Business studies
Direct and inverse proportional allocation
Enter the amount to divide and the quotas: you get each share, the unit value and the check that the parts add back to the total. An inverse split gives more to the smaller quota.
When the split is direct and when it is inverse
A direct split divides an amount into parts proportional to given quotas: the larger the quota, the larger the share. It covers profits split by capital contributed, a bonus split by days worked, service charges split by floor area. The mechanism is always the same: add the quotas, divide the amount by that sum to get the unit value, and multiply the unit value by each quota.
An inverse split reverses the relationship: the larger quota takes the smaller share, because the parts are proportional to the reciprocals of the quotas. It feels backwards until you look at a concrete case. If three departments split a maintenance cost by number of breakdowns, the split is direct. If they split it by distance from the plant instead, it is inverse: the ones further away use the plant less and pay less.
The check is the same every time and should be done every time: the shares must add back to the original amount. If they do not, either rounding is to blame — and the gap is a few cents — or a quota was read wrongly.
Common mistakes
- Using a direct split where an inverse one is needed: the question to ask is whether the largest number should receive more or less. If it should receive less, the split is inverse.
- Dividing the amount by the number of quotas instead of by their sum: that is an equal split, not a proportional one.
- Rounding each share separately without checking the total: with three or more quotas the rounding accumulates and the sum can drift by a few cents, which have to be assigned to one of the shares.
Frequently asked questions
How do you do a direct proportional split?
Add the quotas, divide the total amount by that sum to get the unit value, and multiply the unit value by each quota. With 6,000 and quotas 2, 3 and 5 the unit value is 600 and the shares are 1,200, 1,800 and 3,000.
How do you do an inverse split?
Replace each quota with its reciprocal and proceed as in a direct split. That way the largest share goes to the smallest quota.
When is an inverse split used?
When the quantity being split on is inversely related to the benefit or the responsibility: distance, time elapsed, seniority in some insurance contexts. The practical test is whether a larger number should mean a smaller share.
What is the unit value?
What a single unit of quota is worth: the total amount divided by the sum of the coefficients. Multiplied by each coefficient it returns the matching share, and it is the number that makes checking the arithmetic immediate.
How this calculation works
Direct split of an amount T over quotas q₁, q₂, … qₙ: unit value u = T / (q₁ + q₂ + … + qₙ); share i is Sᵢ = u · qᵢ. Inverse split: replace each quota with its reciprocal, so u = T / (1/q₁ + 1/q₂ + … + 1/qₙ) and Sᵢ = u / qᵢ. In both cases the check is ΣSᵢ = T. An inverse split requires non-zero quotas, since zero has no reciprocal. A compound split, not covered here, multiplies the quotas of two or more criteria together before applying the direct formula.