SOLUTION: a fraction is such if both the denominator and the numerator are increased by 2/3,it becomes 17/20, if the numerator and the denominator are decreased by 1/2 then it becomes 9/11

Question 1072862: a fraction is such if both the denominator and the numerator are increased by 2/3,it becomes 17/20, if the numerator and the denominator are decreased by 1/2 then it becomes 9/11 find the original fraction.

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Let's say the original fraction is , where
= numerator of the original fraction
= denominator of the original fraction
The problem says that
and
Believe or not, that is a system of linear equations.
You can transform those equations into the equivalent equations
and
and the solve the very ugly system.
That system has exactly one solution,
the one pair that satisfies both equations.
If this were real life, you probably would have to solve the ugly system,
and find ugly solutions.
In math problems, you should expect to see signs of intelligent design,
or at least human design, so there has to be a nice answer.
You can also look the problem from another angle
<-->
is the equation of a line with slope
that passes through point (1/2,1/2) and an infinite number of points (x,y).
With that slope, only a small fraction of those points will have positive integer coordinates.
<-->
has the obvious nice solution
-->--> .
There are many other integer solutions,
but you would not expect them to make such a pretty pair.
Similarly,
<-->
has the obvious nice solution
-->--> .
The original fraction is .
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