Question

The ages of saleem and simon are in the ratio 3:4, four years ago their ages were in the ratio 5:7. Find their ages.-

Answer 1

$\textbf{\underline{\underline{Answer :-}}}$

$\text{Saleem is 24 years old}$

$\text{Simon is 32 years old}$

$\textbf{\underline{\underline{Explanation :-}}}$

$\textsf{\underline{\underline{Given :}}}$

Ages of Saleem and Simon = 3:4

Four years ago ratio of ages = 5:7

$\textsf{\underline{\underline{To find :}}}$

The present ages

$\textsf{\underline{\underline{Solution :}}}$

Consider Ratio of present ages = 3x:4x

Consider Saleem's present age as 3x

Consider Simon's present age as 4x

Corresponding fraction of present ages of Saleem and Simon = $\tt{ \dfrac{3x}{4x}}$

So, Equation :

$\sf{\implies \dfrac{3x - 4}{4x - 4} = \dfrac{5}{7}}$

$\sf{\implies7(3x - 4) = 5(4x - 4)}$

$\sf{\implies21x - 28 = 20x - 20}$

$\sf{\implies21x - 20x = - 20 + 28}$

$\sf{\implies \: x = 8}$

${\boxed{\bigstar{\sf\:{x = 8}}}}$

Value of 3x =

$\sf{\implies3 \times 8}$

$\sf{\implies24}$

${\boxed{\bigstar{\sf\:{Saleem \: is \: 24 \: years \: old}}}}$

Value of 4x =

$\sf{\implies4 \times 8}$

$\sf{\implies32}$

${\boxed{\bigstar{\sf\:{Simon\: is \: 32 \: years \: old}}}}$

$\therefore$$\text{Saleem is 24 years old}$

$\text{Simon is 32 years old}$

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

$\sf{\implies \dfrac{(3 \times 8) - 4}{(4 \times 8) - 4} = \dfrac{5}{7}}$

$\sf{\implies \dfrac{24- 4}{32 - 4} = \dfrac{5}{7}}$

$\sf{\implies \dfrac{20}{28} = \dfrac{5}{7}}$

Reduce the numbers 20 and 28 by 4

$\sf{\implies \dfrac{5}{7} = \dfrac{5}{7}}$

${\boxed{\boxed{\sf{\dfrac{5}{7} = \dfrac{5}{7}}}}}$

${\boxed{{\sf\:{LHS = RHS}}}}$

$\therefore$$\text{Saleem is 24 years old}$

$\text{Simon is 32 years old}$

Answer 2

$\sf{\Large {\underline {LINEAR\:EQUATIONS}}}$

Let the present age of Saleem be x and Simon be y years.

Ratio of their ages = 3 : 4

=>$\frac{3} {4}$ = $\frac{x} {y}$

=> 3y = 4x

=> 3y - 4x = 0 ---> ( i )

=> x =$\frac{3y} {4}$

Four years ago,

Age of Saleem = ( x - 4 ) years.

Age of Simon = ( y - 4 ) years.

Ratio of their ages four years ago = 5:7

=>$\frac{( \:x \:-\: 4\:)} {(\: y\: -\: 4\: )}$ = $\frac{5} {7}$

=> 7( x - 4) = 5( y - 4 )

=> 7x - 28 = 5y - 20

=> 7x - 5y = - 20 + 28

=> 7x - 5y = 8 ---> ( ii )

Putting value of x in equation ( ii ),

7( $\frac{3y} {4}$) - 5y = 8

$\frac{21y} {4}$ - 5y = 8

21y - 20y = 8 × 4

$\fbox{y\: = \:32\: years.}$

Putting value of 'y' in equation of 'x',

x = $\frac{3y}{4}$ = 3 × $\frac{32} {4}$

$\fbox{x\: =\: 24\: years.}$

$\sf{\underline{Saleem's \:age\:=\:24\:years.}}$

$\sf{\underline{Simon's \:age\:=\:32\:years.}}$