Question

the ratio of sum and difference of present age of vivek andarvind is 2:1 . four yrs ago this ratio was 3:2 . what will be the ratio of there ages after 12 yrs​

Answer 1

Answer:

Let the present ages of Vivek and Arvind be 'x' and 'y' respectively.

The ratio of sum and difference of present ages is 2 : 1.

The ratio of sum and difference of ages 4 years ago was 3 : 2.

Sum of present ages = x + y

Difference of present ages = x - y

Sum of ages 4 years ago = x - 4 + y - 4 ⇒ x + y - 8

Difference of ages 4 years ago = x - 4 - ( y - 4 )

Difference of ages 4 years ago ⇒ x - y - 4 + 4 ⇒ x - y

Substituting the values according to the question we get:

$\implies \dfrac{x+y}{x-y} = \dfrac{2}{1}\\\\\\\text{Cross multiplying we get:}\\\\\\\implies x + y = 2 ( x - y )\\\\\implies x + y = 2x - 2y\\\\\implies x - 3y = 0 \:\:\: ...(i)$

Using the second condition we get:

$\dfrac{x+y-8}{x-y} = \dfrac{3}{2}\\\\\\\text{Cross multiplying we get:}\\\\\\\implies 2 ( x + y - 8 ) = 3 ( x - y )\\\\\implies 2x + 2y - 16 = 3x - 3y\\\\\implies x - 5y = -16\\\\\implies x = 5y - 16\:\:\: ...(ii)$

Substituting the value of 'x' from (ii) in (i) we get:

⇒ x - 3y = 0

⇒ 5y - 16 - 3y = 0

⇒ 2y - 16 = 0

⇒ 2y = 16

⇒ y = 16/2

⇒ y = 8

Hence the present age of Arvind is 8 years.

Substituting value of 'y' in (ii) we get:

⇒ x = 5y - 16

⇒ x = 5 ( 8 ) - 16

⇒ x = 40 - 16

⇒ x = 24 years

Hence the present age of Vivek is 24 years.

Answer 2

Step-by-step explanation:

Question:

The ratio of sum and difference of present age of Vivek and Arvind is 2:1 . Four years ago this ratio was 3:2 . What will be the ratio of their ages after 12 years​?

Given:

• The ratio of sum and difference of present age of Vivek and Arvind is 2:1

• Four years ago this ratio was 3:2

To find:

• Ratio of their ages after 12 years​

Solution:

• Let x and y be the ages of Vivek and Arvind respectively.

$\begin{gathered}\bf{Sum\:of\:their\:ages = \: x+y} \\ \\ \bf{Difference\:in\:their\:present\:age = \: x-y} \end{gathered}$

Note:

• 4 years ago, Vivek's age will be x-4
• 4 years ago, Arvind's age will be y-4

$\begin{gathered}\pmb{Sum\:of\:their\:age\:4\:years\:ago=\:x-4+y-4} \\ \\ \:\:\: \implies\bf{\pmb{x+y-8}} \\ \\ \bf{Difference\: in\: their \:ages = x-4-(y-4)} \\ \\ \: \: \implies\bf{x-4-y+4} \\ \\ \implies\bf{x-y}\end{gathered}$

$\sf\dfrac{x+y}{x-y}$ = $\sf\dfrac{2}{1}$

Cross multiply:

$\begin{gathered}\implies\sf{x+y=2(x-y)} \\ \\ \implies\sf{x+y=2x-2y} \\ \\ \implies\sf{x-3y=0...(1)} \\ \\ \bf{And\:also,} \\ \\ \sf\dfrac{x+y-8}{x-y} = \sf\dfrac{3}{2} \\ \\ \implies\sf{2(x+y-8)=3(x-y) \\ \\ \: \: \: \: \implies\sf{2x+2y-16=3x-3y} \\ \\ \implies\sf{x=5y-16...(2)}\end{gathered}$

Substitute the x value from (2) in (1):

$\begin{gathered}\sf{ x - 3y = 0} \\ \\ \implies\sf{ 5y - 16 - 3y = 0} \\ \\ \implies\sf{2y - 16 = 0} \\ \\ \:\implies\sf{2y = 16} \\ \\ \: \implies\sf{y = \dfrac{16}{2}} \\ \\ \: \implies\sf{ y = 8}\end$

Substitute y value in (2):

$\begin{gathered}\tt{ x = 5y - 16x} \\ \\ \implies\tt{ 5( 8 ) - 16x} \\ \\ \: \implies\tt{ 40 - 16x = 24 \:}\end$

So, the present age of Vivek: = 24 years

The present age of Arvind: = 8 years

After 12 years,

Vivek"s Age = 24+12 = 36 years

Arvind"s Age = 8+12 = 20 years

So, their ratio after 12 years is:

$\implies\sf\dfrac{36}{20}$ = $\sf\dfrac{9}{5}$ = 9:5