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Question

A fraction is such that if the numerator is multiplied by 3 and the denominator is reduced by 3, we get 811, but if the numerator is increased by 8 and the denominator is doubled, we get 25. Find the fraction.

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Solution

Let the fraction be xy. According to the given conditions, we have
3xy3=1811 and x+82y=23
11x=6y18 and 5x+40=4y
So, we have
11x6y+18=0 (1)
5x4y+40=0 (2)
On comparing the coefficients of (1) and (2) with a1x+b1y+c1=0,a2x+b2y+c2=0
we have a1=11,b1=6,c1=18;a2=5,b2=4,c2=40
Thus, A1b2a2b1=(11)(4)(5)(6)=140.
Hence, the system has a unique solution. Now, writing the coefficients for the cross multiplication, we have
x240+72=y90440=144+30
x168=y350=114
Thus,
x=16814=12;y=35014=25.
Hence, the fraction is 1225.


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