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