Question

Present age of rama and shyam are in the ratio of 4:5, respectively. 5 yrs hence the ratio of their ages become 5:6 respectively. What is ramas present age?

Answer 1

$\boxed{\huge{\underline{\red{Answer}}}}$


$\bold{Let \:present \:age \:of \:Rama \:be \:x}$

And

$\bold{Present \:age \:of \:Shyam \:be\: y}$


$\underline{A.T.Q.}$


$\underline{\underline{Case \:I }}$


$Present\: age\: of \:Rama \:and \:Shyam \:are\: in$ $the\: ratio\: 4:5$

$\dfrac{x}{y}$ = $\dfrac{4}{5}$

x = $\dfrac{4y}{5}$ ___ eq.(1)


$\underline{\underline{Case \:ll}}$


$5\: years\: hence\: the \:ratio \:of\: their \:ages$ $become \:5:6$

$\dfrac{x\:+\:5}{y\:+\:5}$ = $\dfrac{5}{6}$

Cross-multiply them

6(x + 5) = 5(y + 5)

6x + 30 = 5y + 25

6x - 5y = 25 - 30

6x - 5y = - 5

$6 \times \dfrac{4y}{5}$ - 5y = - 5

$\dfrac{24y \:-\:25y}{5}$ = - 5

- y = - 25

$\boxed{y\: =\: 25}$

Put value of y in eq.(1)

x = $\dfrac{4(25)}{5}$

x = $\dfrac{100}{5}$

$\boxed{x\:=\:20}$

So;

$\bold{Present \:age \:of \:Rama \:be \:x = \:20 \:years}$

And

$\bold{Present \:age \:of \:Shyam \:be\: y\:=\:25\:years}$

Answer 2

Let the age of Rama be r and that of Shayam be s

Given that the ratio of their present age is 4 : 5

$\implies \sf r : s :: 4 : 5$

$\implies \sf 5r = 4s\\\\ \implies r = \frac{4}{5}s$

As, product of extremes = product of means.

Now, after 5 years age of Ram will be (r + 5), and that of Shyam will be (s + 5)

Now, their ratio is given as :-

$\implies\sf (r + 5) : (s + 5) :: 5 : 6$

$\implies\sf 6(r + 5) = 5(s + 5) \\ \\ \implies 6r + 30 = 5s + 25 \\ \\ \implies 5s - 6r = 30 - 25 \\ \\ \implies 5s - 6r = 5$

Now, we have concluded that $\sf r = \frac{4s}{5}$

So, Let us substitute the value and solve the equation.

$\sf 5s - 6r = 5 \\ \\ \implies 5s - 6\left(\frac{4s}{5}\right)= 5 \\ \\ \implies 5s - \frac{24s}{5} = 5 \\ \\ \implies \frac{25s - 24s}{5} = 5 \\ \\ \implies \frac{s}{5} = 5$

$\implies \huge{s = 25}$

(By cross multiplication)

So, now,

$\sf r = \frac{4s}{5} \\ \\ \implies r = \frac{4 \times 25}{5}$

$\implies\huge{r = 20}$

$\huge\mathfrak{Answer}$

$\textsf{Ram's age = 20 years}$

$\textsf{Shyam's age = 25 years}$