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Question
The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.
Solution
Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is`x/y`
The sum of the numerator and the denominator of the fraction is `18`. Thus, we have
`x+y=18`
`⇒ x+y-18=0`
If the denominator is increased by 2, the fraction reduces to `1/3`. Thus, we have
`x/(y+2)=1/3`
`⇒ 3x = y+2`
`⇒ 3x -y -2 =0`
So, we have two equations
`x+y -18=0`
`3x -y -2=0`
Here x and y are unknowns. We have to solve the above equations for x and y.
By using cross-multiplication, we have
`x/(1xx(-2)-(-1)xx(-18))=-y/(1xx(-2)-3xx(-18))=1/(1xx(-1)-3xx1)`
`⇒ x/(-2-18)=(-y)/(-2+54)=1/(-1-3)`
`⇒ x/-20=(-y)/(52)=1/4`
`⇒ x/20 = y/52 =1/4`
`⇒ x =20/4,y=52/4`
`⇒ x =5 ,y =13`
Hence, the fraction is `5/13`
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