A gourmet cheese shop sold several orders of English Stilton and Span

Question

﻿A gourmet cheese shop sold several orders of English Stilton and Spanish Manchego yesterday. Customer A purchased 15 pounds of English Stilton and 3.75 pounds of Spanish Manchego for a total of $438.00. If the price for each of these cheeses is proportional to its weight, what is the price of 1 pound of Spanish Manchego?

(1) Customer B purchased 5 pounds of English Stilton and 4 pounds of Spanish Manchego for a total of $214.75.
(2) Customer C purchased 6 pounds of English Stilton and 1.5 pounds of Spanish Manchego for a total of $175.20.

gmatophobia · Accepted Answer

The question premise states that the price of the cheeses is proportional to its weight.

Price = Weight * Constant

Price of 15 pounds of ES cheese =  15*x  ⇒  x  is a constant 
 Price of 3.75 pounds of SM cheese =  3.75*y  ⇒  y  is a constant

15*x + 3.75*y = 438  --- (1)

Statement 1

(1) Customer B purchased 5 pounds of English Stilton and 4 pounds of Spanish Manchego for a total of $214.75.

5*x + 4*y = 214.75

Multiplying by 3 on both sides

15*x + 12*y = (214.75*3)  --- (2)

As we have two different equations, we can find the value of  x  and  y  and use that information to find the value of 1 pound of SM cheese. We don't need to do that while solving the question, however we have the required information to do so.

Hence, equation 1 is sufficient. We can eliminate B, C, and E.

Statement 2

(2) Customer C purchased 6 pounds of English Stilton and 1.5 pounds of Spanish Manchego for a total of $175.20.

6*x + 1.5*y = 175.20

We need to check if this equation is different from the one which is given to use in the premise.

Let's multiply the equation by 2

12*x + 3*y = 350.40

Let's take the coefficient of x. The current coefficient of x is 12, however, we need a coefficient of 15. Hence, we need to add 3 to the coefficient of x.

3 = 1/4th of 12

(12+\frac{12}{4})*x + (3+\frac{3}{4})*y = (350.40+ \frac{350.40}{4})

(12+3)*x + (3+0.75)*y = (350.40+ 87.60)

15*x + 3.75*y = 438

The equation is the same as that of the premise. Hence, this statement provides us with information that we already know and hence doesn't add any additional value.

We can eliminate D.

Option A

MartyMurray · Answer

﻿A gourmet cheese shop sold several orders of English Stilton and Spanish Manchego yesterday. Customer A purchased 15 pounds of English Stilton and 3.75 pounds of Spanish Manchego for a total of $438.00. If the price for each of these cheeses is proportional to its weight, what is the price of 1 pound of Spanish Manchego?

We see that the question stem provides the information that the price of each cheese proportional to its weight. That means that the price goes up evenly as the weight goes up. For example, if the weight of cheese purchased doubles, the price doubles as well.

So, we know that, if the price of a pound of English Stilton is S and the price of a pound of Spanish Manchego is M, the cost of a purchase of English Stilton = Pounds of English Stilton * S, and the cost of a purchase of Spanish Manchego = Pounds of Spanish Manchego * M.

So, the cost of customer A's purchase is 15S + 3.75M = 438.

Statement 1:

(1) Customer B purchased 5 pounds of English Stilton and 4 pounds of Spanish Manchego for a total of $214.75.

The cost of customer B's purchase is 5S + 4M = 214.75.

It appears that we have two different equations and two variables and that, therefore, we can calculate S and M and answer the question. However, let's confirm that the equation for customer B is different from that for customer A.

We can confirm by multiplying the equation for customer B by 3 to get 15S + 12M = 644.25.

We see that we definitely have two different equations and can therefore calculate the values of S and M and answer the question.

Sufficient.

Statement 2:

(2) Customer C purchased 6 pounds of English Stilton and 1.5 pounds of Spanish Manchego for a total of $175.20.

The cost of customer C's purchase is 6S + 1.5M = 175.20.

Let's see whether we have two different equations.

Since 15/6 = 2.5, let's multiply the equation for customer C by 2.5.

We get 2.5 * 6S + 2.5 * 1.5M = 2.5 * 175.20 -> 15S + 3.75M = 438.

So, actually, in the equation for customer C, we have the same equation that we have for customer A, meaning we have only one equation and two variables. So, we don't have enough information to determine what S and M are.

Insufficient.

The correct answer is (A).

