### Solving Problems Using the Principle of Radd

Radd is just a *bit* more complex than `awl but, *insha’Allah*, by the end of this post you will have a pretty good understanding of it.

Lets get started with a very simple example.

**Example 1:** A woman dies leaving behind

- Husband
- 2 Daughters

We know that both of them will inherit, so there is no total exclusion in this case. The husband, however, will be partially excluded to 1/4 while the daughters, of course, will get 2/3. Right off the bat we know that we will need to apply radd in this problem because:

**(1)** 1/4 + 2/3 is less than 1 and

**(2)** there are no `asabaat to claim the residue

First lets at least draw out our initial chart and then, *insha’Allah*, we can work from there:

Subtracting the sum of the portions from the base number, we find that there is a residue of 1 portion left over. This is no surprise, as we already knew that there would be residue due to the fact that the shares add up to less than 1. As explained in the previous post, radd – simply defined – is the redistribution of the residue among the zaawil furoodh categories according to the ratio of their respective portions. In this case, the portions are 3 and 8. This means that residue will be distributed among the husband and daughters in a 3:8 ratio, right? Wrong. Remember, radd is not applied to a spouse. The spouse (in this case, the husband) is not entitled to receive any of the residue. This means that the daughters will get all of the residue. Since there is no other category which can be a recipient of the residue, there is no need to even consider a ratio anymore. The residue, in its entirety, will be redistributed to the daughters.

Now, to represent this in the chart, we need to add a new column, as shown below:

We’ve added a new column to the far right. This additional column is telling us a few things:

- Who the recipient of the residue is (in this case, the 2 daughters)
- What the recipient is receiving (the residue)
- How much the residue is (in this case, only a single portion)

Now, this information may be different for other problems. In this case, only the daughters are getting the residue (since radd is not applied to the husband). It may be that in a different scenario, multiple categories of recipients may have to share the residue (and in that case, we would have to consider the ratio of their portions). It may also be the case that more than 1 portion is being redistributed. However it comes out to be, lets agree to call this far-right column the “radd category”.

We know that all of the residue is going to the daughters, so we can apply radd by simply moving that 1 portion in the radd category over to the daughters category, making their portions increase from 8 to 9. But we won’t do that just yet. Why? You see, we need to treat the radd category like any other category. Meaning, the radd category also has heads and portions, and its heirs also need to be able to share the portions evenly i.e. in whole numbers. In this case, the radd category has 2 heads and 1 portion, and 2 heads cannot share 1 portion. Thus, we need to resolve this category just like we would resolve any other category – by calculating a new base number and new portions. Once we resolve the radd category, only then should we apply radd.

Note: Radd should always be applied *after* resolving all categories. This is the easier way to solve these types of problems.

Keep in mind that sometimes we may have to resolve the radd category along with other categories. *Insha’Allah*, we’ll see this in further examples. In this case, however, only the radd category needs to be resolved.

2 and 1 are tabayun (remember, they cannot be tawafuq because 1 is not considered a common factor). To get the new base number, we multiply the number of heads by the current base number:

2 x 12 ＝** 24 **<——– New base number.

After calculating our new base number of 24, we also calculated our new portions. This time we’ve ended up with 2 portions as residue (with each daughter getting 1 portion of residue). Now all we have to do is apply radd by moving these two portions from the radd category to the actual daughters category, increasing their portions from 16 to 18. The way to show that on the chart is to add an additional “Final Portions” row, like below:

All we’ve done here is simply move the 2 portions of radd over to the daughters category. These 2 portions were added to what the daughters already had, making their final portions 18 (9 each), and leaving the residue at zero. The portions of the husband, of course, remain the same and were simply brought down to the “Final Portions” row.

**Final Answer:** The estate will be divided into 24 equal portions:

The Husband gets 6 portions.

Each Daughter gets 9 portions.

*Insha’Allah*, radd problems will become a lot more intuitive and easy once you practice more examples. So lets move on *insha’Allah*.

