Level two – part 4

Level 2(d): Two categories of heirs cannot share their portions of the estate but the number of heads and corresponding number of portions of BOTH categories CONVERGE i.e. have a common divisor
Rule S: Consider the ADJUSTS of both categories. If they are the same, choose one and multiply it by the base number to obtain the new base number

Example 28

Heirs Mother 4 Uterine brothers 6 Consanguine brothers
Shares 1/6 1/3 Residue
Base number 6
Portions 1 2 3
Number of heads 1 4 6
New base number 12
New portions 2 Each brother = 1 Each brother = 1

4 uterine brothers cannot share 2 portions; but 4 (number of heads) and 2 (number of portions) converge. Their common divisor is 2.
6 consanguine brothers cannot share 3 portions; also their number of heads, 6, and number of portions, 3, converge. Common divisor of 6 and 3 is 3.

Adjust of uterine brothers = 4 ÷ 2 = 2
Adjust of consanguine brothers = 6 ÷ 3 = 2

The two adjusts are the same. The rule says, select any and multiply by the base number, so

New base number = 2 × 6 = 12
New portion of mother: 12 × 1/6 = 2
New portion of 4 uterine brothers: 12 × 1/3 = 4
New portion of 6 consanguine brothers: 12 – (2 + 4) = 6

Rule T: If one ADJUST is a multiple of the other, multiply the higher one by the base number. The result is the new base number

Example 29

Heirs Grandmother 4 Uterine brothers;
4 uterine sisters		 6 Full brothers
Shares 1/6 1/3 Residue
Base number 6
Portions 1 2 3
Number of heads 1 8 6
New base number 24
New portion 4 Each sibling = 1 Each brother = 2

Grandmother is given her 1 portion. She has no problem.
4 uterine brothers and 4 uterine sisters cannot share 2 portions. However, 8 (their number of heads) and 2 (their number of portions) converge. The common divisor of 8 and 2 is 2.

But wait a minute, the category “uterine brothers and uterine sisters” is made up of males and females. And as stated earlier, in this circumstance, males have “2 heads,” therefore, number of heads of 4 uterine brothers and 4 uterine sisters should be 12 not 8! Yes, that’s true. We forgot to mention that that principle applies only when the males in the category will receive twice the portion of the females.  Recall Example 26. The category “grandson and granddaughter” is said to have 3 heads. Why? Because grandson is entitled to two times the number of portions of granddaughter. Thus, he has “2 heads” while granddaughter has “1 head” making 3 heads. That is why their new portions were 4 and 2 respectively. As for uterine siblings, they share their portion of the estate in equal proportions irrespective of gender, so all of them are regarded as having “1 head.” Consequently, number of heads of 4 uterine brothers and 4 uterine sisters is 8.

Adjust of 4 uterine brothers and 4 uterine sisters = 8 ÷ 2 = 4

Similarly, 6 full brothers cannot share 3 portions; nevertheless 6 and 3 converge. Common divisor of 6 and 3 is 3.

Adjust of 6 full brothers = 6 ÷ 3 = 2

Now, the two adjusts, 4 and 2, one is a multiple of the other. Applying the rule,

New base number = 4 × 6 = 24
New portion of grandmother: 24 × 1/6 = 4
New portion of 4 uterine brothers and 4 uterine sisters: 24 × 1/3 = 8
New portion of 6 full brothers: 24 – (4 + 8) = 12
Each uterine sibling gets 1 portion of the estate while each full bother inherits 2 portions.

Rule U: If the ADJUSTS of the two categories that cannot share their portions are PARALLEL, i.e. have no common divisor, multiply both adjusts, then multiply the answer by the base number. This gives the new base number

Example 30

Heirs 6 Full sister 4 Uterine brothers Mother
Shares 2/3 1/3 1/6
Base number 6
Portions 4 2 1

Is there anything intriguing in this example? Probably not obvious. OK, take some time to add up the number of portions. 4 + 2 + 1 = 7. This is greater than the base number. So, what comes to mind? ‘Awl (increment of base number)! But as stated earlier, even if base number is increased, the number of portions of each category of heir is not affected. Hence, 6 full sisters cannot share 4 portions; similarly, 4 uterine brothers cannot share 2 portions. Mother has no problem.

Adjust of full sisters = 6 ÷ 2 = 3 (Common divisor of 6 and 4 is 2).
Adjust of uterine brothers = 4 ÷ 2 = 2 (Common divisor of 4 and 2 is 2).
The adjusts, 3 and 2 are parallel.

New base number = 3 × 2 = 6 × 7 = 42 (Observe that the base number was increased from 6 to 7).

IMPORTANT

Whenever the base number is increased, original shares are not used to determine new portions. New shares are “created” for each category such that the NUMBER OF PORTIONS serves as the numerator while the denominator is the INCREASED BASE NUMBER. Therefore,

New share of 6 full sisters = 4/7
New share of 4 uterine brothers = 2/7
New share of mother = 1/7. Consequently,

New portion of 6 full sisters: 42 × 4/7 = 24
New portion of 4 uterine brothers: 42 × 2/7 = 12
New portion of mother: 42 × 1/7 = 6
Total number of portions: 24 + 12 + 6 = 42!

If the original shares (2/3, 1/3 and 1/6) were used, the total number of portions would have been 49. Confirm that please. As a result, the final table should look like this.

Heirs 6 Full sisters 4 Uterine brothers Mother
Original shares 2/3 1/3 1/6
Base number 6
Portions 4 2 1
Increased base number 7
New shares 4/7 2/7 1/7
New base number 42
New portions Each sister = 4 Each brother = 3 6
Rule V: When the ADJUSTS of both categories of heirs that cannot share their portions of the estate in turn CONVERGE, i.e. have a common divisor, divide any of the adjusts by their common divisor, multiply the solution by the other adjust. Finally multiply the answer by the by the base number. The result is the new base number

Example 31

Heirs 18 Uncles’ sons Mother 2 Uterine brothers;
6 Uterine sisters
Shares Residue 1/6 1/3
Base number 6
Portions 3 1 2
Number of heads 18 1 8
New base number 72
New portions Each son = 2 12 Each uterine = 3

Adjust of 18 full uncles’ sons = 18 ÷ 3 = 6
Adjust of 2 uterine brothers and 6 uterine sisters = 8 ÷ 2 = 4
(The background explanations have been skipped. It is assumed that by now, the reader is conversant with the procedure).
But the adjusts, 6 and 4 converge. Their common divisor is 2. Accordingly,

New base number = 6 ÷ 2 = 3 × 4 = 12 × 6 = 72
Alternatively, 4 ÷ 2 = 2 × 6 = 12 × 6 = 72
New portion of Mother: 72 × 1/6 = 12
New portion of 2 uterine brothers and 6 uterine sisters: 72 × 1/3 = 24
New portion of 18 uncles’ sons: 72 – (12 + 24) = 36
Note that the portions of mother and the uterines have to be determined first before knowing what the residue will be.

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