JeffTargetTestPrep · Answer

If e dollar is the price per pound of English Stilton and s dollar is the price per pound of Spanish Manchego, then we have:

15e + 3.75s = 438

We need to answer the question:

s = ?

Statement One Alone:

=> Customer B purchased 5 pounds of English Stilton and 4 pounds of Spanish Manchego for a total of $214.75.

5e + 4s = 214.75

Since the question stem and the statement equations are not equivalent  , this system of equations would give us a unique value for s.

Statement one is sufficient. Eliminate answer choices B, C, and E.

Statement Two Alone:

=> Customer C purchased 6 pounds of English Stilton and 1.5 pounds of Spanish Manchego for a total of $175.20.

6e + 1.5s = 175.2
(6e + 1.5s = 175.2) × 2.5
15e + 3.75s = 438

Since the question stem and the statement equations are equivalent  , this system of equations would give us multiple possible values for s.

Statement two is not sufficient.

Answer: A

GMATGuruNY · Answer

15x + 3.75y = 438
Since 3.75 = 15/4, multiply by 4 so that the coefficients on the left are both integers.
Resulting equation:
60x + 15y = 1752

Statement 1: 5x + 4y = 214.75
Here, the expression on the left is clearly DIFFERENT from 60x + 15y, since multiplying 5x + 4y by 12 yields 60x + 48y.
Implication:
Statement 1 and the prompt imply two different equations.
the sum for 60x + 15y and the sum for 60x + 48y
Since we have two variables and two distinct linear equations, we can solve.
SUFFICIENT.

Statement 2: 6x + 1.5y = 175.20
Here, multiplying 6x + 1.5y by 10 yields 60x + 15y --> the same expression as that yielded by the prompt.
Implication:
Statement 2 and the prompt both imply the same equation:
the sum for 60x + 15y
Since we have two variables but only ONE linear equation -- and the two variables can be noninteger values -- we cannot solve.
INSUFFICIENT.

A

KarishmaB · Answer

Say price of English Stilton is E and that of Spanish Manchego is S. Given: Customer A 15*E + 3.75S = 438 We need the value of S. (1) Customer B purchased 5 pounds of English Stilton and 4 pounds of Spanish Manchego for a total of $214.75. 5E + 4S = 214.75 I can see that this equation is distinct from the previous given equation. The ratio of 15/5 is not the same as 3.75/4. So the two lines will intersect at one point and give unique values for E and S. Sufficient alone. (2) Customer C purchased 6 pounds of English Stilton and 1.5 pounds of Spanish Manchego for a total of $175.20. 6E + 1.5S = 175.2 This equation also looks different from the given equation 15*E + 3.75S = 438 but I must check the ratios. I know that this is trap - simultaneous equations that look different but are the same. 15/6 = 5/2 To find whether 3.75/1.5 is also 5/2, I will check if 0.75 * 5 is 3.75. It is! Ideally, I must check the ratio of constants also since we could have no solution here but this is a real life case. We have customers A and C who did make these purchases. Hence I know for sure that there must be a solution. So then these two equations MUST BE the same and hence we cannot solve them to get the values of E and S Not Sufficient. Answer (A) Discussion on linear equations: https://youtu.be/Nh77CobN9mQ

siddharth_ · Answer

KarishmaB   Bunuel  can you kindly share your thoughts on this -

'If the price for each of these cheeses is proportional to its weight' 
What's the purpose of this statement? Isn't this something that's always true and rarely mentioned (at least practically and in most GMAT questions)?

Although I right away noticed the 4:1 ratio of QS and 2nd statement while reattempting this post the test, but the statement above confused me immensely while answering this question in the test setting. I understood this as the 'rate' is varying, ie, price per pound varies with weight, in which case the price for 15 pound would be 225p.

KarishmaB · Answer

Yes, it is obvious and yes, rarely mentioned but GMAT does clarify such things sometimes. So don't let it confuse you. 
Even if I were to think that there must be a specific reason here for this statement, what can it be? Can the shop charge differently for each pound of a cheese based on how many pounds you are buying? If that were true, would this statement in the question make any sense: what is the price of 1 pound of Spanish Manchego?
Then the price per pound would vary for each customer and this question would make no sense. So that should make you realize that the question is just over clarifying, nothing else.

MarkGMATMentor · Answer

The key thing to remember in a question like this is the    equation count pattern : To solve a system of linear equations (equations where all variables have an exponent of 1), you need as many  different  equations as variables.

The prompt tells you: 15e + 3.75m = 438

The question asks: What is m?

