### Adjusting the Base Number When Heirs of One Category Cannot Share Portions Evenly

Below is the answer to the problem which was presented at the end of the previous post:

The estate will be divided into 6 equal portions:

The Mother will get 1 portion.

The Father will get 1 portion.

The Son will get 2 portions.

Each Daughter will get 1 portion.

All others are deprived.

Any questions about the above should be posted in the comments section, *insha’Allah*.

As you should already know, the number of portions allotted to an individual heir is calculated by dividing the number of portions given to that entire category (to which that heir belongs) by the number of heads in that category. This will give us the number of portions given to each head, and thus, each individual heir. As a brief review, lets go through some quick examples, *insha’Allah*.

**Quick Example 1:**

A category of 2 sons and 3 daughters are given 14 portions. Applying the above rule, we divide 14 (portions given to the entire category) by 7 (number of heads). This gives us 2 portions per head. Since they inherit in a 2:1 ratio, each son is worth 2 heads, and so each son gets 4 portions. Each daughter, being worth 1 head, gets 2 portions.

**Quick Example 2:**

A category of 3 wives are given 3 portions. This should be an easy one. Each head (i.e. each wife, in this case) gets 1 portion.

**Quick Example 3:**

A category consisting of 1 haqeeqi paternal uncle gets 7 portions. This should be even easier. Since he’s the only one in the category all 7 portions go to him. Applying the rule still gives us the same answer.

As illustrated, knowing the portions given to the whole category is not enough. The solution is only reached once you know the portions given to each heir. This is why, once you have the portions given to each category, you need to ‘scan’ each category to see if you can evenly split the portions among the heirs of that category. If a category consists of a single heir, he gets all the portions given to his category (as shown in example 3 above).

You may recall, from a previous post, that I mentioned that the number of portions allotted to each individual heir must be a whole number. We cannot say that a particular heir is given a certain fraction of a portion, nor can we say that a particular heir is given a decimal number portion. It **must** be a whole number.

Take another look at Quick Example 1 above. Lets change the number of portions given to the siblings to 15. This gives us approximately 2.14 portions for each daughter, and approximately 4.28 portions for each son. This is unacceptable as these are not whole numbers. In such a case, the base number would have to be adjusted.

Similarly, in Quick Example 2, if the wives were given 4 portions, they would no longer be able to evenly share (i.e. with whole numbers). In this case also, the base number would have to be adjusted.

In this post, *insha’Allah*, we will learn how to calculate a new base number when we come across such problems.

The rule is the following:

**When the heirs of a single category cannot share their portions evenly, take a look at the number of heads of that category and the number of portions given to that category: **

**If the two numbers are tabayun, multiply the number of heads by the base number. The result will be your new base number.****If the two numbers are tawafuq, multiply the wafq of the number of heads by the base number. The result will be your new base number.**

Note that this rule applies only when a **single** category of heirs cannot share their portions evenly. Rules for when this happens with multiple categories will be discussed in later posts, *insha’Allah*. Also, the two numbers (heads and portions) can only be tabayun or tawafuq, they cannot be tadakhul or tamathul. This rule applies even when *more* than a single category of heirs cannot share their portions evenly.

A new base number will also change the number of portions given to each category. Once we have the new base number, we use it to get our new portions.

Lets get a better understanding with some examples, *insha’Allah*.

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

- Husband
- 3 Daughters
- 1 Granddaughter
- 2 Akhyaafi brothers
- 1 Akhyaafi paternal uncle
- 1 Haqeeqi nephew

The akhyaafi paternal uncle is a non-heir. The granddaughter and 2 akhyaafi brothers are excluded by the daughters. This leaves us with:

- Husband
- 3 Daughters
- 1 Haqeeqi nephew

The first two are zaawil furoodh with shares of 1/4, and 2/3 respectively. The last one (haqeeqi nephew) is an `asbah, he gets the residue. Our denominators are 4 and 3, which makes our base number 12.

12(1/4) ＝ 3 portions for the husband.

12(2/3) ＝ 8 portions for the daughters.

12-8-3 ＝ 1 portion (residue) for the nephew.

Notice that the heirs of the middle category cannot share their portions evenly amongst themselves. 8 divided by 3 gives us a non-whole number. This means we’ll need to adjust the base number by applying the rule written in bold above. In doing so, we will end up with a new number of portions which will give us a whole number when divided by 3.

Firstly, we take a look at the number of heads of the category we are dealing with and the number of portions given to that category. That’s 3 and 8, respectively. Now we have to figure out whether these two numbers are tabayun or tawafuq. Since they have no common factor (remember, 1 doesn’t count as a common factor), the two numbers are tabayun. According to the rule, now that we’ve established the two numbers to be tabayun, we multiply the number of heads by the base number. This will give us our new base number.

3 x 12 ＝**36 **<——– Our new base number.

Now that we have our new base number, we use it to calculate the new portions. This also means we have to make two additional row’s in our chart. Our new chart will look like this:

Changing the base number also changes the portions allotted to each category, so now have to re-calculate the portions using the **new** base number. The old base number has been abrogated.

