### Determining the Base Number

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The base number is the crux of any inheritance problem. Its imperative that we thoroughly understand what it is and how its calculated.

##### What Exactly is a Base Number?

Firstly, lets give a quick breakdown of a share. A share, as explained in the previous post, is the specific fraction assigned to a zaawil furoodh heir (ex: 1/2; 1/6; 1/8; 1/4 etc).

A share, like any other fraction, is made up of two parts: The numerator and the denominator. The top number is the numerator, and the bottom number is the denominator. Hence, if the share of a daughter is 1/2, the number 1 is the numerator and the number 2 is the denominator. In 2/3, the number 2 is the numerator and the number 3 is the denominator. Hopefully you get the idea.

Now consider the following example: A woman dies leaving behind

• Husband = 1/2
• Mother = 1/3
• 1 Akhyaafi sister = 1/6

The shares of all three heirs have already been given. Lets say this women left behind \$120. How will this money be distributed? What this really is saying is the following: Divide 120 by 2, and Husband gets 1 of those 2 parts. Divide 120 by 3, and the mother gets 1 of those 3 parts. Finally, divide 120 by 6, and the akhyaafi sister gets one of those 6 parts. Mathematically, it will look like this:

120(1/2) ＝ \$60

120(1/3) ＝ \$40

120(1/6) ＝ \$20

Instead of thinking of it as multiplication, we can also think of it as two steps: division followed by multiplication. What we are really doing is dividing 120 by the denominator of the share and then multiplying the answer by the numerator of the share, like this:

120 ÷ 2 x 1 ＝ \$60

120 ÷ 3 x 1 ＝ \$40

120 ÷ 6 x 1 ＝ \$20

So if one of the shares was 2/3, we would do the following:

120 ÷ 3 x 2 which is the same as 120(2/3). Both give the answer of \$80.

But for more advance problems (and when the actual value of the estate is unknown), this may not be so intuitive.  So for convenience sake, we think of the lowest possible number which, when divided by the denominator of each share, gives us a whole number. Essentially what we are looking for is the Lowest Common Multiple (LCM) of all the denominators, and this is called the base number. Once we have the base number, we multiply it with the shares and that gives us the portions given to the corresponding heirs.

In the case above, the base number is 6 (how we know this will be explained later in this post). We can prove this using the definition for base number we provided above. If we divide this number by the denominators of the shares (namely 2, 3 and 6), we should end up with a whole number each time. Lets see:

6 ÷ 2 ＝ 3

6 ÷ 3 ＝ 2

6 ÷ 6 ＝ 1

As you can see, when 6 is divided by any of the denominators of the shares, we end up with a whole number. Of course, there are an infinite amount of numbers that would give us this result, but the reason this is the base number is because it is the lowest possible number you can do this with in this scenario. Accordingly, the following are the portions of each heir:

Husband ＝ 6(1/2) ＝ 3

Mother ＝ 6(1/3) ＝ 2

Akhyaafi sister ＝ 6(1/6) ＝ 1

Final answer: The estate will be divided into 6 equal portions.

The Husband will get 3 portions.
The Mother will get 2 portions.
The Akhyaafi sister will get 1 portion.

##### How to Determine the Base Number

We previously mentioned the term LCM (Lowest Common Multiple). All this really means is that you list the multiples of all the numbers, find the first multiple they have in common, and that will be your LCM or base number. In inheritance, we need to find the LCM of the denominators of the shares. In other words, the LCM of the denominators is the base number.

Lets demonstrate with the same three denominators we’ve been working with: 2, 3, and 6

2: 2, 4, 6, 8, 10, 12, 14, 16, 18….

3: 3, 6, 9, 12, 15, 18, 21, 24….

6: 6, 12, 18, 24, 30, 36, 42….

The red colored numbers (6, 12 and 18) are all the common multiples of these three numbers. Of course, there are an infinite amount of common multiples between these numbers, but the one we want is the smallest, and that is 6. The number 6 is our Lowest Common Multiple (LCM) or base number.

Insha’Allah, now hopefully you understand that LCM and base number are the same thing, so from here on we will exclusively use the term ‘base number’.

But listing out multiples of numbers can sometimes get tedious, especially if you have a lot of numbers to deal with. A faster and easier method of getting the base number of a group of numbers is explained below.

There can be four relationships between any two given numbers. These relationships, in Arabic terminology, are called:

1. Tamathul
3. Tawafuq
4. Tabayun

These four relationships help us determine the base number in inheritance problems. Lets explain each one in detail, insha’Allah.

