1 Morrisons have an offer in the alcohol section, “Six bottles 25% off”. You choose five wine bottles and make the sixth some whisky. The till receipt shows no discount and on the way out you go to check with Customer Services why this is. They point to the picture with the slogan, where ‘bottles’ visibly means wine and champagne but not spirits, so the whisky didn’t count as the sixth bottle. You agree that 25% off applies only to the bottles in the offer. So you ask to go choose a sixth ‘wine’ so as to get the discount. The till receipt shows you chose £7.99, £6.99, £2.79, £3.59, £5.49 and your sixth choice was at £3.99. Who owes who what money?

There is a sense in which you have already paid for the sixth bottle. If you could do the arithmetic fast enough, what could you chose and have no more to pay?

2 You’re wondering which is cheaper of two cereals, the own brand and the famous brand [think Tesco and Kellogg’s, okay?]. The helpful card that shows price per 100g is too blurred to read. The own brand is 85p for what looks to be 800g and the famous one is 65p for 375g. So which is cheaper?

The idea of ‘buy one get one free’ persuades you as customer to buy more product, which is a good thing if this is something you use regularly. From the vendor’s viewpoint it is a good way to move stock, especially if you know there is a change in the market soon (such as the launch of a competing product, shop competition, change in the product). As the retailer you are interested in making some profit and maybe will accept making less. Presumably the cost (to the shop) is less than the three-for-two or two-for-one price (it could be that this is a deliberate loss, called a loss-leader, to entice you into (staying or) shopping. Let us assume no loss is incurred, so we then assume that the minimum price is actually cost (no profit at all, no recovery of the cost of running the business).

3 Let cost of a product be C and the sale price be P and let ‘the deal’ be that you buy N and get one free. Write a formula for the price per unit within the deal as described. Express the % gross profit on a single item sold at the usual price and simplify. [Hint: my result has no P or C in it.]

4 Some Maluka honey from NewZealand was offered in Holland & Barrett as 3 for 2, when a single pot was £39.99 for 500g. Write down the price per kilo on buying any number of pots from 1 to 7. Write a general formula when buying N pots. Find something else priced at £80/kilo and share the information.

5 Our local BodyShop outlet offers a different sort of deal. As you put more products into your basket, so the discount rises. I think the deal is 20% off at three items, 30% at 4 items and 40% off at five items. If all items were £10, write the effective price per item for buying up to six items.

If you run your own shop (imagine, please) then you have bought N copies of an item at a cost, C so C/N is your cost per item. You sell these items at an initial price PN which gains additional discount as the remaining stock shrinks. Eventually you sell the last item at P1 If P1 is close to PN then you will probably have ordered / bought some more. Your gross profit is the difference between cost and sales i.e. it is C-ΣPi. When we are introduced to this idea it is assumed that all Pi are the same, but patently it is not true: think of travel fares or anything that has time-based value, such as newspapers, hotel rooms and this year’s car. So the calculation of profit requires good record-keeping. Price, in some businesses is, at least initially, a fixed multiplier above cost, e.g. cost *2.5, which means cost is 40% of price. A lot of labour is charged at this factor, where the extra money goes into office costs, management costs, labour overheads and so on. £20/hour charged often means £8/hour to the worker. However, when selling there are other considerations, such as having any work rather than no work, having the shelf space available, having any income rather than none. This means that there is what we might call a marginal value on the ‘next’ item for sale. Here are some examples:

6 Your run a(n) hotel with more than a hundred rooms. At 40% occupancy you have covered your costs (you think); at 70% occupancy you need extra staff and if you’re honest at 90% you have problems and may well be losing future repeat business.

(a) You have a customer asking about using your hotel for a conference. At this moment in the negotiations the customer is talking about 60% occupancy. Consider what sort of offer you might make, bearing in mind that your usual occupancy at this time of year is 30% when you make no special deals.

(b) The conference organiser discovers that 30% occupancy is more likely. Your cost per night per room is £60 – there is very little extra cost to you on the grander rooms. Your rates vary from £80 to £150. What is your thinking now?

7 Your airline (you want to be a manager) sells tickets for £200 full price and you offer the following deals: at three or more months advance purchase, 20% off; under three months full price; in the last week you’’re prepared to do some deals. The flight cannot be cancelled unless it is cancelled in both directions, there and back, or wherever it goes. At 60% occupancy you’re happily making money and at 100% you have few additional costs, though you actually prefer 95%, which allows you to provide extra services for people prepared to pay extra for a flight NOW. So your thinking is that you’ll sell space cheaply in the last 24 hours but not in the last hour (of time in which the purchaser can actually make / take the flight). Can you do some research to find what the airlines do?

8 Your building business employs a small number of full-time employees and you employ sub-contractors such as brick-layers, plasterers, plumbers and electricians as needed. You have work right now but none for next month. You have just one enquiry for work, which you have realised will cost you C to do, where C includes labour, plant, materials and overheads. Your employee costs are £E per week and the work would take a week if you used only your full-time staff and worked on nothing else. You are wondering what price, £P to offer. Your overheads are £E/2 (yourself, the office staff, the office). Let’s assume that if you make the price attractive enough then no others will be invited to bid for the work. Write down some conditions (inequalities) to be satisfied, explaining what you think matters.

DJS 20150328

I looked for images illustrating “3 for the price of 2” and found an iPad3 selling for the price of an iPad2.

