The arithmetic a marketer carries
Size a marketing bet as reach × conversion × value − cost, know what a click, an email and a demo cost, and work from the course's shared constants.
On this page
One line of arithmetic sizes most marketing bets: reach × conversion × value − cost. It is multiplication, so it takes minutes, and it is honest only when it runs all the way to a paying customer and the margin that customer returns. Every later lesson computes with the constants fixed below, so a click costs $2 wherever a lesson prices one and does not say otherwise, and a reader who disagrees with a figure knows which line to replace before rerunning the sum.
The one line, and the trap in each term
- Reach is a count of people or exposures (impressions, sends, downloads, visits), never a percentage.
- Conversion is a chain of rates, each taking one count to the next, multiplied out to the step that produces money.
- Value is margin, not revenue, over a stated period: the customer's expected lifetime, or the payback window.
- Cost is everything the bet consumes, including people's hours and tools, not only the media invoice.
The habit is the one the system-design course teaches for capacity, where every number is derivable by the reader and every input is said out loud; see back-of-the-envelope estimation. Here the inputs are rates and prices rather than users and bytes.
Worked: $10,000 of paid social
Assume a self-serve product at $50 a month with 80% gross margin, so $40 of margin a month, and 5% monthly churn. A customer stays 1 ÷ 0.05 = 20 months on average and returns $40 × 20 = $800 of expected margin. Run $10,000 through the course's paid-media rates:
spend $10,000 at a $20 CPM -> 500,000 impressions
clicks 500,000 x 1% CTR -> 5,000 clicks $2 each
sign-ups 5,000 x 3% -> 150 sign-ups $67 each
customers 150 x 20% who pay -> 30 customers $333 each
value 30 x $800 -> $24,000 of expected margin
net $24,000 - $10,000 -> $14,000
Stop the chain one step early and the answer looks wonderful: a $67 sign-up against an $800 customer is twelve to one. This is a common error in marketing arithmetic, and ad platforms invite it, because they report the last event their tracking can see, the sign-up, and not the payment that arrives weeks later in your billing system. Run the chain to the customer and the ratio is $800 ÷ $333 = 2.4. Still positive, and a different conversation. The $67 was never wrong; it was the price of a different thing.
The $14,000 is not cash this month, either. The customer returns $40 in month one and a little less each month after, as churn takes its share. Summed, that is 800 × (1 − 0.95ⁿ) by month n, which first reaches $333 at n = 11: the money spent in January comes back in November. Valued at $800, the chain answers whether the bet ever pays. A channel is usually judged on whether it pays inside the course's 12-month target, at 12-month margin, $368 a customer, and on that basis these 30 return $11,040, $1,040 above the spend. CAC, LTV and payback explains why the payback month is the number that decides, and how the LTV/CAC ratio hides it.
The input the answer is most sensitive to
Every rate in the chain multiplies, so a 10% error in any one of them is a 10% error in customers. The net is a subtraction, and the subtraction amplifies it. Halve the CTR to 0.5% and customers fall to 15, value to $12,000, and the net from $14,000 to $2,000: one input halved, the result down 86%. On 12-month margin the same halving turns $1,040 of profit into a $4,480 loss. When a bet's value is close to its cost, the rate you are least sure of decides it, so name that rate and say how you would measure it before spending against it.
The chain is also linear only for the first dollars: a second $10,000 reaches many of the same people again and buys fewer than 30 customers, a saturation paid media basics prices.
Orders of magnitude worth carrying
Three units come up in nearly every marketing conversation: a paid click, an email open and a demo booked. Between them they span almost five orders of magnitude, from a quarter of a cent to $200, and knowing which order a quoted figure belongs to catches most errors before any arithmetic.
| Unit | Order | The course assumes | Working |
|---|---|---|---|
| An impression | cents | $0.02 | $20 CPM ÷ 1,000 |
| A paid click | dollars | $2 | $0.02 ÷ 1% CTR |
| An email open | a fraction of a cent | $0.0025 | $1 per 1,000 sends ÷ 40% opened |
| A sign-up from paid | tens of dollars | $67 | $2 ÷ 3% |
| A demo booked | hundreds of dollars | $200 | $2 ÷ 1% of visitors who book |
| A deal closed by a rep | thousands of dollars | $6,000 | $1,000 of media plus $5,000 of rep time |
A paid click. Assume $0.50 to $5 depending on the channel and the intent behind the click, and $2 when a lesson needs one figure. Some business-software search terms cost many times that, so price your own before borrowing ours. Module 2's invoicing app buys broad traffic at the bottom of the range, $0.50 a visit, and it converts worse: 2.4% of its sign-ups pay, so its paid CAC is $521, not this chain's $333.
An email open. Sending is nearly free: assume $1 per 1,000 sends, so an open costs a quarter of a cent. A send to 20,000 free users at 40% open, 5% of opens clicking and 10% of clicks upgrading is 20,000 × 0.4 × 0.05 × 0.1 = 40 upgrades, for $20 of sending and an hour of writing at $100: $3 an upgrade. An upgrade is a payment, so the paid figure to compare is the $333 customer, not the $67 sign-up. The cost is not the constraint; the list is. There is no second list of people who already know you to buy at that rate, and every send spends some of the list's attention in unsubscribes. Count clicks rather than opens, too, because mail clients that load images on delivery record opens nobody made. Lifecycle and CRM works the same send.
