A private firm asks whether a project makes money. A public body cannot. A culvert, a noise wall or a flood berm earns nothing, and its payoff lands on people who never see the invoice. Public works are judged instead by setting everything the public gains against everything the public pays, both at the same date.

The benefit–cost ratio: $B/C = \dfrac{B}{C}$. Read aloud: the benefit–cost ratio equals the present worth of benefits over the present worth of costs. $B$ is the present worth of all benefits in dollars: travel time saved, flood damage avoided, collisions prevented, each one priced and discounted to today. $C$ is the present worth of all costs in dollars: construction at time zero, plus every future year of operation and maintenance discounted the same way. The ratio is a bare number, and it is what you solve for. Above 1 the project is justified, because each dollar of cost buys more than a dollar of benefit. Below 1 it is not, however large the benefits look on their own.

Nearly all the work is getting both sides to time zero, and that is machinery you already own. A benefit of the same size every year is a uniform series, so $B = A_b\,\dfrac{1 - (1+i)^{-n}}{i}$, where $A_b$ is the annual benefit in dollars per year, $i$ the discount rate as a decimal and $n$ the study period in years. An annual running cost comes back through the same factor. A first cost is already at time zero and takes no factor at all.

Read backwards, the relation sets a budget: $C = \dfrac{B}{(B/C)}$. If the benefits are worth $2,400,000 and policy demands a ratio of 1.6, the project may cost $1,500,000 in present worth and not a dollar more.

One convention has to be stated every time. The conventional ratio puts every cost under the line. The modified ratio subtracts the annual running cost from the benefits instead, and leaves only the capital below. The two return different numbers for the same project. They always agree about which side of 1 it falls on, but they can rank two projects differently, so a ratio quoted without its convention is half a number. This lesson uses the conventional form throughout.

The named mistake is judging the benefits by their size. Twenty years of $120,000 is $2,400,000 of benefit in the newspaper, and about $1,178,000 at 8% on the evaluation sheet. Undiscounted totals flatter every long-lived project, because the big cost lands first and the benefits arrive late. The second slip is the ratio upside down: cost over benefit passes every bad project and fails every good one.

And the honest limit, in the spirit of the units doctrine. A ratio above 1 says a project is worth doing. It never says the project is the best thing to do. A small scheme at 2.5 can deliver less net benefit than a large one at 1.4, so choosing between alternatives takes an incremental analysis, not a beauty contest between ratios.