Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 4) A line has gradient 5 and passes through (0, 3). Write its equation.
Answer
y = 5x + 3. - Solve 7x = 90 + 2x.
Answer
x = 18: 5x = 90. - What is 70 + 6 × 15?
Answer
160: 6 × 15 = 90, and 70 + 90 = 160.
Words used in this lesson
- model
- an equation or graph that describes a real situation. For example: C = 80h + 200 for a repair.
- rate
- how much changes for each unit. For example: 5 rupees per refill.
- fixed fee
- an amount paid once, whatever the use. For example: a 600 rupee flask.
- break-even
- the point where two costs are equal. For example: 15 uses.
- extrapolate
- use a model beyond the range of the data. For example: predicting day 120 from 10 days.
- interpolate
- use a model inside the range of the data. For example: predicting day 7 from days 0 to 10.
- piecewise
- made of different straight pieces for different ranges. For example: 4 rupees then 7 rupees per unit.
Is the flask worth it?
A bottle of water costs 45 rupees. A steel flask costs 600 rupees and each refill from a tap costs 5 rupees. Someone drinks one bottle a day. Buying bottles costs 45 × (days), and the flask costs 600 + 5 × (days): two linear models. Which is cheaper depends on how many days.
Predict first: after how many days does the flask become the cheaper choice? Use the sliders to test your estimate: set the flask to 600 rupees and move the number of uses until the two costs are equal.
Flask 800 rupees, plus 5 for each refill. After 10 uses: bottles 45 × 10 = 450, flask 800 + 5 × 10 = 850. Bottles are cheaper so far, by 400; the costs meet at 20 uses.
Show me the steps
- Flask 800, 10 uses: bottles 450, flask 850. Bottles are cheaper so far. They meet at 800 ÷ 40 = 20 uses.
- Flask 600, 15 uses: bottles 675, flask 675. The same cost: the break-even.
- Flask 600, 20 uses: bottles 900, flask 700. The flask is now cheaper, by 200 rupees.
- Flask 400, 10 uses: bottles 450, flask 450. A cheaper flask breaks even sooner, at 10 uses.
- Flask 1000, 25 uses: bottles 1125, flask 1125. A dearer flask takes 25 uses to pay for itself.
A linear model is y = mx + c where m is the rate and c is the starting amount. For the flask, m = 5 rupees per refill and c = 600 rupees paid at the start. For bottles, m = 45 and c = 0.
The break-even point is where two models give the same value: set them equal and solve. Here 45x = 600 + 5x gives 40x = 600, so x = 15 uses and a cost of 675 rupees. Before it one plan is cheaper; after it, the other.
Some plans change rate. If the first 20 units cost 4 rupees each and later units cost 7, the model is two straight pieces joined: a piecewise model. Each piece has its own gradient. For 25 units: 20 × 4 + 5 × 7 = 115 rupees.
A common mistake: treating the break-even as the answer to “which is cheaper?”. It is the point where the answer changes. At 10 uses the bottles are cheaper; at 20 the flask is. The break-even is the boundary, and which side you are on depends on how you will use it.
How far can you trust a line?
A model is built from measurements, and it is only as good as the pattern it was built from. A young sunflower was measured for 10 days, and its height fits y = 10 + 4x. Slide the day past the measurements and watch what the model keeps saying.
Day 5: the model gives 10 + 4 × 5 = 30 cm. This day is inside the 10 days that were measured, so the model can be trusted here.
Show me the steps
- Day 5: 10 + 4 × 5 = 30 cm, inside the measured days.
- Day 10: 10 + 4 × 10 = 50 cm, the last measured day.
- Day 20: 10 + 4 × 20 = 90 cm. Twice as far as the measurements went: a prediction.
- Day 60: 10 + 4 × 60 = 250 cm. Six times as far as the measurements: the line still gives a number, and it is only just believable for a sunflower.
- Day 120: 10 + 4 × 120 = 490 cm. Twelve times as far as the measurements: the line still gives a number, but a sunflower stops growing at about 2 to 3 m, so this is not believable.
Using a model inside the measured range is interpolation; beyond it is extrapolation. The arithmetic is the same, but the second is a prediction that the pattern continues. Always ask: does the answer make sense (no negative litres, and no 5 m sunflower), and does the pattern have a reason to continue?
Worked example: the flask
Bottles cost 50 rupees each; a flask costs 640 rupees plus 10 per refill. Find the break-even and say which is cheaper for 30 uses.
- Bottles: y = 50x. Flask: y = 640 + 10x.
- Equal when 50x = 640 + 10x, so 40x = 640 and x = 16. The cost there is 50 × 16 = 800 rupees.
- For 30 uses: bottles 50 × 30 = 1500, flask 640 + 10 × 30 = 940. The flask is cheaper, by 560 rupees.
Your go. The first step is given. Plan A charges 30 rupees per kg. Plan B charges 100 rupees to join and then 10 rupees per kg. Find the break-even.
- Set the costs equal: 30x = 100 + 10x.
- Now solve for x, then find the cost there and check it in both plans.
Check the answer
20x = 100, so x = 5 kg. The cost is 30 × 5 = 150 rupees. Check Plan B: 100 + 10 × 5 = 150. Break-even at 5 kg, 150 rupees.
Check before you go on
1. A bicycle repair shop charges a call-out fee of 200 rupees plus 80 rupees for each hour. Which equation gives the cost C for h hours?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A repair shop's cost is C = 80h + 200 rupees for h hours. What is the cost for 3 hours?
3. Using C = 80h + 200, for how many hours does the shop charge 1000 rupees?
Level 2 · Building
4. A phone battery is modelled by P = 100 − 12.5t, where P is the per cent charge and t the hours. After how many hours is the charge 25%?
5. Plan A costs 20 rupees per unit. Plan B costs 120 rupees to join plus 8 rupees per unit. For how many units do the plans cost the same?
6. A rain tank holds y = 500 − 40x litres after x days. After how many days is it empty? Give it as a decimal.
Level 3 · Stretch
7. A water company charges 5 rupees for each of the first 10 units and 8 rupees for each unit after that. What is the cost of 15 units?
8. A shop sold 200 reusable bags in week 2 and 320 in week 6, and sales rose at a steady rate. Predict the number sold in week 12.
Something to think about
A phone plan charges 300 rupees a month and 2 rupees per text. Another charges 100 rupees a month and 4 rupees per text. Which would you choose, and what would you need to know about the person first?
One way to think about it
They cost the same when 300 + 2x = 100 + 4x, so x = 100 texts. Below 100 texts a month the second plan is cheaper; above it the first is. You would need to know how many texts the person sends: the break-even only turns the problem into a question about their habits.
Exit ticket
9. Plan A costs 15 rupees per unit. Plan B costs 90 rupees to join plus 9 rupees per unit. Someone will use 12 units. Which plan is cheaper, and by how much?