Question

the ratio of the incomes of Amit and Rajiv is 3 : 5 and the ratio of their expenditures is 5:9 if each person saved rupees 5000 then their income of Rajiv is

Answer 1

$\large\bf\underline{Given:-}$

• Ratio of income of Amit and Rajiv is 3:5
• Ratio of their expenditures is 5:9
• each of them saves 5000

$\large\bf\underline {To \: find:-}$

• Income of Rajiv

$\huge\bf\underline{Solution:-}$

Let the income of Amit and Rajeev be 3x and 5x

Let their expenditures be 5y and 9y.

We know that ,

(income - expenditures) = savings.

Savings of Amit

$\leadsto \rm \: 3x - 5y = 5000 \: ....(i)$

Savings of Rajiv

$\leadsto \rm \: 5x - 9y = 5000.........(ii)$

Multiply eq. (i) by 5 and eq. (ii) by 3

we get,

$\rm \: (3x - 5y = 5000) \times 5 \\ \rm \:( 5x - 9y = 5000) \times 3 \\ \: \\ \rm \:15x - 25y = 25000........(iii) \\\rm \: 15x - 27y = 15000..........(iv)$

Solving eq. (iii) and (iv)

$\rm \:15x - 25y = 25000 \\ \rm \:15x - 27y = 15000 \\ \: \: - \: \: \: \: \: \: \: \: + \: \: \: \: \: \: \: \: \: \: - \\ \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \rm \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2y = 10000 \\ \rm \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: y = \cancel\dfrac{10000}{2} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf \: y = 5000$

putting value of y in eq. (iv)

$\longmapsto \rm \: 15x - 27y = 15000 \\ \longmapsto \rm \:15x - 27(5000) = 15000 \\ \longmapsto \rm \:15x - 135000 = 15000 \\ \longmapsto \rm \:15x = 15000 + 135000 \\ \longmapsto \rm \:15x = 150000 \\ \longmapsto \rm \:x = \cancel\dfrac{150000}{15} \\ \longmapsto \bf \:x = 10000$

So, Rajiv's income = 5x = 10000×5

Rajiv's income = Rs 50000

Another method :-

we know that,

Savings = (income - expenditures)

Then,

Income = savings + expenditures.

expenditures of Rajiv = 9y = 9×5000

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 45000

Rajiv's income = 5000 + 45000

Rajiv's income = 50000

Answer 2

Answer:

⋆ DIAGRAM :

$\boxed{\setlength{\unitlength}{.8mm}\begin{picture}(40,22)\thicklines\put(25,15){\vector(1,-1){10}}\put(15,15){\vector(-1,-1){10}}\put(9,18){\sf\large\star \:\underline{Income}}\put(0,1){\sf{Expense}}\put(18,1){+}\put(25,1){\sf{Saving}}\end{picture}}\quad\boxed{\begin{minipage}{5.2 cm}\qquad\qquad\quad\bf{Amit -}\qquad\:\bf{Rajiv -}\\\\\sf Income\quad\:\!= 3\qquad\quad:\quad\qquad5\\Expenses = 5\qquad\quad:\quad\qquad9\\Savings\quad=\:Rs.\:5000\qquad Rs.\:5000\end{minipage}}$

⠀⠀⠀$\rule{160}{0.8}$

Let the Income of Amit and Rajiv be 3x and 5x respectively.

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

$:\implies\sf \dfrac{Amit_{(Income)}-Saving}{Rajiv_{(Income)}-Saving}=\dfrac{Amit_{(Expenditure)}}{Rajiv_{(Expenditure)}}\\\\\\:\implies\sf \dfrac{3x - 5000}{5x - 5000} = \dfrac{5}{9}\\\\\\:\implies\sf (3x - 5000) \times 9 = 5 \times (5x - 5000)\\\\\\:\implies\sf 27x - 45000 = 25x - 25000\\\\\\:\implies\sf 27x - 25x = 45000 - 25000\\\\\\:\implies\sf 2x = 20000\\\\\\:\implies\sf x = \dfrac{20000}{2}\\\\\\:\implies\sf x = Rs.\:10000$

$\rule{150}{1.5}$

☯ $\underline{\boldsymbol{Income\: of \:Rajiv :}}$

$\dashrightarrow\sf\:\:Rajiv_{(Income)}=5x\\\\\\\dashrightarrow\sf\:\:Rajiv_{(Income)}=5 \times Rs.\:10000\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf Rajiv_{(Income)}=Rs.\:50000}}$

⠀

$\therefore\:\underline{\textsf{Hence, Income of Rajiv is \textbf{Rs. 50,000}}}.$