What is the largest number of consecutive odd positive integers that can be added together before the sum exceeds $40$?
We can list the sum of the first $n$ consecutive odd positive integers that are less than $40$: \begin{align*}
1&=1, \\
1+3&=4, \\
1+3+5 &= 9, \\
1+3+5+7 &= 16, \\
1+3+5+7+9 &= 25, \\
1+3+5+7+9+11 &= 36.
\end{align*}So, the six consecutive odd positive integers $1,3,5,7,9$, and $11$ can be added together. Since $1+3+5+7+9+11+13 = 49 > 40$, the answer is $\boxed{6}$.
Let's start by listing some consecutive odd positive integers:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ...
To find the largest number of consecutive odd positive integers that can be added together before the sum exceeds 40, we can start by adding the smallest numbers together until the sum exceeds 40.
1 + 3 = 4 (sum less than 40)
1 + 3 + 5 = 9 (sum less than 40)
1 + 3 + 5 + 7 = 16 (sum less than 40)
At this point, the sum is 16 and adding the next odd positive integer, 9, would make the sum exceed 40. Therefore, the largest number of consecutive odd positive integers that can be added together before the sum exceeds 40 is 3.