A fraction becomes 9/11 , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.

Solution (i) :

Let the fraction be $x/y$ .

According to the question,

$\left( x+2 \right)\text{ }/\left( y+2 \right)\text{ }=\text{ }9/11$

$11x\text{ }+\text{ }22\text{ }=\text{ }9y\text{ }+\text{ }18$

$11x\text{ }\text{ }9y\text{ }=\text{ }-4\text{ }\ldots \ldots \ldots \ldots \ldots ..\text{ }\left( 1 \right)$

$\left( x+3 \right)\text{ }/\left( y+3 \right)\text{ }=\text{ }5/6$

$6x\text{ }+\text{ }18\text{ }=\text{ }5y\text{ }+15$

$6x\text{ }\text{ }5y\text{ }=\text{ }-3\text{ }\ldots \ldots \ldots \ldots \ldots \ldots .\text{ }\left( 2 \right)$

From (1), we get $x\text{ }=\text{ }\left( -4+9y \right)/11~\ldots \ldots \ldots \ldots \ldots ..\text{ }\left( 3 \right)$

Substituting the value of x in (2), we get

$6\left( -4+9y \right)/11\text{ }-5y\text{ }=\text{ }-3$

$-24\text{ }+\text{ }54y\text{ }\text{ }55y\text{ }=\text{ }-33$

$-y\text{ }=\text{ }-9$

$y\text{ }=\text{ }9\text{ }\ldots \ldots \ldots \ldots \ldots \ldots \ldots \text{ }\left( 4 \right)$

Substituting the value of y in (3), we get

$x\text{ }=\text{ }\left( -4+9\times 9\text{ } \right)/11\text{ }=\text{ }7$

Hence the fraction is 7/9.

(ii) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

Solutions:

Let the age of Jacob and his son be x and y respectively.

According to the question,

$\left( x\text{ }+\text{ }5 \right)\text{ }=\text{ }3\left( y\text{ }+\text{ }5 \right)$

$x\text{ }\text{ }3y\text{ }=\text{ }10\text{ }\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ..\text{ }\left( 1 \right)$

$\left( x\text{ }\text{ }5 \right)\text{ }=\text{ }7\left( y\text{ }\text{ }5 \right)$

$x\text{ }\text{ }7y\text{ }=\text{ }-30\text{ }\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\text{ }\left( 2 \right)$

From (1), we get $x\text{ }=\text{ }3y\text{ }+\text{ }10\text{ }\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\text{ }\left( 3 \right)$

Substituting the value of x in (2), we get

$3y\text{ }+\text{ }10\text{ }\text{ }7y\text{ }=\text{ }-30$

$-4y\text{ }=\text{ }-40$

$y\text{ }=\text{ }10\text{ }\ldots \ldots \ldots \ldots \ldots \ldots \ldots \text{ }\left( 4 \right)$

Substituting the value of y in (3), we get

$x\text{ }=\text{ }3\text{ }x\text{ }10\text{ }+\text{ }10\text{ }=\text{ }40$

Hence, the present age of Jacob and his son is 40 years and 10 years, respectively.