compare its allocated shares with the number of heads. If the two figures are mutually prime, multiply the head-count by the root of the problem—including any proportional expansion (ʿawl); the product is the new, sound root. If they are concordant, multiply the co-prime factor (wafq) of the head-count by the root, and the figure that emerges is the correct root.

When the distribution breaks down across two classes, compare the shares of each class with its own head-count. If the two numbers for a class are concordant, reduce that class to its co-prime; otherwise leave it unchanged. Then proceed as follows:

1. If the two head-counts are identical, multiply either of them by the original root with its ʿawl.
2. If one head-count contains the other, multiply the larger of the two.
3. If they are concordant, multiply the co-prime factor of either head-count by the other and then multiply the result by the root.
4. If they are mutually prime, multiply one head-count by the other and then multiply that product by the root. Whatever number results is the valid root.

The same procedure is applied when the break concerns three or four classes, and the breakdown never exceeds four.

When you wish to know the share of each class from the corrected total, take its original entitlement from the root and multiply it by the factor you have just applied to the root; the product is that class’s portion, which you then divide by the number of individuals in the class.

Branch: A man dies leaving heirs, after which one of those heirs dies before the estate is divided. If no one other than the surviving heirs inherits from the second decedent, and their portions from him are identical to what they were to receive from the first estate, treat the second heir as though he had never existed and divide the property among the survivors—such as brothers and sisters, or sons and daughters of whom one dies before distribution.

If the second decedent leaves additional heirs, or if the heirs are the same but their proportions differ, first correct the calculation for the first estate, then that of the second. If the share that fell to the second decedent in the first estate divides evenly across his own calculation, nothing further is required. Otherwise, if the two numbers are concordant, multiply the co-prime factor of the second calculation by the first; if they are not, multiply the entire second calculation by the first. The figure obtained is the root from which both estates are to be divided.

Anyone entitled under the first estate then takes his original share multiplied by the factor applied to that estate, while anyone inheriting under the second estate takes his allocation multiplied by the second decedent’s share in the first estate—or by its co-prime factor, should his calculation or his individual share be concordant with it.