### Adjusting the Base Number When Heirs of Two Categories Cannot Share Portions Evenly

Here is the answer to the problem presented at the end of the previous post:

The estate will be divided into 144 equal portions:

The Maternal Grandmother will get 12 portions.

The Paternal Grandmother will get 12 portions.

The Wife will get 36 portions.

The Akhyaafi brother will get 24 portions.

Each Allaati brother will get 10 portions.

Each Allaati sister will get 5 portions.

All others are deprived.

Any questions about the above solution and how it was reached should be posted in the comments section, *insha’Allah*.

In the previous post we learned how to deal with scenarios in which the heirs of a single category could not share their portions evenly, i.e. in whole numbers. We learned that we simply adjust the base number using the rule provided and re-calculate the portions.

Similarly, when the heirs of *two* categories cannot share their portions evenly, we again adjust the base number using specific rules and re-calculate the portions. The rules this time, however, are a *bit* more complex, but nothing you can’t handle *insha’Allah*.

See the following:

**When the heirs of two categories cannot share their portions evenly, take a look at the number of heads and the number of portions of each category. There can be three possibilities:**

**Heads and portions of one category are tabayun and heads and portions of the other category are also tabayun. In this case, you look at the number of heads in each category:**- If they are tabayun, multiply them together, then multiply the answer by the base number. The result will be your new base number.
- If they are tamathul, pick any one and multiply it by the base number. The result will be your new base number.
- If they are tadakhul, multiply the larger one by the base number. The result will be your new base number.
- If they are tawafuq, multiply the wafq of one by the other, then multiply the answer by the base number. The result will be your new base number.

**Heads and portions of one category are tawafuq and heads and portions of the other category are also tawafuq. In this case, you look at the wafq of the number of heads in each category:**- If they are tabayun, multiply them together, then multiply the answer by the base number. The result will be your new base number.
- If they are tamathul, pick any one and multiply it by the base number. The result will be your new base number.
- If they are tadakhul, multiply the larger one by the base number. The result will be your new base number.
- If they are tawafuq, multiply the wafq of one by the other, then multiply the answer by the base number. The result will be your new base number.

**Heads and portions of one category are tabayun and heads and portions of the other category are tawafuq. In this case, you look at the number of heads in the tabayun category and the wafq of the number of heads in the tawafuq category:**- If they are tabayun, multiply them together, then multiply the answer by the base number. The result will be your new base number.
- If they are tamathul, pick any one and multiply it by the base number. The result will be your new base number.
- If they are tadakhul, multiply the larger one by the base number. The result will be your new base number.
- If they are tawafuq, multiply the wafq of one by the other, then multiply the answer by the base number. The result will be your new base number.

In summary, either they are both tabayun, both tawafuq, or one is tabayun and one is tawafuq. After determining which of the three cases you are dealing with, you apply one of the 4 sub-set rules to get your new base number. Also, notice how the final step in every possibility is to multiply by the base number.

Remember, even if the heads-portions relationship seems tadakhul (ex: 8 and 2), you treat it as tawafuq for the purposes of adjusting the base number. This was also explained in the previous post (see example 4 in previous post).

Lets get a better understanding by working out some example problems *insha’Allah*.

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

- 4 Daughters
- 2 Granddaughters
- 2 Wives
- Mother
- Paternal grandmother
- 1 Maternal aunt
- 3 Haqeeqi nephews
- 2 Allaati nephews

The maternal aunt is a non-heir. The 2 granddaughters are excluded by the daughters. Paternal grandmother is excluded by mother. The allaati nephews are excluded by the haqeeqi nephews. This leaves us with the following heirs:

- 4 Daughters
- 2 Wives
- Mother
- 3 Haqeeqi nephews

The chart, therefore, will look like this:

As you can see, two categories cannot share their portions evenly; 3 portions cannot be divided among 2 wives, and 1 portion cannot be divided among 3 nephews. Both operations give us a non-whole number. To fix this issue, we’ll need to change the base number using the rules provided in the beginning of this post. Once the base number is changed, the new portions it gives us will be able to be divided as whole numbers among the individual heads. As the rule states, we first look at the heads and portions of each category.

The heads and portions of the first category are 2 and 3, respectively. These two numbers are tabayun.

The heads and portions of the second category are 3 and 1, respectively. These two numbers are also tabayun.

Right away, we know that we are dealing with the first scenario listed in the rules. According to that rule, we now look at the heads of each category. The heads are 2 (2 wives) and 3 (3 nephews). These two numbers are, again, tabayun. Since these numbers are tabayun, we apply the first of the four sub-set rules. That is, we now multiply these two numbers together, then multiply the answer with our current base number, and this will give us our new base number.

2 x 3 ＝ 6 ———> 6 x 24 ＝ **144** <——— Our new base number.

Now that we have a new base number, we make a new chart.

Now each individual head is able to receive whole-number portions.

18 ÷ 2 ＝ 9 portions for each wife.

6 ÷ 3 ＝ 2 portions for each nephew.

**Final Answer:** The estate will be divided into 144 equal portions:

Each Daughter gets 24 portions.

Each Wife gets 9 portions.

The Mother gets 24 portions.

Each Haqeeqi Nephew gets 2 portions.

