(The problem has been edited since this response. In the original problem, Quantity A was the sum of integers from 2 to 13.)
There is definitely something wrong with this problem. Assuming l in quantity B means 1, the answer is given as C, but it should be B. Let's prove it. As several people have shown, you can use the average formula, or Sum/(# of items) = Ave, to find the totals of both sides. I would do it more simply in this case, without the formula.
There is a great deal of overlap between the two sides. After all, the integers 2 through 13 are contained inside the integers from 1 through 17. Let's subtract all numbers that are in both sets from both sides. So what numbers are left over? Quantity A is totally contained within Quantity B, so at this point it's got nothing left and is 0. What about Quantity B?
Quantity B has 1, 14, 15, 16, and 17, while Quantity A does not. Adding these we get 63. (A quick way to add the last four numbers would be to add 14 and 17 to get 31, and double that since 15 + 16 must be the same, to get 62, and then adding 1.) Subtracting 34 from 63 will clearly get us something bigger than 0, so the answer should be B, not C.
BONUS PROBLEM: If the answer were legitimately C, what would l have to be? The only way C could be correct is if l were not 1, but some other integer. If we set the two quantities equal to each other we will get:
90 = ((l + 17)/2)(17  l + 1)  34
The two parenthesis on the right represent the average of the integers and the number of integers, respectively. Next, we have:
124 = .5(l + 17)(18  l)
248 = (l + 17)(18  l)
This will be a quadratic equation:
248 = 18l  l^2 + 17x18  17l
248 = l  l^2 + 306
l^2  l  58 = 0
This quadratic can't be factored with integers. 8 comes closest but it doesn't quite work. So basically this proves that there is no way for the two quantities to be equal. There's probably a typo somewhere, even besides the l.
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