In the Muslim law of inheritance has a lot 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, or
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. The process whereby the shares are reduced is called the Doctrine of AUL (Increase) and the process whereby the shares are increased is called the Doctrine of Radd (Return).

The Doctrine of Radd

When in the case the property is left after distribution or we can say when the party of unity divided and the sum is less than unity. After distributing the property of the deceased amongst the Sharers, but there is no heir of the deceased falling under the category of Residuary, the residue of the property reverts back to the Sharers in the proportion of their shares. The Right of the Sharers of revert back of the Residue property to the Sharers in absence of the residuary heirs is called as the Return or Doctrine of Radd.

The Doctrine of Return:

Under the Muslim Law this doctrine comes into action when the residue property returns to the Sharers and not the Distant Kindred in absence of any heir under the residuary category. When in case of more than one Sharer then the property shall be proceeding to be returned in the proportionate shares and if there is only one sharer then the whole of the residue property should be transferred back to the sole sharer. The remaining property or called the residue can not be transferred to distant kindred as long as there is a sharer or residuary alive.

Exception:
The exception to this doctrine is that neither the husband nor the wife is entitled to the return so long as there is another sharer or distant kindred alive.
In the cases of M. A. Chowdry v. S. Banoo, (1878) I. L. R. 3 Calcutta p. 702., and Bafatun v. B. Khanum, (1903) I. L. R. 30 Calcutta (683) the court recognised that the husband or the wife can claim the return of residue property. This however was not recognised in the early orthodox Muslim laws.

If, for example, a male dies leaving his mother whose share is 1/6 and a daughter whose share is 1/2 but no widow. And, if there were two brothers, one of them died leaving two widows. Then the second brother also died leaving behind a widow. It was decided that she would get 1/4 shares as a sharer and remaining 3/4 by the doctrine of Radd.

Illustration:
This Doctrine of Return or Radd can be explained with the help of an illustration where a person dies leaving behind his property. Then the property left after paying for her funeral expenses, debts and legacies is distributed among the Sharers. The Mother and the Daughter of the deceased are alive and fall under the category of Sharer. Then, the mother will be entitled to 1/6th of the property and daughter will be entitled to 1/2 of the property.

The total sum of the properties of Mother and Daughter will be, 1/6 + 1/2 = 2/3. 2/3 is less than the unity and the 1/3 of the property will still remain after distributing the property amongst the Sharers. Therefore, in this case, the doctrine of Return or Radd will apply.
The first step would be to reduce the fractions of the Sharers to a common denominator. Thus, 1/6 + 1/2 = 2/3 = 1/6 + 3/6 (The fractions are reduced in such a form where 6 is the common denominator).

The second step would be to decrease the denominator to make it equal to the sum of numerators and allow the individual numerators to remain the same as they were. Thus, 1/6 + 3/6 will become 1/4 + 3/4 (in this case, the denominator is made equal to 4 because the sum of both the numerators is 4).
Hence, with the help of this doctrine, the Shares of the Sharers are increased proportionately in such a manner that the um of the Shares of the property of Sharers become equal to unity.
 

The doctrine of Aul

When in case, the total sum of the specific shares allotted to various shares exceeds the unity then the doctrine of increase (Aul) comes into the application. In this doctrine, the specific share of each sharer is reduced in a proportionate manner.

The proportionate share reduces in the following manner:

1. The first step is by reducing the shares to a common denominator.
2. The second step is by increasing the denominator to make it equal to the sum of the numerators allowing the numerators to stand as they are.

Illustration:
A woman dies leaving behind her property. Then the property left after paying for her funeral expenses, debts and legacies are distributed among the Sharers. The husband of the deceased women and her two full sisters are alive and fall under the category of Sharer. Then, Husband will be entitled to 1/2 of the property of the deceased and the two full sisters will be entitled to 2/3rd of the property.

Total of 1/2 and 2/3 will be 7/6 which exceeds the unity and hence, in this case, the Doctrine of Increase or Aul will come into application.

The first step would be to reduce the fractions of the Sharers to a common denominator. Thus, 1/2 + 2/3 = 7/6 = 3/6 + 4/6 (The fractions are reduced in such a form where 6 is the common denominator).

The second step would be to increase the denominator to make it equal to the sum of numerators and allow the individual numerators to remain the same as they were. Thus, 3/6 + 4/6 will become 3/7 + 4/7 (in this case, the denominator is made equal to 7 because the sum of both the numerators is 7).

Hence, with the help of this doctrine, the total sum of the shares of the property of Sharers are reduced in a proportionate manner and become equal to unity.

Eg: A. Husband holds share 1/4 = 3/12 reduced to 3/13
Mother holds share 1/6 = 2/12 reduced to 2/13
2 Daughters hold shares 2/3 = 8/12 reduced to 8/13
Total come out with our reducing is 13/12 but after reducing 13/13.