A sandwich shop offers a sandwich menu, a soup menu, and a salad menu.

Question

A sandwich shop offers a sandwich menu, a soup menu, and a salad menu. The number of salads listed on the salad menu is twice the number of sandwiches listed on the sandwich menu and 1 more than the number of soups listed on the soup menu. How many soups are listed on the soup menu?

(1) The total number of choices a customer has when choosing 1 item from each of 2 of the menus is 63.
(2) The total number of choices a customer has when choosing 1 item from each of the 3 menus is 90.

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Official Answer

D

Bunuel · Accepted Answer

(1) indicates that the total number of combinations available when a customer chooses one item from any two of the three menus is 63. Assuming x represents the number of sandwiches, salads are then 2x (twice the number of sandwiches) and soups are 2x - 1 (one fewer than the number of salads). This can be broken down into the following combinations:

1. Sandwich & Salad = x*2x = 2x^2
2. Sandwich & Soup = x(2x - 1) = 2x^2 - x
3. Salad & Soup = 2x(2x - 1) = 4x^2 - 2x

Adding these combinations gives:

2x^2 + (2x^2 - x) + (4x^2 - 2x) = 63
x(8x - 3) = 63

By plugging in factors of 63 for x, we find that only x = 3 works (testing is only necessary up to 7, since already if x = 7, the product exceeds 63). Hence, the first statement is sufficient.

Similarly, (2) indicates that the total number of combinations available when a customer chooses one item from each of the three menus is 90. This implies that:

x*2x(2x - 1) = 90

Again, by plugging in factors of 90 for x, we quickly find that x = 3 (similarly, testing only up to 5 is necessary, since already if x = 5, the product exceeds 90). Thus, the second statement is also sufficient.

Answer: D.

Hope it's clear.

Tukkebaaz · Answer

Yes finally a new/ unsolved one: here it goes - Lets look at the Qn stem: Salad = 2* Sandwich(say x) and Soup = Salad-1 So we have Sandwich : Salad : Soup = x : 2x : 2x-1. S1 - Sufficient 1 item each taken 2 items at a time: Sandwich and Salad +Sandwich and Soup + Salad and Soup = x*2x + x*(2x-1) + 2x*(2x-1)=63 . Note: x cant be fractions, as no of items listed is a whole number. S2 - Sufficient all 3 items taken 1 each: x*2x*(2x-1) =90 . Solving the equation gives x=3 . IMO D. Missed adding this out last time, so updated

Tip: If you see such Qns don't give into the urge to mark 'C' thinking it is a Quadratic/ Polynomial equation as it could be quite possible to solve without additional info. Hope this helps

Kinshook · Answer

Given: A sandwich shop offers a sandwich menu, a soup menu, and a salad menu. The number of salads listed on the salad menu is twice the number of sandwiches listed on the sandwich menu and 1 more than the number of soups listed on the soup menu.

Asked: How many soups are listed on the soup menu?

Let number of salads, number of soup and number of sandwich be x, y & z respectively.

x = 2z = y + 1
y = 2z - 1
y = ?

(1) ﻿﻿﻿﻿The total number of choices a customer has when choosing 1 item from each of 2 of the menus is 63.
xy + yz + zx = 63
2z(2z-1) + (2z-1)z + 2z*z = 63
4z^2 - 2z + 2z^2 - z + 2z^2 = 63
8z^2 -3z - 63 = 0
8z^2 + 21z - 24z - 63 = 0
(z-3)(8z+21) = 0
z = 3
y = 2z - 1 = 5
SUFFICIENT

(2) ﻿﻿﻿﻿The total number of choices a customer has when choosing 1 item from each of the 3 menus is 90.
xyz = 90
2z(2z-1)z = 90
z^2(2z-1) = 45
z = 3
y = 2z -1 = 5
SUFFICIENT

IMO D

PowerHouse · Answer

Sorry, could someone explain why you can use this as simple permutation (vs. it being combination/ pick and having to divide by 2! and 3! respectively for the two scenarios?)

Thanks!

Bunuel · Answer

That's because the order of selection doesn't matter. The statement mentions "The total number of choices...", which implies that the order of selection is not relevant. For instance, selecting sandwich 1 and soup 2 is considered the same choice as selecting soup 2 and sandwich 1.

sgpk242 · Answer

Really struggling on the wording from Statement 1...  Bunuel  can you please break down how you derived an equation from Statement 1?

LamboWalker · Answer

Would really appreciate if someone can help explain how statement (1) can be broken down as the experts above me have done so

chetan2u KarishmaB

chetan2u · Answer

A sandwich shop offers a sandwich menu, a soup menu, and a salad menu. The number of salads listed on the salad menu is twice the number of sandwiches listed on the sandwich menu and 1 more than the number of soups listed on the soup menu. How many soups are listed on the soup menu?
There are three menus with options, that is each menu has certain number of options to choose from.
Sandwich menu: Let the number of options, that is different type of sandwiches be x
Salad menu: Number of salads or options will be 2*x
Soup menu: Number of options in soup be y, where y = 2x-1

we have to find the value of y or 2x-1.

