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Question
The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine 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 denominator of the fraction is 4 more than twice the numerator. Thus, we have
`x+y =2x+4`
`⇒ 2x +4 -x -y=0`
`⇒ x - y + 4=0`
If the numerator and denominator are increased by 3, they are in the ratio 2:3. Thus, we have
` x+ 3 : y+ 3 = 2:3`
`⇒ (x+3)/(y+3)=2/3`
`⇒ 3(x+3)=2/3`
`⇒ 3x + 9 =2y+6`
`⇒ 3x -2y +3=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/((-1)xx3-(-2)xx4)=(-y)/(1xx3-3xx4)=(1)/(1xx(-2)-3xx(-1))`
`⇒ x/(-3+8)=(-y)/(3-12)=1/(-2+3)`
`⇒ x/5=(-5)/(-9)=1/1`
`⇒ x/5 = y/9=1`
`⇒ x=5,y=9`
Hence, the fraction is `5/9`
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