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sm1510
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At a restaurant, a group of friends ordered four main dishes and three 07 Dec 2021, 05:57
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85% (hard)

Question Stats:

52% (02:23) correct 48% (02:23) wrong based on 385 sessions

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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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D
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chetan2u
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At a restaurant, a group of friends ordered four main dishes and three 07 Dec 2021, 23:09
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sm1510
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.
Show more

7 dishes, all with different price.
\(M_1, M_2,M_3,M_4,S_1,S_2,S_3\) is the descending order of cost. We are looking for \(S_1\).

Such questions will invariably either have a statement giving the lowest possible price of most expensive object or otherwise. So, let us work on them.

Total price is 91, so the average price is 91/7 or 13.

maximum possible lowest price.
=> 13 in case there is no restriction of having different costs........13,13,13,13,13,13,13
=> 10 in case there is restriction of having different costs.....16,15,14,13,12,11 and 10 are possible prices for side dishes.

minimum possible highest price.
=> 13 in case there is no restriction of having different costs...............13,13,13,13,13,13,13
=> 10 in case there is restriction of having different costs.....16,15,14,13,12,11 and 10 are possible prices for side dishes.

(1) The most expensive main dish cost $16.
Only possibility is when the prices are 16,15,14,13,12,11 and 10.
So, \(S_1=12\)

(2) The least expensive side dish cost $10
Only possibility is when the prices are 16,15,14,13,12,11 and 10.
So, \(S_1=12\)


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At a restaurant, a group of friends ordered four main dishes and three 28 Dec 2021, 02:11
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Hi,

(1) We know most expensive dish is 16$, so we are left with

75=m_3+m_2+m_1+s_3+s_2+s_1

We know that the main dishes must account for MORE than half of 75, because the side dishes in total cost less than the main dishes in total. So trying out 14+13+12=39 -> 36 left for side dishes, we clearly see that can't work because side dishes at max are 11+10+9<36 -> try 15,14,13 for mains, and we get 42 leaving 32. But that is exactly 12+11+10, so we have a result

(2) Same procedure, even though we don't even have to calculate: We know from (1) that our least expensive side dish was 10, and every dish had to increase by 1$ to reach our 91$ goal. So it must be the same as in (1)
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At a restaurant, a group of friends ordered four main dishes and three 02 Dec 2025, 08:29
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sm1510
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.
Show more
Deconstructing the Question
We have 7 items: 4 Main dishes (\(M\)) and 3 Side dishes (\(S\)).
Total Cost = $91.
Constraints:
1. All prices are distinct integers.
2. \(min(M) > max(S)\) (Every main dish costs more than every side dish).

Let the sorted prices be: \(s_1 < s_2 < s_3 < m_1 < m_2 < m_3 < m_4\).
Target: Find \(s_3\) (the most expensive side dish).

***

Analyze Statement (1)
"The most expensive main dish cost $16."
So \(m_4 = 16\).
To check sufficiency, let's see if the prices are forced or if multiple sets exist.

To maximize the sum of the Main dishes (and thus minimize the Side dishes), we choose the largest possible distinct integers:
\(M = \{16, 15, 14, 13\}\). Sum = 58.
Remaining sum for sides: \(91 - 58 = 33\).
Since \(min(M)=13\), the max side dish must be \(\le 12\).
The largest possible sum for 3 distinct sides \(\le 12\) is \(12 + 11 + 10 = 33\).
This matches perfectly! So one solution is:
Sides: \(\{10, 11, 12\}\). Mains: \(\{13, 14, 15, 16\}\). Here \(s_3 = 12\).

Can we change the prices?
If we lower any Main dish price, the sum of Mains decreases -> the required sum of Sides increases.
Example: Lower \(m_1\) to 12. New Main Sum = 57. Required Side Sum = 34.
But now \(min(M)\) is 12, so \(max(S)\) must be \(\le 11\).
The max sum for sides \(\le 11\) is \(11 + 10 + 9 = 30\).
This is far below the required 34.
Conclusion: It is impossible to form any other set. The solution is unique.
SUFFICIENT

***

Analyze Statement (2)
"The least expensive side dish cost $10."
So \(s_1 = 10\).
Since all 7 prices must be distinct integers, the minimum possible sum for 7 distinct integers starting at 10 is:
\(10 + 11 + 12 + 13 + 14 + 15 + 16\).
Let's verify the sum:
\(Sum = 10 + 11 + 12 + 13 + 14 + 15 + 16 = 91\).

This sum is exactly the Total Cost given in the prompt ($91).
This means there is zero room to increase any price. The prices are locked into this exact sequence of consecutive integers.
The set is unique: Sides \(\{10, 11, 12\}\) and Mains \(\{13, 14, 15, 16\}\).
\(s_3 = 12\).
SUFFICIENT

***

Conclusion
Both statements alone are sufficient.

Answer: D
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