In Muslim Law of inheritance which allots a number or fractional parts of unity to various heirs, it may happen that the fractions when added together may sometimes be 1) equal to unity, 2) more than unity, 3) less than unity. When the sum of fractions is equal to unity, there is no problem. But if it is more or less than unity, the shares of respective heirs are reduced or increased respectively. For this the fragments of property is increased or decreased. The process whereby the shares are reduced (by increase of fragments of property) is called the doctrine of increase (Aul); and the process whereby the shares are increased (by decrease of fragments of property) is called the doctrine of return (Radd).
Doctrine of Aul(Increase)
In Islamic Inheritance Law, if it is found on assigning their respective shares to the Sharers that the total of shares exceeds unity, the share of each Sharer is proportionately diminished by reducing the fractional shares to a common denominator, and increasing the denominator so as to make it equal to the sum of the numerators. Means if the prescribed shares of the heirs add upto more than 100% of the inheritance, so in such case, the prescribed shares are reduced proportionally so that it's 100% of the inheritance while the reduction is shared proportionally according to the shares. According to Mulla-
"Increase (aul)-  If it be found on assigning their respective shares to the sharers that the total of the shares exceeds Unity the shares of each shares is proprotionately diminished by reducing the fractional shares to a common denominator and increasing the denominator so as to make it equal to the sum of the numerator."
For example, a deceased person (female) left a husband, mother and daughters. The husband's prescribed share is a quarter (1/4) of the inheritance because the wife (deceased) has left offspring (daughters). The mother's share is a sixth (1/6) because of the daughters also. Finally, the daughters take two-thirds (2/3) of the inheritance. Now if we add up the prescribed shares: 3/12 + 2/12 + 8/12, we have 13/12 which is 108% of the inheritance. So to correct the percentage to be 100% we put reduction to all the fractions in proportion to their shares. So the husband instead of taking 3/12 equivalent to 1/4, he will take 3/13 which is slightly less than a quarter. Likewise the mother instead of taking 2/12 equivalent to 1/6, she will be reduced to 2/13, a slight reduction from 1/6. Finally the daughters will instead of 8/12 be taking 8/13 which is less than their 2/3 share. Notice how the reduction entered all of the shares in proportion to their original shares. This new correction makes the shares 100% of the inheritance.
We can consider that the value of the property is 166 then according to prescribed shares everyone get-
Husband     3/12 * 166 = 39
Mother       2/12 * 166 = 26
Daughters   8/12 * 166 = 104
Total  = 169
In the above case the numerator ‘13’ is greater than denominator ‘12’ that’s why the total goes to 169 instead 166.
Now we increase the denominator to ‘13’so as to make it equal to the sum of the numerators i.e. ‘13’
Husband    3/13 * 166 = 36
Mother      2/13 * 166 = 24
Daughters  8/13 * 166 = 96
Total = 166
In the above case the numerator ‘13’ and denominator ‘13’ are same that’s why the total goes to 166. The prescribed share of every Sharer reduced proportionally so that it's 100% of the inheritance.
ShiaLaw-  Under Shia Law the excess share directly deducted from the share of daughter or full sister. In the above case if the deceased is governed by Shia Law than the shares are as follows-
Husband     3/12
Mother       2/12
Daughters   7/12 not 8/12
Total  = 12/12
In the above case the numerator ‘13’ is greater than denominator ‘12’ that’s why according to Shia Law excess share is directly deducted from daughters share only.

In Islamic Inheritance Law, if a residue left after satisfying the claims of sharers, but there is no Residuary, the residue reverts to the Sharers in proportion to their shares. This right to reverter is technically called ‘Return’ or ‘Radd’. In this case the denominator is reduced so as to make it equal to the sum of the numerators.  So we can see that Radd, is the opposite of aul, meaning that the prescribed shares of the heirs add up to less than 100% of the inheritance, so in such case, the prescribed shares are increased proportionally so that it's 100% of the inheritance while the increase is shared proportionally according to the shares.
For example, a deceased person left a mother and daughter only. The mother's prescribed share is a sixth (1/6) because the deceased left offspring (daughter). The daughter takes half because she's only one and no brother or cousin is with her. So when we add up the prescribed shares: 1/6 + 3/6, we have 4/6 which is only 66.66% (2/3) of the inheritance. So now we have to give the remaining 33.33% back to the mother and daughter because they are the only inheritors alive. Since the mother and daughter take 4 shares and divide between themselves in a ratio 1:3, which will give the mother a quarter (1/4) and daughter three-quarters (3/4). Notice the shares increased proportionally. The mother instead of taking 1/6 will now take 1/4. The daughter instead of taking half (1/2) will now take three-quarters (3/4). Now if we add up shares: 1/4 + 3/4, we have 100% of the inheritance.
We can consider that the value of the property is 60 then according to prescribed shares everyone get-
1/6 * 60 = 10
3/6 * 60 = 30
Total  = 40
In the above case the numerator ‘4’ is less than denominator ‘6’ that’s why the total goes to 40 instead 60.
Now we decrease the denominator to ‘4’so as to make it equal to the sum of the numerators i.e. ‘4’
1/4 * 60 = 15
3/4 * 60 = 45
Total = 60
In the above case the numerator ‘4’ and denominator ‘4’ are same that’s why the total goes to 60. The prescribed share of every Sharer increased proportionally so that it's 100% of the inheritance.
ShiaLaw-  Under the Shia Law, besides husband or widow, in some exceptional cases, the mother and uterine brother and sister also do not participate in return.

Dr Nupur Goel
Assistant Professor
Shri ji institute of legal vocational education and research
( SILVER law collage )
Barkapur Bareilly