Math CentralQuandaries & Queries


Mary, a student:

can you help,
2/35= 1/x +1/y ; x<y. what is x and y.

Hi Mary,

I think there maybe an error in your given question. As it is written now, there are no two real numbers that would satisfy the solution.

2/35=y/(xy) + x /(xy)

2/35 = (y+x)/(xy)
so 2 = y+x & 35=xy. Since 2-y=x, we can substitute into 35=xy




Using the quadratic formula would result in answers of x = 1-i√34 & y=1+i√34, two complex solutions.

If the equation were 2/35=1/x-1/y, our solution would have two real solutions. If you find a common denominator and simplify the right side of the equation:
2/35=y/(xy) - x /(xy)

2/35 = (y-x)/(xy)

so 2 = y-x & 35=xy. At this point you can either use substitution to find your values of x & y or think "what two numbers subtract to 2 and multiply to 35?"
The numbers 5 & 7 work.

Hope this helps,

Janice & Gary

Mary wrote back

Thank you for your help,
I may have found another solution. PLEASE let me know if this works.

2/35= 6/105= 1/x +1/y x<y

6/105= 5/105 + 1/105

6/105= 1/21 + 1/105 x<y 21<105

or 2/35= 8/140= 1/x +1/y x<y

8/140= 7/140 + 1/140

8/140= 1/20 + 1/140 x<y 20<140


Your answer is correct, our answer is incorrect. We overlooked the fact that a fracion can be written in many equivalent ways. In particular for your first solution you wrote 2/35 as 6/105 and then solved the problem.

Thank you for point out our error.


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