Answer:
To solve this problem, we need to use a systematic approach and logical reasoning. Let's denote the numbers on the hats worn by Linda, Miah, and Noah as L, M, and N respectively.
According to the problem, each person adds the two numbers on the other two hats and comes up with the totals of 11, 17, and 22. This gives us three equations:
1. M + N = 11 (This is Linda's total)
2. L + N = 17 (This is Miah's total)
3. L + M = 22 (This is Noah's total)
We can solve these equations simultaneously to find the values of L, M, and N.
Firstly, let's add all three equations together:
(M + N) + (L + N) + (L + M) = 11 + 17 + 22
2L + 2M + 2N = 50
Dividing through by 2 gives:
L + M + N = 25
Now we have a new equation that we can use to find the values of L, M, and N.
Let's subtract the first equation from this new equation:
(L + M + N) - (M + N) = 25 - 11
L = 14
Subtracting the second equation from the new equation gives:
(L + M + N) - (L + N) = 25 - 17
M = 8
Finally, subtracting the third equation from the new equation gives:
(L + M + N) - (L + M) = 25 - 22
N = 3
So, Linda's hat has the number 14, Miah's hat has the number 8, and Noah's hat has the number 3.
The largest number that is on a hat is therefore 14.