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

- Maternal grandfather
- Mother
- 1 Wife
- 5 Akhyaafi sisters
- 1 Akhyaafi brother
- 2 Haqeeqi brothers
- 2 Haqeeqi sisters
- 1 Allaati sister
- 3 Haqeeqi nephews
- 1 Akhyaafi paternal uncle

Maternal grandfather and akhyaafi paternal uncle are non-heirs. Allaati sister and haqeeqi nephews are excluded by the haqeeqi brothers. This leaves us with the following heirs:

- Mother
- 1 Wife
- 5 Akhyaafi sisters
- 1 Akhyaafi brother
- 2 Haqeeqi brothers
- 2 Haqeeqi sisters

Our initial chart, therefore, looks like the following:

Lets take a look. 2 portions are given to the mother alone. 3 portions are given to the single wife. However, 4 portions cannot be divided among the 6 heads of akhyaafi siblings, nor can 3 portions be divided among the 6 heads of the haqeeqi siblings. These are our two problematic categories. Lets figure out our new base number.

The heads and portions of the first category are 6 and 4, respectively. These two numbers are tawafuq, with a greatest common factor of 2.

The heads and portions of the second category are 6 and 3, respectively. These two numbers are also tawafuq, with a greatest common factor of 3.

Since both sets of numbers are tawafuq, we now know that we are dealing with the second of the three scenarios listed in the rules. According to the rule, we first need to determine the wafq of the number of heads of each category. As a reminder, the wafq of a number is the result you get when you take a pair of tawafuq numbers and divide one of them by their greatest common factor.

We need the wafq of the number of heads of both sets. The number of heads for the first category is 6, and 2 is the greatest common factor.

6 ÷ 2 ＝ 3

The number of heads for the second category is also is 6, and 3 is the greatest common factor.

6 ÷ 3 ＝ 2

The wafq of the heads of the first category is 3, and the wafq of the heads of the second category is 2. Now that we have our two wafqs, we have to figure out the relationship between them, as per the rules. You should quickly realize that 3 and 2 are tabayun. Going back to the rule, we see that we now must apply the first of the four sub-set rules to get our new base number. That is, we multiply the wafqs together, then multiply the answer with our current base number to get the new base number.

3 x 2 ＝ 6 ———-> 6 x 12 ＝ **72 **<———- Our new base number.

Now that we have our new base number, our updated chart is as follows:

24 portions divided by 6 heads gives us 4 portions for each akhyaafi sibling.

18 portions divided by 6 heads gives us 3 portions per head. Each brother is worth two heads in this case, and so each brother gets 6 portions. Each sister, of course, gets 3 portions.

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

The Mother gets 12 portions.

The Wife gets 18 portions.

Each Akhyaafi sibling gets 4 portions.

Each Haqeeqi brother gets 6 portions.

Each Haqeeqi sister gets 3 portions.

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

- Maternal grandmother
- 10 Akhyaafi brothers
- 2 Wives
- 1 Son of haqeeqi paternal uncle
- 2 Daughters of haqeeqi paternal uncle

The daughters of haqeeqi paternal uncle are non-heirs. Everyone else is an heir.

This gives us the following chart:

Upon examining the chart, we realize that the akhyaafi brothers category and the wives category are the two which are problematic. We know this because 4 portions cannot be divided among 10 heads, nor can 3 portions be divided among 2 heads. Of course, changing the base number will solve this problem. As always, we start by considering the heads and portions of each category.

The heads and portions of the first category are 10 and 4, respectively. These two numbers are tawafuq, with a greatest common factor of 2.

The heads and portions of the second category are 2 and 3, respectively. These two numbers are tabayun.

This time, we are dealing with the third scenario listed in the rules: One category is tawafuq while the other is tabayun. We now consider the wafq (of the numbers of heads) of the tawafuq category and the number of heads of the tabayun category and determine the relationship between these two numbers.

The greatest common factor of 10 and 4 is 2, hence we divide 10 by 2 to get our wafq.

10 ÷ 2 ＝ **5** <—— Wafq (of the number of heads) of the tawafuq category.

**2** <——– Number of heads in the tabayun category.

These two numbers (5 and 2) are tabayun. Applying the first of the four sub-set rules, we multiply them together, then multiply the result by the current base number to get our new base number.

5 x 2 ＝ 10 ——-> 10 x 12 ＝ **120** <——- Our new base number.

The new chart, therefore, will look like this:

As you can see, both categories are now resolved. 40 portions can be divided evenly amongst 10 heads, and 30 portions can be divided evenly amongst 2 heads.

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

The Maternal grandmother will receive 20 portions.

Each Akhyaafi brother will receive 4 portions.

Each Wife will receive 15 portions.

The Son of haqeeqi paternal uncle will receive 30 portions.

If you haven’t already, please memorize the rules at the beginning of this post, as well as the one in the previous post. It may seem like a lot to take in, but its not as difficult as it may seem, *insha’Allah*.

In the next post, *insha’Allah*, we will look at problems in which heirs of three or more categories cannot share their portions evenly. But before that, we leave you with a final exercise, the solution to which will be presented in the next post, *insha’Allah*.

A man dies leaving behind:

- 4 Wives
- 2 Allaati brothers
- 2 Allaati sisters

How will the estate be divided and distributed?