Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: a taxi costs 40 rupees plus 12 rupees for each km. How much is a 5 km ride?
Answer
100 rupees. 40 + 12 × 5. - From lesson 5: what is the horizontal asymptote of y = 5/x + 3?
Answer
y = 3. - From lesson 6: solve 24/x = 6.
Answer
x = 4.
Words used in this lesson
- fixed cost
- a cost that is the same however many are made. For example: the setup of a print job.
- average cost
- the total cost divided by the number of items. For example: total ÷ number.
- floor
- a level that a value stays above. For example: a river that stays above 2 metres all year has a floor of 2 metres.
- mean speed
- total distance divided by total time. For example: 120 km in 5 hours.
- target
- the value you want to reach. For example: an average of 9 rupees.
What happens to the cost of one item when you make lots?
Mehwish in Lahore prints posters for a school fair. The setup costs 1500 rupees and each poster costs 12 rupees to print. The average cost per poster is A = (1500 + 12n) ÷ n rupees for n posters.
Predict first: does the average cost per poster keep falling as n grows, and does it ever fall below 12? Use the figure, with its own costs, to test what you think.
Fixed cost 300, 6 per item, 20 items: total 300 + 6 × 20 = 420 and the average is 420 ÷ 20 = 21.
Show me the steps
- F = 300, v = 6, 20 items: the total is 300 + 6 × 20 = 420 and the average is 420 ÷ 20 = 21.
- 50 items: the total is 300 + 300 = 600 and the average is 600 ÷ 50 = 12. The average has fallen towards 6 but is still above it.
- Raise F to 500 with 50 items: the total is 500 + 300 = 800 and the average is 16. The floor is still 6, because it depends only on v.
Average cost = (F + vn) ÷ n = F/n + v. The fixed cost F is shared among more items as n grows, so F/n gets small. The graph is a reciprocal graph moved up by v, with the horizontal asymptote y = v.
The asymptote is a floor: the average gets as close to v as you like but never reaches it. To find how many items give a target average T, solve F/n + v = T for n.
An average speed is not the mean of two speeds. Driving the same distance at two speeds takes longer at the slow one, so the slow speed counts for more. Divide the total distance by the total time.
Worked examples
1. A print shop has a setup cost of 900 rupees and charges 5 rupees per copy. Find the average cost for 150 copies, and the number of copies for an average below 6 rupees.
- Average = 900/150 + 5 = 6 + 5 = 11 rupees.
- 900/n + 5 < 6 means 900/n < 1, so n > 900: more than 900 copies.
2. Zawadi cycles 60 km at 30 km/h and then 60 km back at 20 km/h. Find her mean speed.
- The times are 60/30 = 2 hours and 60/20 = 3 hours, so the total time is 5 hours.
- Mean speed = 120 ÷ 5 = 24 km/h, not the average 25 of the two speeds.
3. With the same print shop, find the number of copies that gives an average of exactly 8 rupees.
- 900/n + 5 = 8, so 900/n = 3.
- n = 900 ÷ 3 = 300 copies.
Your go. The first step is given. A workshop has a setup cost of 600 shillings and each item costs 3 shillings. Find the average cost for 200 items, and the number of items for an average of 4 shillings. The average is 600/n + 3.
- Put n = 200 into 600/n + 3.
- Solve 600/n + 3 = 4 for n.
Check the answer
600/200 + 3 = 6 shillings. And 600/n = 1 gives n = 600 items.
With a GDC (optional)
A GDC can graph an average cost such as y = (900 + 5x)/x. Use the table of values with large x to see y get close to 5, and the intersect tool with y = 6 to find the number of items for that target. The graph may draw a near-vertical line at x = 0: it is not part of the model, because a number of items cannot be 0 or negative.
The same answers come from solving 900/x + 5 = 6 by hand, and a table only confirms them. Everything on this page can be done without a GDC.
Check before you go on
1. A taxi firm's average cost per trip is C = (3500 + 40n)/n rupees for n trips. For how many trips is the average exactly 40 rupees?
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 bakery has a fixed cost of 1200 naira and each loaf costs 8 naira to make. Find the average cost per loaf for 40 loaves.
3. The average cost per item is (500 + 12n)/n rupees. How many items give an average of 14 rupees?
Level 2 · Building
4. A lorry drives 90 km at 45 km/h and then another 90 km at 30 km/h. What is its mean speed over the whole journey, in km/h?
5. The average cost per item is (2400 + 15n)/n rupees. A firm wants the average to be below 20 rupees. What is the smallest whole number of items it must make?
Level 3 · Stretch
6. The average cost per item is F/n + v. It is 16 shillings for 50 items and 12 shillings for 100 items. Find F.
7. The average cost per ticket is (3000 + 25n)/n rupees. The promoter wants the average to be at most 1.2 times the 25 rupees that each ticket costs to make. What is the smallest number of tickets?
Something to think about
A manager says: “If we just make enough items, the average cost will drop to the cost of making one.” Explain to her what the graph shows, and whether more items always help.
One way to think about it
More items always lower the average, but it can only approach the cost of one item, because the fixed cost is shared and never disappears. Past some point the extra saving is tiny, so other changes, such as lowering the cost of each item, matter more.
Exit ticket
8. A workshop's average cost per item is (1800 + 30n)/n rupees. A buyer asks for an average of exactly 30 rupees, or else exactly 36 rupees. Which reply is right?