**Example 2: **A woman dies leaving behind

- Paternal grandmother
- Maternal grandmother
- 1 Haqeeqi sister
- Mother
- 2 Nieces

The 2 nieces are non-heirs. Both grandmothers are excluded by the mother. So that only leaves us with:

- 1 Haqeeqi sister
- Mother

Our initial chart looks like this:

Remember, the mothers share this time is 1/3 because their is only one sibling present. Two or more siblings, however, would partially exclude the mother to 1/6.

Of course, its easy to see why this is a radd problem. The shares add up to less than 1, which means a residue is left over when the sum of portions is subtracted from the base number. There are no `asabaat inheriting to claim this residue, so radd must be applied.

In the previous example, only one category was entitled to receive the residue. In this example, however, both categories (i.e. the sister *and* the mother) are entitled to radd. The way this is done is that we consider the portions of each category: the residue will be divided among them with respect to the ratio of their portions . In this case, the residue will be distributed among the sister and mother in a 3:2 ratio. Before moving on, lets draw out our new chart, with the radd category:

Take look at the radd category. This time there are two recipients. Keep in mind that even when radd is applied to multiple categories of heirs (like in this example), they will all still be put into one radd category, as shown above.

The radd category here is telling us a few things:

- Both the sister and mother are getting the residue
- They will share the residue in a 3:2 ratio
- The sister gets the greater part of the ratio
- The residue is 1 portion

Notice how we’ve written out the heirs in the radd category, “Sister : Mother”. Right under we’ve written their ratio of ‘”3:2″. This is because we want to stay organized and know who is getting what part of the ratio. Lined up, we can tell that the sister will get the “3” (i.e. the greater part), while the mother will get the “2” (i.e. the lesser part).

To understand how this will work, think about how a son and daughter inherit together. We know by now that a son and daughter inheriting together share the residue in a 2:1 ratio, meaning the son is worth 2 heads (hence, he gets twice as many portions as his sister), and the daughter is worth 1 head (hence, she gets half as many portions as her brother), making that entire category worth 3 heads. Of course, if there are multiple sons and multiple daughters, the heads will increase accordingly.

The same is done here. The sister and mother in the radd category inherit the residue in a 3:2 ratio, and hence the sister is worth 3 heads while the mother is worth 2 heads. This means that the entire radd category now has 5 heads. For every 5 portions, the sister will get 3 and the mother will get 2.

One thing, however, should be kept in mind. Even if there were multiple sisters in this case, the number of heads would still be 5; they would NOT increase, unlike in the case of multiple sons and daughters inheriting together. This is because when multiple sons and daughters are inheriting together, **each** son gets twice as much as **each **daughter i.e. the residue is divided among *each individual heir* of the category. But when it comes to radd, the residue is divided among each *category*, and not among each individual heir. So even if there were multiple haqeeqi sisters in this scenario, all of them would be in one category, and hence all of them together (i.e. as a category) would be worth 3 heads. *Insha’Allah*, this will be further understood in example 3.

Now we know that the radd category has 5 heads and 1 portion. 1 portion cannot be divided among 5 heads, so we need to resolve this category before we apply radd.

We resolved the radd category by calculating a new base number of 30, followed by new portions. The radd category ended up with a total of 5 portions, which gives us 1 portion per head. Since the sister (i.e. her category) is worth 3 heads, she gets 3 portions, while the mother (i.e. her category) who is worth 2 heads, gets 2 portions.

The final step is to add the “Final Portions” row, which we did in the same chart above, and add the radd portions to each category. The sisters portions increased to 18 (15 + 3 radd), while the mothers increased to 12 (10 + 2 radd), leaving the residue at zero.

**Final Answer:** The estate will be divided into 30 equal portions:

The Haqeeqi sister will get 18 portions.

The Mother will get 12 portions.

**Example 3:** A man dies leaving behind

- 2 Wives
- Mother
- 7 Akhyaafi brothers

Everyone inherits.

Initial chart is below:

The wives get their full share of 1/4 because there are no children, mother gets 1/6 due to the presence of the 7 siblings, and the 7 akhyaafi siblings get 1/3.

Subtracting the portions from the base number, we find our residue this time to be 3.

