MBAlad wrote:

Fig,

Help me out a little. Firstly, I've done the first part of the question two ways and come up with two different answers:

First:

10x + 20y = k(x +y) ---> 10x - kx = ky - 20y ---> x (10 - k) = y (k - 20)

For x and y to be positive k<10 or k<20???? Where'd I go wrong????

The other way I arranged it was the same as you:

<=> (10x+20y) = k*(x+y)

<=> (k-10)*x = (20-k)*y

How did you go from here to working out it's 18?

I think I'm up too late!!

By your first way, it's a little harder to view the elements

... It makes us deal with an equation with negative factors in front. But it's possible.

x (10 - k) = y (k - 20)

Imagine now:

o If k = 21 the equation is turned to be -11*x = y than x and y cannot both be positive. >>>> k < 20.

o If k = 9 the equation is turned to be x = -11*y than x and y cannot both be positive. >>>> k > 10.

Then, 10-k and k-20 suggest a sort of symetry with a center at 15. To confirm this feeling, we can take k=15.

x (10 - 15) = y (15 - 20) <=> x = y : Bingo there is a symetrical behaviour at 15.

So now we can try k = 12,

x (10 - 12) = y (12 - 20) <=> 2 x = 8 y >>>> x > y : the opposite of what we are looking for.

So k = 18

By the second way, we are using positive factors in front of x & y.

(k-10)*x = (20-k)*y

Here, we can observe without calculation that:

> at left side of the equation, k must be greater than 10 (otherwise we have a negative factor)

> at right side, k must be fewer than 20 (otherwise we have a negative factor)

One more time, k-10 and 20-k suggest a symetry at 15. Then, similar to the first way but without calculation, we can see that greater k is fewer x is.

Thus, as we want x < y and as k = 15 is a symetry point, k must be equal to 18.