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प्रश्न
The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.
उत्तर
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 12. Thus, we have
`x+y=12`
`⇒ x+y-12=0`
If the denominator is increased by 3, the fraction becomes `1/2`. Thus, we have
`x/(y+3)=1/2`
`⇒ 2x=y+3`
`⇒ 2x -y -3=0`
So, we have two equations
`x+y-12=0`
`2x-y-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/(1xx(-3)-(-1)xx(-12))=(-y)/(1xx(-3)-2xx(-12))=1/(1xx(-1)-2xx1)`
`⇒ x/(-3-12)=(-y)/(-3+24)=1/(-1-2)`
`⇒ x/-15=(-y)/21=1/-3`
`⇒ x/15=y/21=1/3`
`⇒ x=15/3,y =21/3`
`⇒ x=5,y=7`
Hence, the fraction is `5/7`
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