First, we need to calculate the total cost of the skateboard and helmet.
The cost of the skateboard is:
\[
$45.50
\]
The cost of the helmet is:
\[
$18.25
\]
Now, let's calculate the total cost:
\[
\text{Total cost} = 45.50 + 18.25 = 63.75
\]
Next, we know that Jason earned \( \frac{1}{3} \) of the total cost by mowing lawns. We can find that amount:
\[
\text{Amount earned by mowing lawns} = \frac{1}{3} \times 63.75 = \frac{63.75}{3} = 21.25
\]
To find out how much Jason earned by babysitting, we subtract the amount earned by mowing lawns from the total cost:
\[
\text{Amount earned by babysitting} = 63.75 - 21.25 = 42.50
\]
Thus, Jason earned $42.50 babysitting.