First one:

Say you take out 10 coins from the pile into another group, so:

Group 1:

H: n (we don't know the number of heads left so lets just say it has n heads)

T: 90 - n (there are 90 coins in total in group 1, so number of tails must be the difference of 90 and number of heads left)

Group 2:

H: 10 - n (group 1 has n heads and since there are only 10 heads, there must be the difference of that in Group 2)

T: 10 - (10 - n) = n (same logic as before, there are 10 coins in group 2, so the difference must be between number of coins left and number of heads)

What you get is essentially that there are the same number of heads in group 1 as tails in group 2, which is n. Now take all coins in group 2 and flip them, and you now have same number of heads in both groups.

The second problem is a about what you know to be true essentially. If all the boxes are labelled incorrectly that is the truth you use to solve the problem.

Actual:

1: Apple

2: Oranges

3: Both

Incorrect labeling possibilities:

1: Oranges/Both

2: Apple/Both

3: Apple/Oranges

Example:

1. Oranges
2. Both
3. Apple

So for picking out, you pick from one that says 'Both', since whatever you get from it, is the label for that box. In this case you pick an orange from box 2 and you label it correctly as 'Oranges'. Box 3 says apple, so you know its not apples, so the possibilities are 'oranges' or 'both', but since you've already determined the oranges box as box 2, you know box 3 has to be 'Both' and box 1 to be 'Apple'.