# Inheritance arithmetic (“inherithmetic”)

The interesting aspect of inheritance is the arithmetic component. We say arithmetic NOT mathematics because the operations involved are addition, subtraction, multiplication and division only. So even those who dislike mathematics I believe do not find these four basic operations tasking.

In any inheritance problem, the aim is to determine the MINIMUM BASE NUMBER that will enable the estate to be distributed among the heirs such that each will get his/her PORTION, WITHOUT a remainder or decimal. To start with, let’s define some terms (as used in this text):

1. Share: – the fraction of the estate an heir is entitled to inherit such as ½, 1/8, 2/3 and so on.
2. Portion: – the number of segments of an estate an heir will receive. This MUST NECESSARILY be a whole number, not a fraction or number with decimal.
3. Base number (aslul mas-ala): – as stated above, it’s a WHOLE NUMBER that facilitates the distribution of the estate in accordance with the shares of ALL the heirs and also generates the portion of each.

A numeric example will clarify the concepts. Assuming a father gives his 3 children, ‘A’, ‘B’ and ‘C’ £120 to share in proportions of 1/3, 1/6 and ½ respectively, how much will each child receive?  Literally, the question is saying that £120 should be divided into 3, then ‘A’ gets 1 part out of the 3. Then £120 should be divided by 6, from which ‘B’ receives 1 part out of 6. Similarly, ‘C’ is entitled to 1 out of 2 parts of £120. Therefore,

‘A’ receives £120 ÷ 3 = £40
‘B’ gets £120 ÷ 6 = £20
‘C’ is given £120 ÷ 2 = £60

To check whether we are correct, we add up what each child receives: £40 + £20 + £60 = £120. This is how a deceased’s estate is distributed. But practically, the estate is made up of houses, cars, clothes, books, etc., and in most cases the total worth is not available. Therefore for convenience, we determine a number which can be divided by each of the DENOMINATORS of the shares under consideration. Note that every ‘fraction’ has a NUMERATOR (number on top of the slash) and a denominator (number at the bottom of the slash). In this case, 1, 1 and 1 are the numerators of 1/3, 1/6 and ½; while 3, 6 and 2 are the denominators. Now, what number can be divided by 3, 6 and 2 WITHOUT a remainder or decimal?

Let’s assume 3. So,
3 ÷ 3 = 1          3 ÷ 6 = 0.5       3 ÷ 2 = 1.5

Its clear 3 is not the number we are looking for because when divided by 6 and 2, the solutions have decimals. What if we consider 6 and 12?

6 ÷ 3 = 2          6 ÷ 6 = 1          6 ÷ 2 = 3
12 ÷ 3 = 4        12 ÷ 6 = 2        12 ÷ 2 = 6

Both 6 and 12 give us good solutions (i.e. with no decimals), so which one do we choose? The minimum. Consequently, our base number in this problem is 6. What this means is that the estate should be divided into 6 portions. ‘A’ takes 2 (1/3 of 6), ‘B’ gets 1 (1/6 of 6) and ‘C’ receives 3 (½ of 6). So, differentiating between SHARE and PORTION, the shares of ‘A’, ‘B’ and ‘C’ are 1/3, 1/6 and ½ respectively; while their portions are 2, 1 and 3 respectively. Hopefully the definition of portion as the “number of segments of an estate an heir will receive” now makes more sense.

1. Number of heads: – this is the number of heirs IF they are of the same gender. Hence, the number of heads of 2 sons is 2; the number of heads of 9 granddaughters is 9. As simple as that. But if the heirs are of mixed gender, a male has “2 heads” while a female has 1. This is because a male gets double the share of a female. So, the number of heads of 3 full brothers and 4 full sisters is 10; likewise the number of heads of 12 sons and 5 daughters is 29.
2. Category: – a single heir makes a category if he/she inherits a share of the estate alone while 2 or more heirs make a category if they are to distribute a share of the estate among themselves. For instance, if the surviving heirs of a deceased are wife and son, we have 2 categories of heirs since the wife has a share (1/8) and the son also has a share (residue). Wife, father and 2 daughters; this is 3 categories given that the 2 daughters will share ½ of the estate equally. Husband, 2 consanguine brothers, 4 consanguine sisters; this is 2 categories. Consanguine brothers and sisters will share the residue in a ratio of 2 to 1. Grandfather, 3 daughters, grandson and 5 granddaughters; how many categories? 3.