The income of A is $$\frac{2}{3}$$ of B's income and the expenditure of A is $$\frac{3}{4}$$ of B's expenditure. If $$\frac{1}{3}$$ of the income of B is equal to the expenditure of A, then the ratio of the savings of A to those of B is:

Solution(By Examveda Team)

Income of A is equal to $$\frac{2}{3}$$ of income of B.
$$\eqalign{ & A = B \times \frac{2}{3} \cr & \frac{A}{B} = \frac{{2x}}{{3x}} \cr} $$
A's expenditure is equal to $$\frac{3}{4}$$ B's expenditure
$$\eqalign{ & A = B \times \frac{3}{4} \cr & \frac{A}{B} = \frac{{3y}}{{4y}} \cr} $$
A's income is equal to $$\frac{1}{3}$$ B's income
3x × $$\frac{1}{3}$$ = 3y
x = 3y
Saving ratio of A and B
= (2x - 3y) : (3x - 4y)   [∵ x = 3y]
= (6y - 3y) : (9y - 4y)
= 3y : 5y
= 3 : 5

Alternate solution
\[\begin{array}{*{20}{c}} {}&{{\text{A}}\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{B}}} \\ {{\text{Income}} \to }&{\,\,\,\,\,\,\,{2_{ \times 3 = 6}}\,\,\,{3_{ \times 3 = 9}}} \\ {{\text{Expenditure}} \to }&{3\,\,\,\,\,\,\,\,\,\,\,\,\,\,4} \\ {{\text{Saving}} \to }&{\overline {\underline {\,\,3\,\,\,\,\,:\,\,\,\,\,5\,\,} } } \end{array}\]