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Main dish and side dish

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Main dish and side dish

by j_shreyans » Mon Oct 06, 2014 9:12 am
At a restaurant, a group of friends ordered four main dishes and three side dishes at a total cost of $91. The prices of the seven items, in dollars, were all different integers, and every main dish cost more than every side dish. What was the price, in dollars, of the most expensive side dish?

(1) The most expensive main dish cost $16.

(2) The least expensive side dish cost $10.

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by fcabanski » Mon Oct 06, 2014 9:57 am
Make a list of prices, and see if it adds to 91. Consider the constraints: the price of the highest main or lowest side, and that all the prices must be integers. Start from the given price. If the price doesn't add to $91, make adjustments.

1: The main dishes are 16, 15, 14, and 13 dollars. The side dishes are all cheaper than any main dish: 12, 11, 10 dollars. The total is 16+15+14+13+12+11+10 = 91 dollars. There is no other choice for the costs, because the total must be 91. If any of the side dishes are less, then the total is less than 91. If any of the main dishes (other than the $16 dish) are less, then also the side dishes prices must go down, and the total is less than 91.

1 is sufficient.

2: The side dishes are 10, 11, and 12 dollars. The main dishes are 13, 14, 15, and 16 dollars. This is the same price list as 1, but it forces the prices from the other side (anything else will make the total too high). The price is 91. If any of the side dishes is more expensive, or if the main dishes are more expensive, then the total is more than 91.

2 is sufficient.

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Last edited by fcabanski on Mon Oct 06, 2014 12:13 pm, edited 1 time in total.
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by [email protected] » Mon Oct 06, 2014 12:09 pm
Hi j_shreyans,

DS questions of this type are always meticulously designed, so you have to take note of the details and be prepared to do some "brute force" math to prove what the answer is. The explanation provided by fcabanski proves that both Fact 1 and Fact 2 are sufficient because there is only one way to get to a total of $91. The only way to prove it though was to say "what is the highest I can make each of these items?" and "what is the lowest that I can make each of these items?" and then do the necessary math.

Variations of this type of question CAN have insufficient (re: multiple) answers though. For example...

2 main dishes and 2 sides dishes have a total cost of $52. The prices are all different integers and every main dish costs more than every side dish. What is the price of the more expensive side dish?

(1) The most expensive main dish cost $16

(2) The least expensive side dish cost $10

Here, there are multiple ways to get a total of $52 under the given constraints. Can you prove how?

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by Matt@VeritasPrep » Sun Oct 12, 2014 10:09 pm
A tip for anyone who's wondering how to make that deduction under test conditions: since the question tells us that each dish is an integer and that there are seven dishes, we might notice that $91 = 7 * 13. This sort of number sense is critical on the GMAT, and should show you right away that we can only have seven dishes sum to $91 if the average of the set is $13. Since the GREATEST value is $16 and each price is different, this means the LEAST value must be $10 and that we must have the prices $10 though $16.
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by GMATGuruNY » Mon Oct 13, 2014 3:02 am
For a trickier version of this problem, check here:

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