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What a discounted sale contributes, the gain from new customers, what an unfenced or leaky discount gives away on existing sales, and the net effect on the month.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will work out what a discounted sale contributes, count a discount's gain from new customers and what it gives away on sales that were happening anyway, set the two against each other, and say how fencing the discount, a leak through the fence and spare capacity change the answer.
You know a price cut comes entirely out of contribution, and that extra customers are only worth something if there is capacity to serve them. A discount is a price cut with a boundary: the question is who is inside it. This lesson counts a discount's two effects in dollars — what the new sales add, and what is given away on sales that would have happened anyway — and then looks at the design choices that decide which effect wins: the fence, how well it holds, and the spare room behind it.
| Term | What it means |
|---|---|
| Gain | What the new sales a discount brings contribute. |
| Giveaway | The discount paid out on sales that would have happened anyway. |
| Fenced discount | One that reaches only some sales — new customers, quiet hours — so its giveaway is small. |
| Leak | Customers outside the fence who use the discount anyway. |
| Net effect | The gain less the giveaway: what the discount does to the month. |
| Spare capacity | The units the business could still make and serve. |
Corner Bean sells 3000 flat whites a month at 4 dollars; each costs 2. The café has room for 1200 more. Alexa is considering 25 percent off, which she expects to bring 800 new sales.
The gain. At 3 dollars each new cup contributes 3 − 2 = 1. Eight hundred of them add 800 a month.
The giveaway. If the discount is for everyone, each of the 3000 cups that were selling anyway now brings in a dollar less: 3000 given away.
$$\text{net} = \text{new sales} \times \text{discounted contribution} - \text{existing sales} \times \text{discount in dollars}$$
Here the net is 800 − 3000 = −2200 a month. The café would be busier, the baristas would work harder, and the month would end 2200 worse. Offered only to new customers, the same 25 percent adds 800 and gives away nothing.
Notice the halving hidden in the gain: a quarter off the price took half the contribution of each discounted cup, because the whole dollar came out of the 2 of contribution and none out of the 2 of cost.
Another way: steps
Another way: table
Four discounts, each offered to everyone.
| Offer | Gain from new sales | Given away | Net |
|---|---|---|---|
| Flat whites, 25 percent off | 800 | 3000 | −2200 |
| Screens, 10 percent off | 960 | 800 | 160 |
| Lunch plates, 10 percent off | 1260 | 960 | 300 |
| Veg boxes, 20 percent off | 255 | 576 | −321 |
A small discount with a big response can pay; a big one rarely does.
Write the offer exactly. How much off, for whom, when, and for how long. Ten percent off is not an offer until it says to whom.
Estimate the new sales honestly. New customers, not regulars moving from full price. The estimate is the weakest figure in the test, so it is worth asking what it would take for the discount to pay, and whether that many new customers is believable.
Work the gain and cap it. New sales times the discounted contribution, but never more than spare capacity times the same.
Work the giveaway. The existing sales the discount reaches — all of them for an open offer, the leak for a fenced one — times the discount in dollars.
Subtract, and decide. A positive net pays on these figures; a negative one does not, unless there is a named reason, such as bringing customers who will return at full price.
Check the answer by working the month twice, with and without the offer: total contribution at full price, then total contribution with the new sales and the discounted existing ones. The difference must equal the net effect. If it does not, a sale has been counted at the wrong price.
The giveaway is the part a design can shrink. A first-visit offer, an off-peak price, a code for one catering client: each keeps the discount away from sales that were happening at full price. The fence is never perfect — some regulars will find the code — but a leaky fence still beats none.
The gain has a ceiling too: spare capacity. Fixit Mobile has room for 40 more screens a month. A discount that would bring 60 new customers adds at most 40 sales; the other 20 wait, or leave annoyed. And a discount offered when the bench is already full adds no sales at all. It only gives money away.
Turning the test round gives the number a discount must bring just to stand still:
$$\text{new sales needed} = \frac{\text{giveaway}}{\text{discounted contribution}}$$
For Corner Bean's 25 percent off for everyone, the giveaway is 3000 and each new cup contributes 1, so the café needs 3000 new sales a month — more than its spare capacity of 1200 — just to break even on the offer. That single number usually settles it faster than any forecast of how popular the offer will be.
For a fenced offer, the giveaway is only the leak, and the number needed is small. That is the whole case for fencing: it turns an offer that needs an impossible response into one that pays with a modest one.
