Compound Partnership — Capital Multiplied by Time

When partners keep their capital invested for different lengths of time, dividing profit by capital alone would be unfair — a large sum invested for two months has clearly done less earning than a smaller sum left in for the whole year. A compound partnership therefore divides profit in the ratio of each partner's capital multiplied by the time it stayed invested. This product — capital times the number of time-units — is often called the partner's capital-months (when time is measured in months) or the monthly equivalent of capital.

For partners investing capitals $C_1, C_2, \dots$ for times $t_1, t_2, \dots$ respectively, $\text{Profit ratio} = C_1 t_1 : C_2 t_2 : \dots$ Each partner's share is then found from this ratio exactly as in a simple partnership: $\text{Partner's share} = \frac{(C\,t)\text{ of that partner}}{\text{sum of all } C\,t \text{ products}} \times \text{Total profit}.$

Time must be expressed in the same unit for every partner before the products are formed — mixing months for one partner and years for another is the single most common source of error in this topic. It does not matter whether that common unit is months or years, because the ratio depends only on the relative sizes of the products, not their absolute value; months are usually the most convenient choice. …