Question

One ninth of an estate is left to a sister, one- fourth to a son, one- sixth to a brother and the remainder, which is $575000$ to the widow. What is the value of the estate?

Answer 1

Hint: Here we can simply find the fraction in terms of assumed value of estate and then by adding the shares and subtracting resultant from the given remaining share.

Complete step-by-step answer: Let the value of the estate be $x$
One-ninth of an estate= $\dfrac{1}{9} \times (x) = \dfrac{x}{9}$
$\therefore $So share left to sister =$\dfrac{x}{9}$
One-fourth of an estate=$\dfrac{1}{4} \times (x) = \dfrac{x}{4}$
$\therefore $So share left to son=$\dfrac{x}{4}$
One- sixth of an estate=$\dfrac{1}{6} \times (x) = \dfrac{x}{6}$
$\therefore $So share left to brother=$\dfrac{x}{6}$
$\therefore $The total share of sister, son and brother =$\dfrac{x}{9} + \dfrac{x}{4} + \dfrac{x}{6}$
Taking LCM of (9, 4, 6) which is 36
$ \Rightarrow \dfrac{{4x + 9x + 6x}}{{36}} = \dfrac{{19x}}{{36}}$
Remaining estate=$x - \dfrac{{19x}}{{36}} = \dfrac{{36x - 19x}}{{36}}$
$\therefore $Remaining estate=$\dfrac{{17x}}{{36}}$
Also it is given that remaining estate 575000 is given to a widow so we will equate our solved remaining estate with the given remaining estate.
$ \Rightarrow \dfrac{{17x}}{{36}} = 575000$
$ \Rightarrow x = \dfrac{{575000 \times 36}}{{17}}$
$\therefore x = 1217647$
$\therefore $The final value of estate is 1217647


Note: Alternative method: -Here in this question we can also apply the percentage method to find out the value of the estate.
 For example: - If we calculate half value’s percentage then it will come as follows:-
$ \Rightarrow \dfrac{1}{2} \times 100 = 50\% $ (So from $100\% $if one half is deducted then half will be left i.e.$50\% $)
Let the total estate be $100\% $
One-ninth of an estate=$\dfrac{1}{9} \times 100\% = 11.11\% $
One-fourth of an estate=$\dfrac{1}{4} \times 100\% = 25\% $
One- sixth of an estate=$\dfrac{1}{6} \times 100\% = 16.67\% $
Total share=$11.11\% + 25\% + 16.67\% = 52.78\% $
Remaining share=$100\% - 52.78\% = 47.22\% $
$ \Rightarrow 47.22\% = 575000$
$1\% = \dfrac{{575000}}{{47.22}} = 12177.04$
$100\% = 1217704$
$\therefore $The final estate value is 1217704
Since it is an approximation method so try to use only when options are given and value closed to the answer should be considered otherwise the first method is valid for all such types of questions. The difference is only due to the numbers after decimal which we have approximated because they were coming repetitive.