A food-truck owner restocks supplies before a weekend festival. The owner can buy items priced at $7, $9, $20, $35, and $60. If the owner buys exactly $200 worth of these items, then which of the following could be the total number of items purchased?
I. 26
II. 27
III. 28A. Only I
B. Only II
C. Only III
D. Only I and III
E. I, II, and III
Take $7 as the starting price for each item.
If all n items cost $7, the total is 7n. We then check whether the remaining amount needed to reach $200 can be obtained by replacing some $7 items with more expensive ones.
Compared with a $7 item:
$9 adds $2
$20 adds $13
$35 adds $28
$60 adds $53
I. 26 items
26 * $7 = $182.
So we need an extra $18 to get to $200.
We can get $18 by replacing nine $7 items with nine $9 items:
9 * $2 = $18.
So 26 items is possible: 17 items priced at $7 and 9 items priced at $9, for a total of 26 items costing exactly $200.
II. 27 items
27 * $7 = $189.
So we need an extra $11 to get to $200.
Also, some items must be $7 items, since even 27 items at $9 each would cost 27 * $9 = $243, already more than $200.
Now, every time we replace a $7 item with a $9 item, we add $2. Replacing it with a $20, $35, or $60 item adds at least $13, which is already more than the $11 we need.
So the only usable increase is $2, and no number of $2 increases can total $11.
Thus, 27 items is not possible.
III. 28 items
28 * $7 = $196.
So we need an extra $4 to get to $200.
Replace two $7 items with two $9 items:
2 * $2 = $4.
So 28 items is possible: 26 items priced at $7 and 2 items priced at $9, for a total of 28 items costing exactly $200.
Answer: D.