Businesses that sell to other businesses meet two more kinds of discount. A volume discount — a lower price for a larger order — is tested the same way: the gain is the extra units the customer orders because of it, and the giveaway is the discount on the units they would have ordered anyway. An early-payment discount — two percent off for paying in ten days — is not about volume at all. Its gain is cash arriving sooner, which the cash-cycle lessons value; its giveaway is two percent of every invoice paid early. It is worth offering when the business needs the cash sooner more than it needs the two percent.
Most discounts are judged on the month they run, and most of their real effect comes after. Three things can happen when the offer ends, and the test should say which is expected.
New customers stay at full price. Then the offer was a way of finding them, and its cost — the discounted contribution given up in the offer period, plus any giveaway — should be set against what those customers contribute over the months they stay. This is the case that makes most introductory offers worth running.
New customers leave when the price returns. Then the offer bought a busy month and nothing else, and the test of this lesson is the whole story: gain less giveaway, for the offer period only.
Regulars expect the discount to come back. An offer that runs too often teaches customers to wait for it, and full-price sales fall in the weeks between. That is a giveaway spread over the whole year, and it is the reason many businesses run offers rarely, briefly, and only to people who have not bought before.
A simple habit catches all three: count how many of the offer's customers are still buying at full price a month after it ends, and whether regulars' full-price sales dipped in the weeks before and after it. Those two figures say whether the offer was worth what it cost.
A small gym has 400 members paying 40 dollars a month. Each member costs about 4 a month in towels, cleaning and the booking app, so each contributes 36. The gym could take 150 more members before the classes are full. For January the owner considers half price for three months.
Offered to everyone, the giveaway is 400 × 20 = 8,000 a month for three months: 24,000. Each new member at 20 contributes 16. To stand still the gym would need 8,000 ÷ 16 = 500 new members — far more than its 150 places. The open offer cannot pay.
Fenced to new members, it gives nothing away on the 400. If it brings 100 new members, they add 100 × 16 = 1,600 a month for the three months. A leak — say 20 existing members who cancel and rejoin as new — gives away 20 × 20 = 400 a month, leaving 1,200.
The real test is what happens in April, when the offer ends. If 60 of the 100 stay at full price, each contributes 36 a month from then on: 2,160 a month. The offer was a way of finding members, and the owner writes down the three things it must achieve — 100 joiners, a leak of no more than 20, and 60 staying — and checks each at the end of April.
Airlines, hotels and cinemas fence discounts carefully: cheap seats that must be booked weeks ahead, off-peak prices, fares that require a Saturday night away. Each fence keeps the discount from reaching customers who would pay full price — business travelers, peak-hour audiences — while filling capacity that would otherwise be empty. It is this lesson's arithmetic applied thousands of times a day.
Any sale is better than no sale. A discounted sale that takes the discount off ten full-price sales is worse than none.
A 10 percent discount needs 10 percent more customers. It needs enough to cover the giveaway at a reduced contribution, usually far more.
The gain is the new revenue. It is the new contribution; revenue includes the cost of making what was sold.
Discounting fills a busy business. It only fills spare capacity.
A fence means no giveaway. Some regulars always get through; count the leak.
The offer month is the whole story. What new customers do after the price returns, and whether regulars learn to wait for the next offer, usually matter more than the month itself. Count both a month after the offer ends, before running another, and write the two figures beside the offer's own test.
A discount that fills the room is a success. Only if the contribution it brings exceeds what it gives away to customers who would have come anyway.
Find what a discounted screen contributes: 80 less 8, less 48.
$72 - 48 = 24$
The discount comes out of the 32 of contribution.
Find the gain from 40 new screens.
$40 \times 24 = 960$
All 40 spare slots filled.
Find the giveaway if everyone gets it.
$100 \times 8 = 800$
The regular screens each lose 8.
Subtract the giveaway from the gain.
$960 - 800 = 160$
Positive, but only because every spare slot fills.
Work the fenced version.
$960 - 0 = 960$
New customers only: the whole gain is kept.
Find what a discounted cup contributes: 4 less 1, less 2.
$3 - 2 = 1$
Half the usual 2.
Find the gain from 800 new customers.
$800 \times 1 = 800$
New sales at the lower contribution.
Find the giveaway if it were for everyone.
$3000 \times 1 = 3000$
Every regular cup.
Count the regulars who find a new-customer code: 10 percent.
$3000 \times 0.10 = 300$
The leak.
Find what the leak gives away.
$300 \times 1 = 300$
Regulars who would have paid full price.
Subtract the giveaway from the gain.
