Can someone please explain this solution? It make no sense to me whatsoever. Greg did not have time in the video to explain or take any questions. Thanks!
Since x and y are positive numbers and x + y = 1, then we can see that x and y must be between 0 and 1. If either were greater than 1, then x+y > 1 (plug in numbers to see).
Thus, we have that 0<x<1 and 0<y<1.
We see that we are trying to find the possible values of 100x+ 200y. Let’s try to multiply the inequality 0<y<1 and the equation x + y = 1 by 100 on both sides.
i) 0<100y<100
ii) 100x+100y=100
We want 100x + 200y. We see that 100x + 200y = 100x + 100y + 100y = (100x + 100y) + 100y = 100 + 100 y.
I used (ii) from above to make the last substitution.
Thus, we see that 100x + 200y = 100 + 100y. We can bound 100 + 100y from below and above using 0<100y<100. Since 0<100y<100, then by adding 100 to each part of the inequality, we get 0 + 100 < 100 y + 100 < 100 + 100 → 100 < 100 + 100y< 200.
Thus, 100 + 100y is between 100 and 200 exclusive. Answers (II) and (III) work.