1) Tamathul

This one is simple. Two numbers are tamathul when they are the same. So the relationship between 3 and 3 is tamathul; 4 and 4 are tamathul; 11 and 11 are tamathul.

The base number of two numbers that are tamathul is simply the number itself. The base number of 3 and 3 is 3. The base number of 4 and 4 is 4. Simple.

This one is also simple. Two numbers are tadakhul when one is a multiple of the other.

Examples: 3 and 6; 2 and 8; 4 and 12

The base number of two numbers that are tadakhul is the larger number. Therefore, in the above three examples, the base numbers would be 6, 8 and 12 respectively.

3) Tawafuq

Two numbers are tawafuq when they share a factor. A factor is a number by which another number can be divided, and the result will be a whole number. Factors of the number 24, for example, are 24,  12, 8, 6, 4, 2 and 1.

If you divide 24 by any of those numbers listed above, the result will be a whole number.

So when two numbers have a common factor, they are tawafuq. Lets list the factors of 12 and 16, and see if we find a common factor:

12: 1, 2, 3, 4, 6 and 12

16: 1, 2, 4, 8 and 16

So 12 and 16 have two common factors: 2 and 4. However, what we are looking for is the Greatest Common Factor (GCF), and that here is 4.

Now, to find the base number of these two numbers, we do the following: Divide one of the two numbers by the GCF. The resulting number is called the wafq. The wafq should now be multiplied by the other number.

12 ÷ 4 ＝ 3 —–> 3 x 16 ＝ 48

and alternately,

16 ÷ 4 ＝ 4—–> 4 x 12 ＝ 48

As you can see, no matter which of the two numbers you choose to divide by the GCF, the result will be the same. Our base number in this case is 48.

One thing, however, should be noted. All numbers have 1 and the number itself as factors. But when it comes to faraa’idh, 1 is not considered a common factor, ever.

4) Tabayun

Two numbers are tabayun when they are neither of the three above. Meaning, if two numbers are not tamathul, tadakhul, or tawafuq, then they must be tabayun.

Lets look at the numbers 5 and 6. Obviously they are not tamathul. They are also not tadakhul. They do not share any factors at all (remember, 1 doesn’t count as a common factor) so they are not tawafuq either. These two numbers are tabayun.

To find the base number of two numbers that are tabayun, simply multiply them. Thus, our base number in this case is 30 (5 x 6 ＝ 30).

Know these four relationships well, as they are means to determining the base number, and the base number is the crux of all inheritance problems.

Now lets take it up a notch. How do you find the base number of more than two numbers? Its simple, really. Lets say you have four numbers: A, B, C and D. All you have to do is find the base number of A and B. Then find the base number of that (i.e. the previous base number) and C. Then find the base number of that and D. This process can continue infinitely. The final result will be the base number of all the numbers.

Lets take you back to the example we gave in the beginning of the post. The heirs were:

• Husband = 1/2
• Mother = 1/3
• 1 Akhyaafi sister = 1/6

The denominators are 2, 3 and 6. Lets try to find the base number of these three numbers using the four relationships we’ve learned above. Remember, the answer should come out to 6.

2 and 3 are tabayun. Their base number, therefore, is 6. Now we find the base number of this result (6) and the remaining number, 6.
6 and 6 are obviously tamathul. Therefore, their base number is 6. The base number of all three numbers is 6. Alternately, you can go in a different order:

3 and 6 are tadakhul. Their base number is the larger number, or 6. Now we find the base number of this result (6) and the remaining number, 2.
2 and 6 are also tadakhul, meaning their base number is again 6. The base number of all three numbers is 6. Or the final way to do it:

2 and 6 are tadakhul. Their base number is 6. Now we find the base number of this result (6) and the remaining number, 3.
3 and 6 are tadakhul, and their base number is therefore 6. The base number of all three numbers is 6.

No matter which order you go in, the result will always be the same. Once you run out of numbers, the final result is the base number of all the numbers.

Below we’ve written some example problems for you to try out. For each problem, you are given a list of numbers, and all you have to do is find their base number. Calculators may come in handy for some.

(A) 3 and 8.

(B) 2 and 10

(C) 8, 6 and 6

(D) 10, 12, and 15

(E) 3, 5, 6, and 14

(F) 1, 4, 8, 12, 14 and 17

Insha’Allah, answers will be provided in the next post.