36(1/4) ＝ 9 portions for the husband.

36(2/3) ＝ 24 portions for the daughters.

36-9-24 ＝ 3 portions (residue) for the nephew.

As you can see, the method for calculating portions is still the same as always: multiply base number by the share of the category.

Here is the benefit of adjusting our base number: now the portions allotted to the 3 daughters can be evenly divided among them. Since there are 24 portions given to the category, and there are 3 heads in this category, each head – i.e. each daughter – gets 8 portions.

**Final Answer:** The estate will be divided into 36 equal portions:

The Husband will get 9 portions.

Each Daughter will get 8 portions.

The Haqeeqi nephew will get 3 portions.

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

- 3 Wives
- 5 Granddaughters
- 2 Great-granddaughters
- 3 Akhyaafi brothers
- 5 Allaati sisters
- 4 Haqeeqi nieces (daughters of late haqeeqi brother)

The haqeeqi nieces are non-heirs, just as all nieces are non-heirs. The 2 great-granddaughters are excluded by the 5 granddaughters. This application is essentially the same as **Rule #6, **except on lower generations**. **The akhyaafi brothers are excluded by the granddaughters. This leaves us with the following heirs:

- 3 Wives
- 5 Granddaughters
- 5 Allaati sisters

The wives share 1/8, while the granddaughters share 2/3. The allaati sisters share the residue as they are `asabaat. Looking at the denominators of the shares (8 and 3), we find our base number to be 24. From here, calculating the portions of each category is simple. Our chart, so far, looks like this:

24(1/8) ＝ 3 portions for the wives.

24(2/3) ＝ 16 portions for the granddaughters.

24-3-16 ＝ 5 portions for the allaati sisters.

As you can see, the heirs of the first category (the wives) can share there 3 portions evenly amongst themselves. The heirs of the third category can also share there 5 portions evenly. The heirs of the second category, however, cannot share there portions evenly. 16 portions cannot be shared evenly (i.e. in whole numbers) amongst 5 heads.

As with the previous example, we’ll need to change our base number to fix this problem. Lets apply the rule again. We’re dealing with the second category:

Number of heads ＝ 5

Number of portions ＝ 16

5 and 16 have no common factors, which means they are not tawafuq, which means they must be tabayun. Now that we’ve established the two numbers to be tabayun, we multiply the number of heads (5) by the base number (24) to get our new base number.

5 x 24 ＝ **120** <—— Our new base number.

Now we have to re-calculate our portions using this new base number. The new chart will look like this:

120(1/8) ＝ 15 portions for the wives.

120(2/3) ＝ 80 portions for the granddaughters.

120-15-80 ＝ 25 portions for the allaati sisters.

Now the heirs of all three categories can share their portions evenly.

The first category has 15 portions and 3 heads.

15 ÷ 3 ＝ 5 portions per head.

The second category has 80 portions with 5 heads.

80 ÷ 5 ＝ 16 portions per head.

The third category has 25 portions with 5 heads.

25 ÷ 5 ＝ 5 portions per head.

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

Each Wife will get 5 portions.

Each Granddaughter will get 16 portions.

Each Allaati sister will get 5 portions.

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

- Paternal grandfather
- Maternal grandfather
- 3 Wives
- 1 Daughter
- 6 Granddaughters
- 2 Haqeeqi brothers
- 1 Haqeeqi paternal uncle

Maternal grandfather is a non-heir. Haqeeqi brothers and paternal uncle are excluded by the paternal grandfather. This leaves us with:

- Paternal grandfather
- 3 Wives
- 1 Daughter
- 6 Granddaughters

Paternal Grandfather ＝ 1/6 + Residue

3 Wives ＝ 1/8

1 Daughter ＝ 1/2

6 Granddaughters ＝ 1/6

As you can see, all heirs are zaawil furoodh (with paternal grandfather also being an `asbah, since he belongs to both groups). Firstly, notice that when we add up all the shares, the result is less than 1. This means that we will have residue leftover. This residue will go to the paternal grandfather (in addition to his 1/6), since he is also an `asbah.

Looking at the denominators of the shares, we work out our base number to be 24. From here, its just a matter of simple arithmetic to calculate the portions.

24(1/6) ＝ 4 portions for the paternal grandfather.

24(1/8) ＝ 3 portions for the wives.

24(1/2) ＝ 12 portions for the daughter.

24(1/6) ＝ 4 portions for the granddaughters.

When we subtract the sum of all portions from the base number, we find that there is 1 portion left over. This was expected, as we already figured out that there would be residue since the shares do not add up to 1. Normally we would immediately add this 1 portion to the paternal grandfathers 4 portions, to make a total of 5 portions. However, this time we have to do something else first. The heirs of the last category are not able to share their portions evenly. 4 portions cannot be shared by 6 heads, so we need to fix this first. Once we figure out a new base number to fix this problem, and calculate our new portions, only then will we add the residue to the grandfathers portions. It’s simply easier to add the residue at the very end.

Lets figure out a new base number.