12 – 3 – 2 – 4 ＝ **3**

Lets add the radd category to our chart:

Radd will applied to the mother and the akhyaafi brothers category. Radd is not applied to spouses, hence the wives are not included in the radd category.

The ratio of the portions of the mother and akhyaafi brothers category is 2:4, which can be reduced to 1:2, and this is the ratio in which the residue will be distributed among them. The mother (i.e. her category) represents 1 head, while the akhyaafi brothers category represents 2 heads. Note: In conjunction with what was explained in the previous example, its not that each akhyaafi brother represents 2 heads, its that the *entire category* of akhyaafi siblings represents 2 heads. So even if there were twice as many akhyaafi siblings (male or female), they would all still represent 2 heads because they are still a single category.

This time the radd category does not need resolving, but two others do: The wives category and the akhyaafi brothers category.

*Insha’Allah*, lets resolve these two categories first, then we can apply radd.

The heads and portions of both categories are tabayun, therefore we multiply the heads together, then multiply the result by the current base number to end up with the new base number.

2 x 7 ＝ 14

14 x 12 ＝ **168** <—– New Base Number

As you can see, with our new base number and new portions, all categories are resolved. We ended up with a residue of 42 portions in the radd category, which gives us 14 portions per head. Thus, the mother’s category (which is worth 1 head) is assigned 14 portions, while the akhyaafi brothers category (worth 2 heads) is assigned 2 x 14 ＝ 28 portions.

Next we simply moved those assigned radd portions to the appropriate category. After adding 14 portions to the mothers category, the mother now has a total of 42 portions. After adding 28 portions to the akhyaafi brothers category, the akhyaafi brothers now have a total of 84 portions (12 for each brother). The portions of the wives category stay the same. The residue, of course, is now zero.

**Final Answer:** The estate will be divided into 168 equal portions:

Each Wife receives 21 portions.

The Mother receives 42 portions.

Each Akhyaafi brother receives 12 portions.

**Example 4: **A woman dies leaving behind

- Paternal grandmother
- Maternal grandmother
- Maternal grandfather
- 1 Daughter
- 4 Granddaughters
- 2 Akhyaafi brothers
- 1 Paternal aunt

The maternal grandfather and paternal aunt are non-heirs. The akhyaafi brothers are excluded by the daughter and granddaughters. This leaves us with:

- Paternal grandmother
- Maternal grandmother
- 1 Daughter
- 4 Granddaughters

The shares of the daughter and the granddaughters have to add to 2/3. Since the daughter gets 1/2, that means the granddaughters will get 1/6.

Next we add in the radd category to our chart:

This time three categories are entitled to receive radd. Ratio is 3:1:1, with the daughters category representing 3 heads, and the granddaughters and grandmothers category each representing 1 head. This is the ratio in which these three categories will receive the residue.

However, before we can apply radd and redistribute the residue, there are 3 categories (out of the 4 we have now including the radd category) that need need to be resolved: The granddaughters category, the grandmothers category, and the radd category.

We chose to first resolve the granddaughters and grandmothers category. The heads and portions of both categories are tabayun, hence we look at the heads: 4 and 2. These two numbers are tadakhul, so we keep the larger one – in this case, the 4 – and call it **X**.

The remaining category – the radd category – has 5 heads (3 + 1 + 1 ＝ 5) and 1 portion. These two numbers are tabayun, therefore we say that **Y** ＝ 5.

4 and 5 (**X** and **Y**) are tabayun, multiplying them gives us 20. Now the final step is to simply multiply this result by the current base number:

20 x 6 ＝ **120** <—– New base number.

Using our new base number we calculated new portions and got the above chart, with all categories resolved.

The radd category receives 20 portions, which gives us 4 portions per head. So it follows that the daughter – worth 3 heads – will receive 4 x 3 ＝ 12 portions. The other two categories are each worth 1 head, hence they will get only 4 portions.

The final step is to just apply radd and move the radd portions over to their appropriate categories. The radd portions are added to the portions already given to give us the final portions of each category.