Statement 1

This tells you: 5e + 4m = 214.75

This is  different  than the original equation because there's no way to manipulate it to get 15e + 3.75m = 438.

Because it's different, you have two equations and two variables; thus, you can solve for m.  Sufficient.

Statement 2:

This tells you: 6e + 1.5m = 175.20

If you multiply this equation by 2.5, you will get 15e + 3.75m = 438.

This is just the original equation "in disguise". You only have one equation and two variables; you cannot solve. There are many possibilities for m.  Insufficient.

The answer is A.

By recognizing this pattern, you can limit your calculations, and solve with speed and confidence.

I made a full walkthough here: 
 https://www.youtube.com/watch?v=SKwzz56xAzE

sourabhgx · Answer

MartyMurray 
 KarishmaB

price of each type cheese is proportional to its weight. why we have not reduced price of each cheese block proportionality to its weight?

as per question - 15 x + 3.75 y = 438

if price of each cheese type is factorized in its proportion then -

statement 1 - 5*(5x/15) + 4*(4y/3.75) = 214.75

statement 2 - 6*(6x/15) + 1.5*(1.5y/3.75) = 175.20

please help

MarkGMATMentor · Answer

Hi  sourabhgx ,

If I may, I'll try to answer your question --
 
You may be misinterpreting the use of the word "proportional". When the question says that "t he price for each of these cheeses is proportional to its weight", all this means is that the price increases proportionally as more cheese is purchased. For example, if one type of cheese is priced at $10 per pound, then the price of 1 pound is 1*10 = $10, 2 pounds is 2*10 = $20, 3 pounds is 3*10 = $30, etc.

But we don't know the price per pound. As you have it above, the price per pound is represented by x for English Stilton and y for Spanish Manchego. So the price of 1 pound of English Stilton is 1*x = 1x, 2 pounds is 2*x = 2x, 3 pounds is 3*x = 3x, etc., and similar for Spanish Manchego using y instead of x.

So, there is no need to "reduce" the price proportionally. Since x and y are prices  per pound , the proportion is already addressed. In Statement 1, the price of 5 pounds of English Stilton is 5*x = 5x and the price of 4 pounds of Spanish Manchego is 4*y = 4y, yielding a total of $214.75:

5x + 4y = $214.75

And similar for Statement 2.

Then you can test for sufficiency by comparing to the equation gained from the prompt, as I and others have explained above.

Hopefully that helps clear things up.

sourabhgx · Answer

Hello  MarkGMATMentor , Thanks for valuable insight.

AmishaJ · Answer

Why are we NOT taking into account - the number of cheese per type?

MartyMurray · Answer

Although the stem mentions "cheeses," the question concerns only the pounds and price per pound of each type of cheese, not with the numbers of items or pieces of cheese.

egmat · Answer

Hi AmishaJ,

Your instinct comes from the phrase  "the price of each cheese is proportional to its weight,"  which makes it sound like weight needs to be built into the equation in some special, extra way. That's the exact spot to fix.

"Proportional to weight" is already fully captured by writing weight × price-per-pound.  Let S = price of  1 pound  of Stilton and M = price of  1 pound  of Manchego. Because price scales evenly with weight:

- cost of Stilton = (pounds) × S
- cost of Manchego = (pounds) × M

So Customer A gives  15 S +  3.75 M =  438 . The number of pounds ( 15  and  3.75 )  is  the count per type - it's the multiplier sitting right in front of each price. You are already taking it into account.

Where the double-counting creeps in:  if you also divide the price by the weight (like writing  5  × ( 5 S/ 15 )), you're inserting the weight a second time. Weight belongs in the equation exactly  once  - as the multiplier. The per-pound price itself is a fixed constant; it does not change as you buy more.

A quick everyday check

Say apples cost  $2 per pound  (a constant). Then:

-  3  pounds cost  3  ×  $2  =  $6 
-  5  pounds cost  5  ×  $2  =  $10

The weight appears once as the multiplier. Nobody writes  3  × ( 3  ×  $2 ) - that would count the  3  pounds twice and give the wrong total.

Same here: each customer's equation is just (pounds of Stilton)·S + (pounds of Manchego)·M = total. Once you set them up that plain way, Statement ( 1 ) gives a genuinely new equation (so you can solve for M), while Statement ( 2 ) collapses back into Customer A's equation - which is why the answer is  A .

Answer: A

A gourmet cheese shop sold several orders of English Stilton and Span

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