Number of heads ＝ 6

Number of portions ＝ 4

Remember, the two numbers can either be tabayun or tawafuq, they cannot be tamathul or tadakhul. In this case, 6 and 4 are clearly tawafuq as they have a common factor of 2. According to the rule, if the two numbers are tawafuq, we multiply the wafq of the numbers of heads, by the base number. The result will be our new base number.

A quick reminder: wafq is the number you get when you divide one of two tawafuq numbers by their common factor. In this case, we need the wafq of the number of heads. We multiply this wafq by 24 (our current base number), and the result will be our new base number.

6 ÷ 2 ＝ 3 <—– wafq of 6.

3 x 24 ＝ **72** <——- Our new base number.

Now, just like in the last two problems, we use this new base number to calculate our new portions.

After calculating the new portions, we subtracted the sum of all the portions (12 + 9 + 36 + 12) from 72, and found that the residue is now 3 portions. Because all the numbers (base number and portions) increased proportionally, the residuary portions also increased from 1 to 3. This is why its better to wait until the end to add the residual portions to the grandfathers original portions. After receiving the residue, the paternal grandfather has a total of 15 portions.

The granddaughters can also now share there portions evenly, with each granddaughter getting 2.

**Final Answer:** The estate will be divided into 72 equal portions:

The Paternal grandfather will get 15 portions.

Each Wife will get 3 portions.

The Daughter will get 36 portions.

Each Granddaughter will get 2 portions.

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

- Paternal grandmother
- Maternal grandmother
- 1 Wife
- 2 Haqeeqi brothers
- 8 Haqeeqi sisters
- 3 Allaati brothers
- 3 Allaati sisters
- 2 Akhyaafi brothers
- 2 Akhyaafi sisters
- 2 Haqeeqi nephews (sons of haqeeqi brother)

The allaati siblings and haqeeqi nephews are excluded by the haqeeqi brothers. This leaves us with:

- Paternal grandmother
- Maternal grandmother
- 1 Wife
- 2 Haqeeqi brothers
- 8 Haqeeqi sisters
- 2 Akhyaafi brothers
- 2 Akhyaafi sisters

Both grandmothers ＝ 1/6

Wife ＝ 1/4

Akhyaafi siblings ＝ 1/3

Haqeeqi siblings ＝ Residue in 2:1 ratio

Please note that each individual grandmother does not get 1/6, rather both grandmothers **together** share 1/6 equally. Similarly, all akhyaafi siblings share 1/3 equally amongst themselves.

Our denominators are 6, 4 and 3, which makes our base number 12. Our chart, therefore, will look like this:

After calculating the portions, we find that the heirs of the first three categories can all share their portions evenly. The heirs of the last category, however, cannot share their portions evenly. We now have to find a new base number by applying the rules we’ve learned.

Number of heads ＝ 12

Number of portions ＝ 3

Remember, the number of heads is 12 (not 10), because each brother is counted as two heads in this case.

At first glance, one is tempted to say that these two numbers are obviously tadakhul. And this is true, 12 is in fact a multiple of 3. But remember the rule: when dealing with a single category, the number of heads and the number of portions can only be either tabayun or tawafuq. So even though these two numbers are primarily tadakhul, for the purpose of finding a new base number we will treat them as tawafuq. All tadakhul numbers are also tawafuq. In this case, 3 is the common factor between 3 and 12. Now lets apply the rule we’ve learned:

12 ÷ 3 ＝ 4 <——- wafq of 12. (Remember, wafq of the number of heads is taken, NOT the number of portion)

4 x 12 ＝ **48** <—— Our new base number.

Now that we have a new base number, we re-calculate our portions.

48(1/6) ＝ 8 portions for the two grandmothers.

48(1/4) ＝ 12 portions for the wife.

48(1/3) ＝ 16 portions for the akhyaafi siblings.

48-8-12-16 ＝ 12 portions for the haqeeqi siblings.

Notice, now the heirs of all four categories can share their portions evenly. In the last category, we now have 12 heads with 12 portions, and this gives us 1 portion per head. Of course, since they share in a 2:1 ratio, each brother gets 2 and each sister gets 1.

**Final Answer:** The estate will be divided into 48 equal portions:

The Paternal grandmother gets 4 portions.

The Maternal grandmother gets 4 portions.

The Wife gets 12 portions.

Each Akhyaafi sibling (whether male or female) gets 4 portions.

Each Haqeeqi brother gets 2 portions.

Each Haqeeqi sister gets 1 portion.

In all these problems, the heirs of only a single category were unable to share their portions evenly. In the next post *insha’Allah*, we’ll learn how to deal with problems in which the heirs of *two* categories are unable to share their portions evenly.

But before that, lets end with a final problem for you to work out. The answer will be provided in the next post, *insha’Allah*.

A man dies leaving behind:

- Maternal grandmother
- Paternal grandmother
- 1 Wife
- 1 Akhyaafi brother
- 3 Allaati brothers
- 6 Allaati sisters
- 2 Haqeeqi paternal uncles
- 5 Allaati nephews (sons of Allaati brothers)

How will the estate be divided and distributed?