$800 - 300 = 500$
The fenced offer still pays, by 500 rather than 800.
Find a box's discount at 20 percent of 24.
$24 \times 0.20 = 4.80$
The discount in dollars.
Find a discounted box's contribution.
$24 - 4.80 - 9 = 10.20$
Down from 15.
Find the gain from 25 new boxes.
$25 \times 10.20 = 255$
Within the 30 spare.
Find the giveaway on the 120 regular boxes.
$120 \times 4.80 = 576$
If everyone gets it.
Subtract the giveaway from the gain.
$255 - 576 = -321$
The open offer loses 321 a month.
Find the new boxes needed to stand still.
$576 \div 10.20 \approx 57$
Nearly double the spare capacity.
Redesign the offer.
$\text{first box only: } +255$
Fenced to new customers, it adds 255 a month.
Find what a discounted sale contributes.
$40 - 30 = 10$
Discount the price, then take off the cost.
Find the gain and the giveaway if everyone gets it.
$45 \times 10 = 450; \quad 60 \times 10 = 600$
Both effects in dollars.
Subtract the giveaway from the gain.
Fixit Mobile sells battery swaps at $50$ dollars, each costing $30$ to make. It is considering $20$ percent off. Complete the sentence in dollars.
At the discount, each one sells for p dollars and contributes c dollars.
Complete the worked solution: a shop sells $24$ items a month at $17$ dollars, each costing $9$. A discount of $3$ dollars for everyone would bring $33$ new sales. What does it do to the month's contribution?
Find what a discounted sale contributes.
$17 - 3 - 9 =$ d
The whole discount comes out of contribution.
Find the gain from the new sales.
$33 \times (\text{discounted contribution}) =$ g
New sales at the lower contribution.
Find the giveaway on existing sales.
$24 \times 3 =$ w
Every sale that was happening anyway loses the discount.
Take the giveaway from the gain.
$(\text{gain}) - (\text{giveaway}) =$ n
The discount pays only if this is above zero.
Say what fencing would do.
$\text{new customers only} \Rightarrow \text{no giveaway}$
The gain alone would remain.
Fixit Mobile sells $60$ battery swaps a month at $50$ dollars, each costing $30$ to make, and has room for $50$ more. It is considering $20$ percent off, which it expects to bring $45$ new sales a month. If the discount is offered only to new customers, how many dollars a month does it add to contribution?
Answer:
Corner Bean sells $1200$ lunch plates a month at $8$ dollars, each costing $3$ to make, and has room for $300$ more. It is considering $10$ percent off, which it expects to bring $300$ new sales a month. This time the discount is for everyone. Fill in its two effects on the month's contribution and the net change, in dollars. Write a fall with a minus sign.
| Dollars | |
|---|---|
| Added by the new sales | |
| Given away on sales already happening | |
| Net change in the month's contribution |
Corner Bean sells $3000$ flat whites a month at $4$ dollars, each costing $2$ to make, and has room for $1200$ more. It is considering $25$ percent off, which it expects to bring $800$ new sales a month. Offered to everyone, does the discount raise the month's contribution?
A florist offers $1$ dollars off a first order, where each bouquet normally contributes $8$ dollars. The offer brings $51$ new customers a month. But $20$ percent of her $230$ regular customers a month also find the code and use it. Fill in what the offer does to the month's contribution, in dollars.
| Amount | |
|---|---|
| Gain from new customers | |
| Regulars who use the code | |
| Given away through the leak | |
| Net change in the month's contribution |
A hair salon's cut contributes $38$ dollars after products, fees and the stylist's hourly pay, and it does $283$ cuts a month with room for more. An offer of $10$ dollars off would bring $18$ new clients a month. Fill in what the offer does if it is for first visits only, and if it is for everyone.
| Amount | |
|---|---|
| Contribution of a discounted cut, dollars | |
| Net change, first visits only | |
| Given away on regulars, if for everyone | |
| Net change, if for everyone |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Fixit Mobile sells battery swaps at $50$ dollars, each costing $30$ to make, and has room for $50$ more a month. A discount of $20$ percent is offered to new customers only. However many new customers come, what is the most it can add to the month's contribution, in dollars?
Answer:
You can test a discount in dollars on both sides, including a fence that leaks. Tell someone why a busier month can end with less contribution. Next: the tax a business collects on its sales and hands on.
16. Your turn: price 50, cost 30, 60 sold, 20 percent off, 45 new sales, step 3
$450 - 600 = -150$
Fence it, or do not run it.