Question

33
2. The present ages of Vijay and Gautam are in the ratio 2:3. Five years hence, their ages
will be in the ratio 3:4. Find their present ages.​

Answer 1

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

Present age of Vijay is 10 years, Present age of Gautam is 15 years

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

Given :- Ratio of present ages of Vijay and Gautam = 2 : 3

Ratio of ages of Vijay and Gautam 5 years hence (After) = 3 : 4

To find :- Their present ages

Solution :-

Ratio of present ages of Vijay and Gautam = 2 : 3

Let the present age of Vijay be 2x and Gautam present age be 3x

5 years hence

Vijay's age = 2x + 5

Gautam's age = 3x + 5

Ratio of ages of Vijay and Gautam 5 years hence (After) = 3 : 4

According to the question :-

We can form an equation

Equation formed :-

$\boxed{ \tt \dfrac{2x + 5}{3x + 5} = \dfrac{3}{4}}$

By Cross multiplication :-

$\tt (2x + 5)4 = 3(3x + 5)$

$\tt 8x + 20 = 9x + 15$

$\tt 8x - 9x = 15 - 20$

$\tt \cancel-x = \cancel - 5$

$\tt x = 5$

Present age of Vijay = 2x = 2(5) = 10 years

Present age of Gautam = 3x = 3(5) = 15 years

So present age of Vijay is 10 years, present age of Gautam is 15 years.

$\mathfrak{\large{\underline{\underline{Verification:-}}}}$

To check whether the answer is correct or not substitute their present ages in the equation that is formed to solve.

$\tt \dfrac{10 + 5}{15 + 5} = \dfrac{3}{4}$

$\tt \dfrac{15}{20} = \dfrac{3}{4}$

$\tt \dfrac{15 \div 5}{20 \div 5} = \dfrac{3}{4}$

$\tt \dfrac{3}{4} = \dfrac{3}{4}$

Answer 2

• $\textbf{Let present age of Vijay be 2M}$ and $\textbf{present age of Gautam be 3M.}$

» $\underline{\bold{Five\: years\: hence..}}$

• $\underline{Age \:of \:Vijay \:=\: 2M\: + \:5}$

and

• $\underline{Age\: of \:Gautam \:=\: 3M \:+\: 5}$

» After five years.. ratio of their (Vijay and Gautam) ages will be 3:4.

$\implies$ $\dfrac{2M\:+\:5}{3M\:+\:5}$ = $\dfrac{3}{4}$

• Cross-multiply them..

$\implies$ 4(2M + 5) = 3(3M + 5)

$\implies$ 8M + 20 = 9M + 15

$\implies$ 8M - 9M = 15 - 20

$\implies$ - M = - 5

$\implies$ M = 5

_____________________________

• We have to find the present ages of Vijay and Gautam.

So, put the value of M in their let ages.

Present age of Vijay = 2M

$\implies$ 2(5)

$\implies$ 10 years.

Similarly

Present age of Gautam = 3M

$\implies$ 3(5)

$\implies$ 15 years.

______________________________

$\textbf{Present age of Vijay is 10 years and}$

$\textbf{present age of Gautam is 15 years.}$

________________ [$\bold{ANSWER}$]

______________________________

✡ From above calculations we have value of M is 5.

Put value of M in $\dfrac{2M\:+\:5}{3M\:+\:5}$ = $\dfrac{3}{4}$ this equation.

$\implies$ $\dfrac{2(5)\:+\:5}{3(5)\:+\:5}$ = $\dfrac{3}{4}$

$\implies$ $\dfrac{10\:+\:5}{15\:+\:5}$ = $\dfrac{3}{4}$

$\implies$ $\dfrac{15}{20}$ = $\dfrac{3}{4}$

$\implies$ $\dfrac{3}{4}$ = $\dfrac{3}{4}$

____________ [HENCE VERIFIED]