Question

The wealth of a person A equals the sum of that of 8 and C. If he distributes half of his wealth between B and C in the ratio 2 : 1 then the wealth of B equals the sum of that of A and C. Then the fraction of the wealth that A should distribute between B and C in the ratio1:2 so that the wealth of C equals the sum of thatof A and B is

• A. 1/2
• B. 2/3
• C. 3/4
• D. 1

Answer 1

The correct option is D 1

Let wealth A, B, C are x, y, z
Given A=B + C
x = y + z ... (i)
A distribute half of his wealth to B & C
$\left(\frac{x}{2}\right)\to 2:1$ i.e. $\frac{2}{3}$ & $\frac{1}{3}$
Now, $A=\frac{x}{2}$
$B=y+\frac{2}{3}\left(\frac{x}{2}\right)$
$C = \displaystyle z + \frac{1}{3} \left ( \frac{x}{2} \right )$
Now, B = A + C
$y+\frac{x}{3}=\frac{x}{2}+z+\frac{x}{6}$
$y-z=\frac{x}{3}$
$3y-3z=x$ ............. (ii)
Let t be fraction that A should distribute and the ratio of distribution is 1 : 2. $\frac{1}{3}<\frac{2}{3}$
Now, $A=(1-t)x$
$B=y+\frac{tx}{3}$
$C=z+\frac{2tx}{3}$
$$\displaystyle z + \frac{2tx}{3} = y + \frac{tx}{3} + (1 - t) x$
$z+\frac{tx}{3}=y+x-tx$
$3z+tx=3y+3x-3tx$
$4tx=3(y-z)+3x$
$4tx=3x+x$
$4tx=4x$
$t=1$
So, A would distribute his whole wealth to B and C.
$\therefore$ Fraction is 1.