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One of my friends gave me this apparently easy-looking problem which I do not know how to crack. The problem is to find the values of "*" where

$$5 \frac {3}{*} \times 3 \frac {*}{2}=19\text{ ?}$$

I can rearrange the problem as $5 \frac {3}{x} \times 3 \frac {y}{2}=19$ and I have to find the values of $x,y.$ Now $5 \frac {3}{x} \times 3 \frac {y}{2}=19 \implies \frac {5x+3}{x} \times \frac {6+y}{2}=19.$ Now, I do not know which way to go?

Can someone point me in the right direction? Thanks in advance for your time.

EDIT: Here "*"-s are not same. Infact I know the answer but do not know how to get it. Here $x=7,y=1.$

Michael Hardy
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learner
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  • If $*$ is not the same, how about naming them just $x$ and $y$? – Asaf Karagila Mar 30 '13 at 10:12
  • The question was exactly what I posted.I did not know how to express it otherwise. – learner Mar 30 '13 at 10:17
  • You do know how to express it otherwise--that's what you did when you "rearranged" the problem. Inform your friend that (s)he shouldn't use one variable to mean two separate things at the same time. – Cameron Buie Mar 30 '13 at 10:25
  • It surely has been a mistake from his part. To eliminate the confusion about "*" I said in the question "find the values of..." – learner Mar 30 '13 at 10:34

2 Answers2

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I assume $x, y$ are supposed to be positive integers (otherwise there are infinitely many real solutions for $x, y$). If $y \geq 2$, then $19=(5+\frac{3}{x})(3+\frac{y}{2})>5(3+\frac{2}{2})=20$, a contradiction. Thus $y=1$, so $x=7$.

Ivan Loh
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On simplification, $y=\frac{8x-18}{5x+3}$

Now, $$8(5x+3)-5(8x-18)=114$$

So, $5x+3$ must divide $114$ and as $x>0, 5x+3>3$

Now the factors of $114$ are $1,2,3,6,19,38,57,114$

So, $5x+3$ can be $6,19,38,57,114$

$6\equiv1\pmod 5,19\equiv4, 38\equiv3,57\equiv2,114\equiv4$

So, $5x+3$ can be $38\implies x=7,y=1$

lab bhattacharjee
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