Discounts, markups and profit: formulate the problem, choose fractions, percentages or ratios, solve, and say what the answer means in dollars.
A $200 bike is 20% off, and then a coupon takes a further 10% off the new price. Is that the same as 30% off?
Money problems are ratio and percentage problems with a context. The habit is the same three steps:
Formulate: what is known (price, discount, cost), what is unknown, and which representation (fraction, decimal, percentage, ratio) fits?
Solve: choose an efficient calculation, estimate first, and use a calculator where it helps.
Interpret: say what the answer means, in dollars, with a sensible amount of rounding, and justify why you chose that representation.
Two traps: a percentage is always of something (the original amount for profit or discount), and percentages of different amounts cannot be added.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Two discounts
Why is 10% of the new price less than 10% of $200?
TryTry it on the board and compare the result with your prediction.
30% off $200
Now set 20%, then use 10% off that price.
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Set the discount to 30%. What is the sale price? Is it the same as your predicted price for two discounts?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Formulate, solve, interpret
Known: a $60 jumper is 25% off. Unknown: the sale price. What is it, in dollars?
Ben finds the percentage profit on a scarf. Click the line where his working goes wrong.
ModifyChange the problem on purpose and see what happens.
New money problems
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A game costs $120 and has 15% off. What is the sale price?
Use the board. This ticks when you get there.
You earn $20 an hour and get a 5% pay rise. What is your new hourly rate, in dollars?
A $50 top is marked up by 40% and then sold at 20% off the new price. Estimate first: will the final price be more or less than $50? Then work it out, in dollars.
PractiseA few on your own, to check you have it.
Money models
A $240 prize is shared: 41 to Ann, and the rest equally among 3 others. How much does each of the other three get, in dollars?
A bike costs $400. You have saved 65% of it. How many more dollars do you need?
A pack of 12 pens costs $9, while one pen costs $1. What percentage cheaper is each pen in the pack?
Review: share $84 in the ratio 3:4. What is the bigger share, in dollars?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A $75 sale
Then write one sentence for a shopper explaining your sale.
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Need a hint?
The sale price is price × (1 − discount).
Another hint
Try $100 with 25% off.
Choose a price and a discount so the sale price is exactly $75.
Use the board. This ticks when you get there.
For teachers
Students model money problems (discounts, rises, markups, profit and sharing) by formulating what is known and unknown, choosing a representation, solving, and interpreting and justifying the answer in dollars (AC9M7N09, nodes 1 to 3). About 45 minutes.
I can say what is known and unknown in a money problem and choose how to model it.
I can solve problems with fractions, percentages and ratios, including percentages in two steps.
I can write my answer in dollars and explain why I chose that representation.
Before class
Do the Run board yourself first. Students need percentages of amounts (Percentages), sharing in a ratio, and rounding for money.
Three nodes in one lesson: formulate (known, unknown, representation), solve and interpret in dollars, and justify the choice. The two traps are using the wrong base (profit as a percentage of the selling price, Ben) and adding percentages that are of different amounts (20% then 10% is not 30%).
Calculators are fine. The estimate comes first, and the answer is written to the cent with units.
Running the lesson
Predict: students who chose $140 added 20% and 10%; $150 and $160 are other slips.
Run: ask why 10% of $160 is less than 10% of $200.
Investigate: have students say 'of what?' aloud for each percentage in the jumper and the scarf.
Modify: for the markup then sale, ask whether the final price is more or less than $50, and why.
Make: share the pairs and ask which shop would you prefer, and why.
The hard part (on purpose)
Knowing what the percentage is of. Ask 'of which amount?' every time a percentage appears, especially the second one in a two-step problem.
When students get stuck
A student divides profit by the selling price (Ben's slip). Ask: What did the shop spend, and what is the profit compared with? Fix: Profit as a percentage is out of the cost, the original amount.
A student adds 40% and subtracts 20% to get +20%. Ask: Is the 20% off of $50 or of the new price? Fix: Do it in steps: × 1.4, then × 0.8.
A student gives an answer without units. Ask: 33.6 what? Fix: Write dollars to the cent: $33.60.
Stretch and support
Stretch: A shop marks up its cost by 25%, then offers a 20% sale on the new price. Does it make a profit? Does the shop ever lose money with a 25% markup followed by a bigger sale?
Support: Use the bar for each step, and write 'of $…' next to every percentage.
Where it leads
Next: Modelling with ratios, with mixes, scale drawings and models in measurement.
Where it leads: GST, simple and compound interest, profit and loss, and financial literacy in Years 8 to 10.
What each step is for
two discounts: Committing shows whether they add the percentages (30% off, $140), apply them in turn ($144), or take both off the original.
30% off $200: Setting 30% off $200 gives $140, the single-step answer to compare with the two-step $144.
formulate, solve, interpret: Profit percentage is out of the cost, the original; Ben divides by the selling price. A fraction is easier to compare when both discounts are in the same form.
new money problems: A discount on the board, a pay rise, a markup then a discount, and the answer written the way a shop would: each asks for the model, then the units.
money models: A fraction of a prize, a saving goal as a percentage, a bulk saving, and a ratio review.
a $75 sale: Choosing a price and a discount that give exactly $75 shows many models fit and each says something different about the shop.
Now you can
Say what is known and unknown in a money problem and choose how to model it.
Solve problems with fractions, percentages and ratios, including percentages in two steps.
Write my answer in dollars and explain why I chose that representation.