http://cemc.uwaterloo.ca/resources/potw/2017-18/English/POTWC-17-NN-08-P.pdf click on link and answer below (DETAILED ANSWERS ONLY)

DannyBatman8
Nov 10, 2017

#1**+2 **

*There are 15 Blue Jays and 14 Orioles that wish to rest on the branches of three older trees. Each of the trees will have at least 4 Blue Jays and 2 Orioles. However, no tree may have more Orioles than Blue Jays in its branches. Determine the largest number of birds that can be in one tree.*

This is how I would "think through" the problem.

Each tree will have at least 4 Blue Jays and 2 Orioles.

Let's go ahead and "put" 4 Blue Jays and 2 Orioles in each tree.

1^{st} tree: 4, 2 2^{nd} tree: 4, 2 3^{rd} tree: 4, 2 remaining: 3, 8

Now let's try to put as many of the remaining birds in the first tree as we can.

Let's put the rest of the Blue Jays in the first tree.

1^{st} tree: 7, 2 2^{nd} tree: 4, 2 3^{rd} tree: 4, 2 remaining: 0, 8

Now...no tree may have more Orioles than Blue Jays, so the maximum number of remaining Orioles that we can put in the first tree is 5 .

1^{st} tree: 7, 7 2^{nd} tree: 4, 2 3^{rd} tree: 4, 2 remaining: 0, 3

And it doesn't matter how we distribute the remaining Orioles.

The largest number of birds that can be in one tree is 14 .

hectictar
Nov 10, 2017

#2**+3 **

There are 15 Blue Jays and 14 Orioles that wish to rest on the branches of three older trees. Each of the trees will have at least 4 Blue Jays and 2 Orioles. However, no tree may have more Orioles than Blue Jays in its branches. Determine the largest number of birds that can be in one tree.

Put 4 Jays and 4 Orioles in each tree

So......we have used 24 and there are 5 left - 3 Jays and 2 Orioles

Put the 3 Jays in each of the trees and 2 Orioles in any of two trees

So....we will have

5 Jays - 5 Orioles 5 Jays - 5 Orioles 5 Jays - 4 Orioles

Take 1 Jay and 1 Oriole from each of the last two trees and put them in the first tree

So we have this arrangement

7Jays - 7 Orioles 4 Jays - 4 Orioles 4 Jays - 3 Orioles

Each tree has at least 4 Jays and 2 Orioles....and no tree contains more Orioles than Jays

And I agree with hectictar.....14 is the greatest number in any tree !!!

CPhill
Nov 10, 2017