the new price.

1. Find the final price. Is the jacket back to 200$?
2. What single percentage change has actually occurred?
3. Is the advertisement dishonest? Discuss.

Socratic scaffolding:

Prompt	Purpose
Compute the chain.	; .
So the “discounted” price is…?	187.50$ — actually below the original.
The single multiplier?	— a decrease overall.
Is the ad lying?	The off the current price is arithmetically true. Whether the impression is honest is another matter — this is why consumer law regulates “was/now” pricing.
Looking back	Same structure as Lessons 66 and 68: percentage of what decides everything.

Answers: 1. 187.506.25%$ decrease; 3. Discussion — arithmetic truth versus created impression.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A retailer buys at 3650%$. Find the selling price.
2. Add GST to a 240$ quote.
3. A price of 121$ includes GST. Find the GST component.
4. Find the simple interest on 9004%2$ years.
5. Reasoning. Why is the GST inside a shelf price one-eleventh of it, not one-tenth?

Answers: 1. 54$264121 \div 1.1 = 110= $11$72110%10110\tfrac{10}{110} = \tfrac{1}{11}$.

Common Misconceptions

Misconception	How to pre-empt it
Removing GST by subtracting of the shelf price.	The one-eleventh derivation, done slowly with 66$.
Believing then returns the start.	The misleading-discount inquiry — third visit to this idea, now with a consumer edge.
Quoting a margin as a percentage without a base.	Always ask “percentage of cost, or of price?” — they differ.
Simple interest compounding by accident (interest on interest).	Simple interest pays on the original principal only; the formula makes explicit.
Comparing offers by discount size instead of final price.	Problem 2: the smallest sticker price lost.
Dropping cents in money answers.	Money keeps two decimal places throughout.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A 6030%30%$. Find the final price.

Answer

54.609%$ overall decrease.

E2 (AMC Junior style). Simple interest of 63$7003$ years. Find the annual rate.

Answer

p.a.

E3 (Challenge). After a discount and then adding GST, an item costs 187$. Find the original pre-discount, pre-GST price.

Answer

E4 (Challenge). Which grows more in years: 5006%$5204%$ p.a. simple interest? By how much?

Answer

First: 560520 + 41.60 = $561.60$1.60$ — the head start narrowly beats the higher rate over this horizon.

E5 (Challenge). A store’s ” off” sale price of 8135%$ from cost. Find the cost.

Answer

Marked price ; cost 100$.

Homework

1. Find the selling price: (a) cost 2875%$15040%$8.40150%$.
2. Add GST: (a) 120$55.50$1240$.
3. Each price includes GST — find the pre-GST price and the GST: (a) 33$275$52.80$.
4. Find the simple interest and final balance: (a) 6005%2$15003%4$8504.2%18$ months.
5. A store buys at 6450%20%$. Find the final price and the profit as a percentage of cost.
6. Rank these final prices for the same headphones: X — 18920%$16510%$139$ plus GST.
7. A bill of 92.40$ includes GST. Find the GST.
8. Reasoning. A friend says ” off, then GST added, is the same as GST first, then off.” Are they right? Justify with multipliers.
9. Challenge. An investment of P5%$13803P$.

Answers: Q1 — (a) 49$210$21$132$61.05$1364$30$3$250$25$48$4.80$60$660$180$1680I = 850 \times 0.042 \times 1.5 = $53.55$903.5564 \times 1.5 \times 0.8 = $76.80$12.80 = 20%$151.20$148.50$152.9092.40 \div 11 = $8.400.7 \times 1.1 = 1.1 \times 0.7 = 0.77P \times 1.15 = 1380P = $1200$.