Once the radd portions are distributed to their appropriate categories, the residue drops to zero.

**Final Answer:** The estate will be divided into 120 equal portions:

The Daughter will get 72 portions.

Each Granddaughter will get 6 portions.

The Paternal grandmother will get 12 portions.

The Maternal grandmother will get 12 portions.

**Example 5: **A man dies leaving behind

- 1 Haqeeqi sister
- 5 Allaati sisters
- 1 Akhyaafi sister

All will inherit.

Lets form our first chart:

The allaati sisters get 1/6 due to the presence of the single haqeeqi sister. This is the same as the daughter-granddaughters rule applied in example 4.

Lets move on and add in the radd category to our chart above:

All three categories will receive radd.

We have a residue of 1 left over, which will have to be redistributed among 5 heads in a ratio of 3:1:1. Before we can apply radd, however, we need to resolve the allaati sisters’ and radd category.

The heads-portions of both categories are tabayun. They both have 5 heads, making them tamathul. Hence, we multiply 5 by the current base number to get our new base number:

5 x 6 ＝ **30 **<—— New base number.

Our new chart is below:

With all categories resolved, we end up with a residue of 5. This gives us 1 portion per head. Since the haqeeqi sister category is worth 3 heads, she will get 3 portions. The allaati sisters category as well as the akhyaafi sister are each worth 1 head, therefore they will each receive 1 portion.

Of course, these portions are added to what each category already has; and so the haqeeqi sister ends up with 18 portions, and the allaati sisters category as well as the akhyaafi sister end up with 6 portions each.

However, these are not really the final portions. Take a look at the chart again. Notice that after applying radd, the allaati sisters category again needs to be resolved. This will happen sometimes: You may resolve certain categories, and then after applying radd you may need to resolve them again because the added portions of radd no longer allow even division. In these types of cases, we do what we would normally do: resolve the category by calculating a new base number, and use the new base number to calculate new portions.

In this case, the heads and portions of the allaati sisters category are 5 and 6, respectively. These two numbers are tabayun, so we multiply 5 by the current base number:

5 x 30 ＝ **150** <—— New base number. You can think of this as our n*ew* new base number, although it doesn’t really matter, as long as you know that this is now our current base number and the old one is scrapped.

Using this new base number, we now calculate our new (and final) portions.

Our final chart look likes this:

What’s this? Have we done the math wrong? Take a look:

150(1/2) ＝ 75

150(1/6) ＝ 25

150(1/6) ＝ 25

These are the numbers we should have gotten as our final portions, right? Actually, not exactly.

When we apply radd, we are also changing the shares of the categories. So while the original share of the haqeeqi sister was 1/2, her new share, after receiving radd, is 18/30 (or 3/5 reduced), which is more than 1/2. Similarly, the share of the allaati sisters category is 6/30 (or 1/5 reduced) and the share of the akhyaafi sister is also 6/30. Essentially, applying radd has slightly increased the shares of all the categories to which radd was applied, and this needs to be kept in mind if the need ever arises to resolve a certain category after having applied radd to it.

So when we adjust our base number *after* having applied radd, we need to make sure to use the new, modified shares to calculate our final portions. In this case, its as below:

150(18/30) ＝ **90**

150(6/30) ＝ **30**

150(6/30) ＝ **30**

This is how we got our final portions. Of course, since there are 5 Allaati sisters, each one gets 6 portions. The residue is still zero.

**Final Answer:** The estate will be divided into 150 equal portions:

The Haqeeqi sister receives 90 portions.

Each Allaati sister receives 6 portions.

The Akhyaafi sister receives 30 portions.

*Insha’Allah*, we hope that this post on radd was easy to understand. We now leave you with a final example problem to do on your own. The solution to it will be given in the next post, *insha’Allah*.

A man dies leaving behind:

- 4 Wives
- Paternal grandmother
- 1 Paternal aunt
- 1 Akhyaafi brother
- 2 Nieces
- 1 Daughter of haqeeqi paternal uncle
- 2 Daughters of haqeeqi paternal uncle’s son.

How will the estate be divided and distributed?