(1) ﻿﻿﻿﻿The total number of choices a customer has when choosing 1 item from each of 2 of the menus is 63.
Each of the 2 menus means that you have to pick all possible pair from the three menus:-
 Sandwich & Salad:  Pick one item from each, so x*2x or  2x^2  ways
 Sandwich & Soup:  Pick one item from each, so x*(2x-1) or  2x^2-x  ways
 Salad & Soup:  Pick one item from each, so 2x*(2x-1) or  4x^2-2x  ways
Total ways =  2x^2+2x^2-x+4x^2-2x=63......8x^2-3x=63......8x^2-3x-63=0......8x^2-24x+21x-63=0.....(8x+21)(x-3)=0 .
Thus x = -21/8 or 3. x cannot be negative, so x=3
Sufficient.

(2) ﻿﻿﻿﻿The total number of choices a customer has when choosing 1 item from each of the 3 menus is 90.
Here, total = product of all items in each menu =  x*2x*(2x-1) = 90 ......x^2(2x-1)=45=9*5 = 3^2(2*3-1) 
Hence, x=3
Sufficient

D
 LamboWalker  the method would be same as Bunuel and others did above. I am just expanding it a bit more.

KarishmaB · Answer

Yeah, I was a little uncertain too upon reading statement 1 before realising what it must mean. (1) The total number of choices a customer has when choosing 1 item from each of 2 of the menus is 63. You have to select 2 of the 3 menus and select 1 item from each menu - this much is clear. Now, does it mean that when we select any 2 menus and then select 1 item from each, I get 63 options? Well this is not possible because number of items are distinct in each menu. It is still possible that 2 menus have the same number of items - soups and sandwiches - but the third cannot for sure. So number of choices cannot be the same in each case. This means "total number of choices" means that we select 2 menus from the 3 and 1 item from each menu; then we select another conbination of 2 of the 3 menus and 1 item from each and add to the previous choices; again we select the last combination of 2 menus and 1 item from each and add to the previous total. This is what gives us "total number of choices". So if sandwiches are w, salads 2w and soups (2w-1), I can pick sandwich and salad menus and select 1 from each in w*2w ways. I can pick salad and soup menus and select 1 item from each in 2w(2w - 1) ways and I can select sandwich and soup menus and select 1 from each in w(2w-1) ways. So w*2w + 2w(2w - 1) + w(2w-1) = 63 which is 8w^2 - 3w - 63 = 0 Since product of the roots is negative, one root will be positive and one negative. Hence this statement will be sufficient alone. Here is a video on selections: https://youtu.be/tUPJhcUxllQ Here is a video on quadratic equations: https://youtu.be/QOSVZ7JLuH0

deveshkhare · Answer

After initially considering statement 1 sufficient using the same logic as you, I re-considered my answer after carefully re-reading the question - "choosing 1 item from each of 2 of the menus is 63". This statement sounds like it does not imply all combinations of 2 out of 3 menus but only one specific combination, so I concluded that statement 1 is insufficient. Could you clarify the wording  Bunuel

Bunuel · Answer

I don’t know what else I can add to the solution I've provided, except that "Choosing 1 item from each of 2 of the menus" means exactly what I’ve already described.

MartyMurray · Answer

Here's statement (1)

(1) ﻿﻿﻿﻿The total number of choices a customer has when choosing 1 item from each of 2 of the menus is 63.

It could be read as being about one specific combination.

At the same time, the only possible factors of 63 are 1, 3, 7, 9, 21, and 63.

Also, the passage indicates that the numbers of the item types are x, 2x, and 2x - 1.

Notice that no two factors of 63 can be two of x, 2x, and 2x - 1.

By noticing that, we can determine that statement (1) is not saying that one specific combination produces 63 items. Having determined that, we can seek to read statement (1) in a way that fits what the passage says.

Statement (1) could have been written more clearly. Fortunately, though, only the reading intended by the question writer fits the information presented by the passage.

jaggraf · Answer

Can Bunuel or someone explain this response? I am confused about this response, shouldn't we still have to divide by 2! and 3!. If order does not matter or it's irrelevant, doesn't that mean we use the combination formula, not the permutation? If sandwich 1 and soup 2 are considered the same choice as selecting soup 2 and sandwich 1. Then we don't want to double-count combinations and divide by a factorial.

GMATGuruNY · Answer

The number of salads listed on the salad menu is twice the number of sandwiches listed on the sandwich menu and 1 more than the number of soups listed on the soup menu.
 These parameters are satisifed by the following cases:

sandwich options - salad options - soup options
............1......................2...................1
............2......................4...................3
 ............3......................6.................. 5  
............4......................8...................7

Statement 2:
Only the green case yields 90 ways to a combine sandwich, salad and soup:
3*6*5 = 90
Thus, the number of soup options = 5
SUFFICIENT.

Statement 1:
Check whether the green case yields 63 ways to combine 2 food items, each from a different menu.

In the green case:
Number of sandwich-salad combinations = 3*6 = 18
Number of sandwich-soup combinations = 3*5 = 15
Number of salad-soup combinations = 6*5 = 30
Total ways = 18+15+30 = 63

Clearly, any other case will yield a result less than 63 (since it will combine from fewer food choices) or greater than 63 (since it will combine from a larger number of food choices).

Implication:
Only the green case is viable.
Thus, the number of soup options = 5.
SUFFICIENT.

D

A sandwich shop offers a sandwich menu, a soup menu, and a salad menu.

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GMAT Data Sufficiency (DS) Questions

A sandwich shop offers a sandwich menu, a soup menu, and a salad menu.

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