A demo booked. If 1% of paid visitors book a demo, each demo costs $2 ÷ 1% = $200 in media, and at a 20% close rate a customer costs $1,000 before anyone has spoken to them. Then the rep: assume $150,000 a year loaded, closing 30 deals a year, so $5,000 of rep time a deal and about $6,000 all in. A $1,200-a-year contract returns $960 of margin a year against that $6,000, more than six years to pay back before churn, which is most of the case PLG versus sales-led makes.
The course's shared constants
Every lesson uses these, and every exercise restates the ones it relies on in its situation. They are assumptions, chosen to be plausible and easy to multiply, not measurements of any market. Replace any of them with your own figure and the arithmetic survives.
| Name | Assumption | Used in |
|---|---|---|
| currency | US dollars | every module |
| a month | 30 days; 22 working days of 8 hours, so 176 working hours, 2,112 a year | every monthly figure |
| gross margin | 80% of revenue, for a software product | every value |
| the default customer | $50 a month, so $40 of margin; 5% monthly churn, so a 20-month average life and $800 of expected margin; module 2 names it the invoicing app | modules 0, 2, 3, 4, 5, 6, 7 |
| payback target | 12 months on expected margin, so a CAC ceiling of 800 × (1 − 0.95¹²) ≈ $368 at the default customer | modules 0, 2, 3, 4, 7 |
| paid media | $20 CPM and 1% CTR, so $2 a click; 3% of clicks sign up, so $67 a sign-up; 20% of sign-ups pay, so $333 a customer | modules 0, 2, 3, 4, 5, 6, 7 |
| $1 per 1,000 sends; 40% open, 5% of opens click, 10% of clicks convert | modules 0, 3, 5 | |
| demos | 1% of paid visitors book; 20% of demos close | modules 0, 2 |
| a sales rep | $150,000 a year loaded, 30 closed deals a year, so $5,000 a deal; $75 an hour over about 2,000 hours, the 2,112 less about 14 days of leave | modules 0, 2, 5, 6 |
| a marketing hour | $100 loaded: a founder's hour, or a senior marketer's | modules 0, 1, 3, 4, 5, 6, 7 |
| a written post | $400, four of those hours | modules 0, 3, 5, 7 |
| a generated post | under $1 of model cost, plus a $50 review of 30 minutes | modules 0, 3, 5, 7 |
| a test | two-sided α = 0.05 and 80% power | modules 0, 4, 5, 6 |
The test settings have one consequence worth carrying now: detecting a 10% relative lift on a 2% conversion rate, 2.0% to 2.2%, takes about 81,000 visitors in each arm. Most marketing tests are far smaller than that, which is why so many winners fail to repeat; holdouts, geo tests and power works the formula. How to cost one model call is in where the money goes; at the illustrative $3 and $15 per million input and output tokens in the four numbers, a 1,500-word post from a 3,000-token brief costs about $0.04, so under a dollar holds across several drafts.
Converting to another currency. The figures are in US dollars because the benchmarks the course draws on are published in them. To convert, replace every figure a local market sets with your market's own: a CPM is set by a local auction and a salary by a local labour market, so neither a $20 CPM nor the $100 hour converted at the exchange rate is the figure in your market. Convert only what you actually pay in dollars, such as model and software costs, once, at one exchange rate, before the sum. A comparison survives a slightly wrong rate, but not two different ones.
In practice
When a media seller or an agency quotes a price, it is almost always for the first step of the chain: a CPM, a cost per click, a cost per lead. Ask for the next step down, then the one after: cost per lead, cost per qualified lead, cost per paying customer. Ask what the second $10,000 buys at the same targeting. A seller who can answer only the first question is selling reach, which is fine, but then the conversion rates are yours to estimate and yours to be wrong about.
With the team, write the chain as one line with every rate labelled as given or assumed. A disagreement then lands on a rate ("I don't believe 20% pay") rather than on the conclusion, and it can be settled by measuring that rate.
The mistake that costs money is comparing a cost per sign-up with a customer's lifetime value. $67 against $800 says twelve to one; $333 against $800 says 2.4; and at 2.4, a halved click-through rate leaves $2,000 of lifetime margin above the spend and a loss inside the payback window.
Check yourself
1. A newsletter sponsor quotes $3,000 for one send to 60,000 subscribers and reports a 40% open rate with 5% of opens clicking. Using the course's rates from the click onwards, what does a customer cost, and do you buy the send?
60,000 × 40% × 5% = 1,200 clicks, $2.50 each. Then 1,200 × 3% × 20% = 7.2 customers, so $3,000 ÷ 7.2 ≈ $417 a customer, above the $368 ceiling; 800 × (1 − 0.95ⁿ) first reaches $417 at n = 15. Not at $3,000. Offer about 7.2 × $368 ≈ $2,650, and ask for the measured click counts of the sponsor's last three sends, since the open rate includes opens nobody made.
2. Your rep-assisted CAC is $6,000 and your contract is $1,200 a year at 80% margin. Which input do you change first, and what does it have to reach for a 12-month payback?
Payback is $6,000 ÷ $960 a year, more than six years before churn. The rep's $5,000 is five-sixths of the cost, so it is the input to move. Either the contract grows until a year's margin covers $6,000, which on an annual contract paid up front is $6,000 ÷ 0.8 = $7,500 a year, or the rep leaves the motion. Without a rep, self-serve costs $333 a customer on this lesson's chain or $521 on module 2's funnel, which $80 of margin a month at 5% churn pays back in month 5 or month 8. At $1,200 a year, drop the rep.