"The Legacy" Problem

[zenith20]

Another puzzle from Shakuntala Devi, i wouldnt understand the solution:

"The Legacy"
When my uncle in Madura died recently, he left a will, instructing his
executors to dividehis estate of
Rs 1,920,000 in this manner:
Every son sould receive three times as much as a daughter, and that every
daughter should get
twice as much as their mother. What is my aunt's share?

the answer is:
Rs. 49200(10/13)

-----------------------------------
my problem is i can't get the answer! cause there's not cleared the number
of each son/daughter.
could you please explain it a bit more to me?

 

[Deleted member 4993]

Another puzzle from Shakuntala Devi, i wouldnt understand the solution:

"The Legacy"
When my uncle in Madura died recently, he left a will, instructing his
executors to dividehis estate of
Rs 1,920,000 in this manner:
Every son sould receive three times as much as a daughter, and that every
daughter should get
twice as much as their mother. What is my aunt's share?

the answer is:
Rs. 49200(10/13)

-----------------------------------
my problem is i can't get the answer! cause there's not cleared the number
of each son/daughter.
could you please explain it a bit more to me?

Does the answer look something like:

$\displaystyle 49200\frac{10}{13}$

.

 

[zenith20]

Does the answer look something like:

$\displaystyle 49200\frac{10}{13}$

.

Yes, Subhotosh Khan, That's the answer, and i can't get it

 

[soroban]

Hello, zenith20!

Sorry, I don't get it either . . . Is there some missing information?

The Legacy: When my uncle in Madura died recently, he left a will
instructing his executors to divide his estate of $1,920,000 in this manner:

Every son should receive three times as much as a daughter,
and that every daughter should get twice as much as their mother.

What is my aunt's share?

The answer is: .$\displaystyle 49200\tfrac{10}{13}$ . . . . How?

Let: mother's share = $\displaystyle x$ dollars.

Then each daughter gets $\displaystyle 2x$ dollars.

And each son gets $\displaystyle 6x$ dollars.


If there are $\displaystyle S$ sons, they get a total of $\displaystyle 6Sx$ dollars.

If there are $\displaystyle D$ daughters, they get a total of $\displaystyle 2Dx$ dolars.


So we have: .$\displaystyle 6Sx + 2Dx + x \:=\:1,920,000$

. . . That is: .$\displaystyle (6S + 2D + 1)x \;=\;1,920,000$ .[1]



At first, I assumed there is an integer solution,

On the left side of [1], note that $\displaystyle (6S+2D+1)$ is an odd integer.

On the right side: .$\displaystyle 1,920,000 \:=\:2^{10}\cdot3\cdot5^4$
. . Hence, the odd factors are: .$\displaystyle 1, 3, 5, 15, 25, 75, 125, 375, 625, 1875$


For most of the factors, there are multiple solutions.

For example: .$\displaystyle 6S + 2D + 1 \:=\:15 \quad\Rightarrow\quad 6S + 2D \:=\:14 \quad\Rightarrow\quad 3S + D \:=\:7$

. . . .Hence: .$\displaystyle (S,D) \;=\;(0,7),\;(1,4),\;(2,1)$

So that approach is a dead end . . .



$\displaystyle \text{Then we are told that: }\;x \;=\;49200\tfrac{10}{13} \;=\; \frac{639,610}{13}$ . impossible!


Substitute into [1]:

. . $\displaystyle (6S + 2D+1)\cdot\frac{639,610}{13} \;=\;1,920,000 \quad\Rightarrow\quad 6S + 2D + 1 \;=\;\frac{13}{639,610}(1,920,000)$


\(\displaystyle \text{And we have: }\;\underbrace{6S + 2D + 1}_{\text{This is an integer}} \;=\;\underbrace{39.02378012\hdots}_{\text{This is not!}}\)


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


Suppose there is a typo in the given answer.

$\displaystyle \text{Suppose it was actually: }\;49230\tfrac{10}{13} \:=\:\frac{640,000}{13}$


Substitute into [1]:

. . $\displaystyle (6S + 2D+1)\cdot\frac{640,000}{13} \;=\;1,920,000 \quad\Rightarrow\quad 6S + 2D + 1 \:=\:39$

Then we have: .$\displaystyle 6S + 2D \:=\:38 \quad\Rightarrow\quad 3S + D \:=\:19$


Then the number of sons and daughters would be:

. . \(\displaystyle (S,D) \;=\;(1,16),\2,13),\3,10),\4,7).\5,4),\6,1)\)


But why would we choose any of those?

 

[zenith20]

Dear Soroban,

Thanks for your kindness, i also think there's something missing.

most of Shakuntala's solution to her Puzzles are in brief, and we cannot get if it's kinda misspelling or not!

thank you anyway