1 The five bottles come to £26.85 charged. The six bottles total £30.84, less 25% is £23.13, so there’s no charge for the sixth bottle and you’re owed £3.72.

0.25(£26.85+B)=B => B=(0.25x£26.85)/0.75 = £26.85/3 = £8.95

You might view this as a targeted problem. In general, adding the last bottle for no additional charge would give 0.75(T + X) = T would have no additional charge, so X=P/3 suggests X=£8.95. Check; new total for 6 bottles is £26.85+£8.95= £35.80, 75% of which is £26.85.

25% off a collection means that, if you've added up the prices of all but the last, then there is no charge for that last bottle if it is a third of the running total. Once you see the 25% connect with the third, this could be quite a quick piece of arithmetic. Or you could spend less; do you really need to drink alcohol?

2 Working in pence per 100 grams: 85/8 = 10.6 but 65/3.75 is 17.3, much more expensive. This is obvious, since 800g is more than twice 375g, while 85p is nowhere near twice 65p. Giving the answer should be very quick; it is explaining why that takes longer. School maths demands shown working; real life demands only that you're right. This means that a checking routine becomes more important.

3 Pay for N, walk off with N+1, so NP=(N+1)C, yes? Gross profit is (P-C)/C as a %, which I think gives us 1/N as profit, or 100/N %. Interesting how the algebra all cancels out.

4 Rounding up to the nearest penny the price per kilo for 1 to 5 is {79.98, the same but silly because the next pot is free, 53.32, 59.99, 63.94 (silly), 53.32, 57.13} Formula is P [N-int(N/3)/(N/2)] = 2P - (2Pint(N/3)/N.

Check when N=19 formula gives P*(19-int(19/3))/(19/2) = P*13*2/19 = £54.72 and at N=20 formula gives P*(20-int(20/3))/(20/2) = P*14/10=£55.99 (but silly, as the 21st pot would be free). There should be an interesting conversation about other materials at £80 /kg.

5 Pairings (number of items, price per item): (1,£10), (2,£10), (3,£8), (4,£7), (5,£6), (6,£6). This, surely, makes you wonder about the shop mark-up and its margins. Perhaps Head office is most interested in moving product, so that £6 per item represents a fair margin but they want to effectively penalise you for buying less?

6a Potentially 90% occupancy (30% usual + 60% conference) is into the problem zone, so you will be needing extra staff if you can close the deal. Your usual custom is at risk, so you need to do a deal with other local hotels for surplus, you need to keep the conference attendees away from your regulars and maybe offer the regulars a significant discount for the disruption – or an ‘upgrade’ at no charge. What you really want is for the conference to need only 40-50% occupancy. So your pricing represents balance between the work you have to do and the gain you seek to make. You want a sizeable deposit to cover booking the extra staff and you want a deal that reflects where your problems are – or one that gives you enough funds to solve any problem. Overflow is a big problem if the local competition is below your standard. You would consider setting an upper limit on conference numbers but it might be best to attend to the non-conference customer problem first; I’d suggest that once the conference deal is sealed the non-conference customer is given a heavy discount with a warning and help finding alternatives (a change of date for the same discount or a change of venue with a subsidy / upgrade offer).

6b 30% regular business plus 30% conference for no extra work is perfect; your staff is busy but not overloaded. This is a 20% margin of gain. I’d suggest (again) a discount / upgrade / separate floor to the non-conference customers and something similar to the conference, with conditions based on volume. At 60% total occupancy you can space and separate the conferees from the other customers. You might well make an offer that is very attractive at 25-40% occupancy and a concern is to cover your costs sufficiently well if the conference fails to attract as much as 25% occupancy.

7 I found and have experienced a variety of ideas. There is a time period before the flight in which tickets are cheap but it may well be in the zone after costs are covered and before crowding is recognised, say between 80 and 95% occupancy, when every extra sale represents rather little extra cost (extra fuel & food, very little extra staffing or occupancy cost). At close to 100% sale there are other issues to do with over-booking (a general assumption that 4% of customers miss their flight encourages 104% sale of spaces) and issues of selling the ‘last’ space, including providing upgrades to a few customers so as to share out the staffing load (or to even the weight distribution in the plane, perhaps?). See Flight Experience, essay 61.

8 Your costs every week are 3E/2 plus any materials and plant used. If the contract is not agreed your costs are 3E/2. So your optimum price for P is somewhere bigger than C, but any income is better than none. No income means paying 3E/2 and doing maintenance tasks in the builder’s yard, which is much the same as employing yourself. The equivalent of no income (but having work) is when P=C-3E/2. So P > C-3E/2, just a little more than the materials and plant, expresses that having work is better than not having work, while P>C makes genuine profit.

An almost immediate consequence of this pattern of working is that the specialists (e.g. the plumber) have to charge you more than if they were employees, because they need their total pay to average out to a reasonable sum – they don't have guaranteed work. In effect, you have to pay for them to take the risk of there being less work about. It is worse for a specialist who has to work outside, for there are days when work is not possible (mortar fails when close to freezing, for example). Quite a lot of days are lost to bad weather in the winter, so there is value in having indoor work available.

I’ve seen this happen, being the client involved. Prices quoted went from a minimum of 0.5C to a maximum of 2.5C. I called the company in with the minimum and persuaded them to work for nearer C. They still went out of business while working for me, but only because the boss died. I then employed the workers directly, which meant everyone was less upset. It was certainly weird having a conversation with the boss where I was trying to push his price upwards to where I thought he'd survive!