Five factors are on the shelf already: a single sum moved forward and back, a series moved forward and back, and capital recovery. The sixth closes the family, and it answers the saver's question rather than the borrower's: what equal deposit, every period, grows to a stated sum by a stated date?

The sinking fund factor: $(A/F) = \dfrac{i}{(1+i)^{n} - 1}$, written (A/F, i, n) in the tables. Read aloud: A given F equals i over, one plus i to the n, minus one. $A$ is the equal deposit at the end of every period in dollars. $F$ is the future sum the fund must hold the moment the last deposit lands, also in dollars. $i$ is the rate the account is credited per period, as a decimal, and $n$ is the number of deposits. The factor itself is a bare fraction per period with no currency on it. Multiply the target by it and you have the deposit: $A = F\,\dfrac{i}{(1+i)^{n} - 1}$, and $A$ is what you solve for.

It is the compound-amount factor from Uniform series turned upside down, and nothing more. That factor took a known deposit and returned the pile. This one takes a known pile and returns the deposit.

Two rails on the value. The factor is always smaller than $1/n$, because the interest does part of the saving. And it is exactly the capital recovery factor minus $i$: $(A/P) = (A/F) + i$. Read that as a sentence about owning anything. The annual cost of an asset is the interest on the money tied up in it, plus a deposit toward replacing it. At 6% over 5 years the tables print 0.2374 and 0.1774, and the gap between them is 0.06 to the last digit.

Salvage belongs in this lesson. If the worn-out unit will fetch $S$ dollars on the day its replacement is bought for $C$ dollars, the fund only has to reach $F = C - S$. Build it to the full price and the ratepayers have been overcharged in every year of the asset's life.

The named mistake is dividing the target by the years. $F/n$ is the deposit for an account that pays nothing. It always over-deposits, and over twenty years at 6% it asks for 84% more than the fund needs. The opposite slip is reaching for capital recovery, which charges the fund the very interest it should be earning.

The six, side by side. (F/P) and (P/F) move one sum. (F/A) and (A/F) trade a series for a future sum. (P/A) and (A/P) trade a series for a present sum. Each pair is a reciprocal, all six are built from $(1+i)^{n}$, and naming the two dates on the cash-flow diagram tells you which one you need before any algebra starts.