Section
If all the heirs are residuary agnates, the estate is divided equally. When they are exclusively male or exclusively female, each takes the same share; when both sexes are present, every male is counted as the equivalent of two females, and the weighted head-count becomes the root of the problem (aṣl al-masʾala).

When a fixed-share heir appears—whether he has one share or two identical shares—the root is simply the denominator of that fraction: the denominator for one-half is 2, for one-third 3, for one-quarter 4, for one-sixth 6 and for one-eighth 8.

If two fixed shares with different denominators occur, three possibilities arise:

1. Contained (mutadākhil). If one denominator subsumes the other, the root is the larger of the two, as with one-sixth and one-third, whose root is 6.
2. Concordant (mutawāfiq). If they share a common factor but neither contains the other, multiply the co-prime part (wafq) of either by the other; the product is the root. Thus one-sixth and one-eighth give a root of 24.
3. Mutually prime (mutabāyin). If they have no common factor, multiply each denominator by the other; the product is the root, as with one-third and one-quarter, which yield 12.

The fundamental roots are therefore seven: 2, 3, 4, 6, 8, 12 and 24.

Among these, those subject to ʿawl (proportional expansion) are as follows:
• The root 6 may expand to 7 in a case of a husband and two full sisters; to 8 when their mother is added; to 9 when a uterine brother is added; and to 10 when a second uterine brother is included.
• The root 12 may rise to 13 with a wife, mother and two full sisters; to 15 upon adding a uterine brother; and to 17 with a second uterine brother.
• The root 24 may be pushed up to 27 in a case involving two daughters, both parents and a widow.

When the two denominators are identical, no further work is needed. If they differ and the greater is exactly divisible by the lesser two times or more, they are overlapping (mutadākhil), like 3 with 6 or 9. If they can be exhausted only by a third number, they are concordant (mutawāfiq) in the proportion represented by that number—for example, 4 and 6 are concordant through one-half. If no number except 1 exhausts them, they are mutually prime (mutabāyin), as with 3 and 4. Every overlapping pair is necessarily concordant, though the converse does not hold.

Sub-section: Once the root has been calculated, if the shares divide evenly among the heirs, the matter is settled; but if the distribution breaks down for any class