Question

The ratio of daily income of X and Y is 3:4 and the ratio of their daily expenses is 9:11 and ratio of their savings is 2:3. If their total saving per day is Rs 75, find their daily income.
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Answer 1

$\sf\large\underline\green{Given:}$

$\rm{\implies Ratio\:of\:x\:and\:y\:_{(incomes)}=3:4}$

$\rm{\implies Ratio\:of\:x\:and\:y\:_{(expenses)}=9:11}$

$\rm{\implies Ratio\:of\:x\:and\:y\:_{(savings)}=2:3}$

$\rm{\implies x\: and\:y\:_{(savings)}=Rs.75\:per\:day}$

$\sf\large\underline\green{To\: Find:}$

$\rm{\implies Daily\: income\:of\:x\:and\:y=?}$

$\sf\large\underline\green{Solution:}$

• At first calculate the savings of x and y by assuming their savings be 2x and 3x. After we have to set up equation. Before setting up equation we have to also assume that their daily income be x and their daily expenses be y then solve the equation:]

$\rm{\implies Savings\:_{(x+y)}=75}$

$\rm{\implies 2x+3x=75}$

$\rm{\implies 5x=75\implies x=15}$

• Now calculate their savings separately here:]

$\rm{\implies Savings\:of\:(x)=2x=2*15=Rs.30}$

$\rm{\implies Savings\:of\:(y)=3x=3*15=45}$

• Then setting up equation here with the help of given clue in the question. As above we assume their income be 3x and 4x and their expenses be 9y and 11y:]

$\sf\large\underline\red{Calculation\:for\:(x):}$

$\rm{\implies Income- expenses=savings}$

$\tt{\implies 3x-9y=30-----(i)}$

$\sf\large\underline\purple{Calculation\:for\:(y):}$

$\rm{\implies Income- expenses=savings}$

$\tt{\implies 4x-11y=45-----(ii)}$

• In eq (i) multiplying by 4 and in eq (ii) multiplying by 3 then subract to eachother:]

$\tt{\implies 12x-36y=120}$

$\tt{\implies 12x-33y=135}$

• By solving we get, here

$\tt{\implies -3y=-15\implies y=5}$

• Now putting the value of y=5 in eq (I)

$\tt{\implies 3x-9y=30}$

$\tt{\implies 3x-4*9=30}$

$\tt{\implies 3x-45=30}$

$\tt{\implies 3x=30+45}$

$\tt{\implies 3x=75\implies x=25}$

$\tt\huge{Hence,}$

$\sf\purple{\implies Daily\: income\:(x)=3x=3*25=Rs.75}$

$\sf\blue{\implies Daily\: income\:(y)=4x=4*25=Rs.100}$

Answer 2

$\bf\bold\green{\underline{\underline{✧ Question :- }}}$

• The ratio of daily income of X and Y is 3:4 and the ratio of their daily expenses is 9:11 and ratio of their savings is 2:3. If their total saving per day is Rs 75, find their daily income.

$\bf\bold\purple{\underline{\underline{✧ Given :- }}}$

Ratio of :-

• Income/day for X : Y = 3:4
• Expenses/day of X : Y = 9:11
• Savings/day of X : Y = 2:3
• Savings/day of X and Y = ₹75.

$\bf\bold\blue{\underline{\underline{✧ To \: find :- }}}$

• Daily income of X and Y i.e., ??

$\bf\bold\pink{\underline{\underline{✧ Solution:- }}}$

• Let's take the savings as 2a and 3a.
• According to the question, the savings and the and it's ratio is equal.

Hence, we can write it as :-

$\bf \implies \: 2a + 3a = 75 \\ \bf \implies \: 5a = 75 \\ \bf(dividing \: both \: side \: by \: 5)\\ \bf \therefore \: a = 15$

$\bf\bold\purple{\underline{\underline{✯ Now,:- }}}$

As per as calculation :-

Savings :-

$\bf \bold \orange{\underline{income \: - expenses = { \purple{ \: savings}}}}$

$\bf \: For \: X \implies \: 3x - 9y = 30 \:\:\:\:\:\:\:.....(i)\\ \bf \: For \: Y\implies \: 4x - 11y = 45\:\:\:\:\:......(ii)$

Mutiplying equation (i) and (ii) by 4 and 3 respectively.

We get $\bf \implies \: 12x - 36y = 120 \\ \bf \implies \: 12x - 33y = 135$

✯ Subtraction of both equations:-

$\bf \: \cancel{12x} - 36y = 120 \\ \bf \: {\underline{ \cancel{12x} - 39y = 135}} \\ \bf \: - 3y = - 15 \\ \bf \: \therefore y = 5$

✯ Putting the value of 'y' on eq. (i) :-

$\bf \implies 3x - 9y = 30 \\ \bf \implies 3x - (9 \times 5) = 30 \\ \bf \implies\: 3x - 45 = 30 \: \\ \bf \implies3x = 75 \\ \bf \therefore \: x = 25$

$\huge\bold{ \mathfrak { \underline{ \purple A \pink \blue n \green s \red w \pink e \orange r :-}}}$

• $\rm\bold{\underline{Daily\: income\: of \:X\: is\:3x \:= \:3 \:× \:25\: = \:₹75.}}$
• $\rm\bold{\underline{Daily\: income \:of \:Y\: is \:4x \:=\: 4 \:× \:25 \:= \:₹100.}}$