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Giyo
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M24-04 29 Jul 2023, 01:49
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Bunuel
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Official Solution:


The price of Mixture A is $2 per gallon and the price of Mixture B is $5 per gallon. Is $9 enough to buy \(x\) gallons of Mixture A and \(y\) gallons of Mixture B?

The question asks whether \(2x+5y\leq{9}\).

(1) The ratio of \(x\) to \(y\) is 2 to 1.

This implies that \(\frac{x}{y}=\frac{2}{1}\) or that \(x=2y\). Thus, the question becomes: is \(2(2y)+5y\leq{9}\) or is \(y\leq{1}\). Since we don't know that, this statement is not sufficient.

(2) $10 is NOT enough to buy \(2x\) gallons of Mixture A and \(\frac{y}{2}\) gallons of Mixture B.

So, we are told that \(2*2x+5*\frac{y}{2} \gt 10\), or that \(8x+5y \gt 20\). If \(x=0\) and \(y=5\), then the answer is NO (\(2x+5y > 9\)) but if \(x=3\) and \(y=0\), then the answer is YES (\(2x+5y\leq{9}\)). Not sufficient.

(1)+(2) From (1) we know that \(x=2y\). Substitute this in (2): \(8(2y)+5y \gt 20\), from which we can deduce that \(y \gt \frac{20}{21}\). However, this is still not sufficient to determine whether \(y\leq{1}\). Not sufficient.


Answer: E

i have a doubt. in statement 1y<=1. So at the maximum y=1 then x=2 so the cost is 5+4 =9. so we can purchase using 9. so why it is not A? please help me to understand ...

The ratio of \(x\) to \(y\) is 2 to 1 does NOT meant that the least value of y is 1. For instance it can be that x = 4 and y = 2, or x = 10 and y = 5, and so on.
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I completely agree with you...but in that equation we get y<=1... So naturally the maximum value of y is 1.. am I right...this is is making me confused

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Bunuel
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M24-04 29 Jul 2023, 01:53
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Giyo
I completely agree with you...but in that equation we get y<=1... So naturally the maximum value of y is 1.. am I right...this is is making me confused

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No, we did not get that y <= 1, the question became: is \(2(2y)+5y\leq{9}\) or is \(y\leq{1}\)? It's not a fact we derived, it's how the question transformed.
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M24-04 29 Jul 2023, 02:01
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Bunuel
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Official Solution:


The price of Mixture A is $2 per gallon and the price of Mixture B is $5 per gallon. Is $9 enough to buy \(x\) gallons of Mixture A and \(y\) gallons of Mixture B?

The question asks whether \(2x+5y\leq{9}\).

(1) The ratio of \(x\) to \(y\) is 2 to 1.

This implies that \(\frac{x}{y}=\frac{2}{1}\) or that \(x=2y\). Thus, the question becomes: is \(2(2y)+5y\leq{9}\) or is \(y\leq{1}\). Since we don't know that, this statement is not sufficient.

(2) $10 is NOT enough to buy \(2x\) gallons of Mixture A and \(\frac{y}{2}\) gallons of Mixture B.

So, we are told that \(2*2x+5*\frac{y}{2} \gt 10\), or that \(8x+5y \gt 20\). If \(x=0\) and \(y=5\), then the answer is NO (\(2x+5y > 9\)) but if \(x=3\) and \(y=0\), then the answer is YES (\(2x+5y\leq{9}\)). Not sufficient.

(1)+(2) From (1) we know that \(x=2y\). Substitute this in (2): \(8(2y)+5y \gt 20\), from which we can deduce that \(y \gt \frac{20}{21}\). However, this is still not sufficient to determine whether \(y\leq{1}\). Not sufficient.


Answer: E

i have a doubt. in statement 1y<=1. So at the maximum y=1 then x=2 so the cost is 5+4 =9. so we can purchase using 9. so why it is not A? please help me to understand ...

The ratio of \(x\) to \(y\) is 2 to 1 does NOT meant that the least value of y is 1. For instance it can be that x = 4 and y = 2, or x = 10 and y = 5, and so on.
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still i am not getting it..whatever value we put x=2y and if we substitute it in 1 we get y<=1... right
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pudu
still i am not getting it..whatever value we put x=2y and if we substitute it in 1 we get y<=1... right
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No. Again, you are mixing what is given and what is asked.

The question asks whether \(2x+5y\leq{9}\).
(1) says that the ratio of \(x\) to \(y\) is 2 to 1.

If x = 1 and y = 1/2, the answer to the question whether \(2x+5y\leq{9}\) would be YES!
If x = 10 and y = 5, the answer to the question whether \(2x+5y\leq{9}\) would be NO!

Two different answers. Hence, not sufficient.
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Bunuel
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Bunuel
Official Solution:


The price of Mixture A is $2 per gallon and the price of Mixture B is $5 per gallon. Is $9 enough to buy \(x\) gallons of Mixture A and \(y\) gallons of Mixture B?

The question asks whether \(2x+5y\leq{9}\).

(1) The ratio of \(x\) to \(y\) is 2 to 1.

This implies that \(\frac{x}{y}=\frac{2}{1}\) or that \(x=2y\). Thus, the question becomes: is \(2(2y)+5y\leq{9}\) or is \(y\leq{1}\). Since we don't know that, this statement is not sufficient.

(2) $10 is NOT enough to buy \(2x\) gallons of Mixture A and \(\frac{y}{2}\) gallons of Mixture B.

So, we are told that \(2*2x+5*\frac{y}{2} \gt 10\), or that \(8x+5y \gt 20\). If \(x=0\) and \(y=5\), then the answer is NO (\(2x+5y > 9\)) but if \(x=3\) and \(y=0\), then the answer is YES (\(2x+5y\leq{9}\)). Not sufficient.

(1)+(2) From (1) we know that \(x=2y\). Substitute this in (2): \(8(2y)+5y \gt 20\), from which we can deduce that \(y \gt \frac{20}{21}\). However, this is still not sufficient to determine whether \(y\leq{1}\). Not sufficient.


Answer: E

i have a doubt. in statement 1y<=1. So at the maximum y=1 then x=2 so the cost is 5+4 =9. so we can purchase using 9. so why it is not A? please help me to understand ...

The ratio of \(x\) to \(y\) is 2 to 1 does NOT meant that the least value of y is 1. For instance it can be that x = 4 and y = 2, or x = 10 and y = 5, and so on.
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okay...so x=6 and y=3...still if we put x=2y i the given equation we get y<=1....so it is trap....i think i understood...thank you for such a patient listening... :)
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I think this is a high-quality question and I agree with explanation.
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I’m so confused. Where are multiple choice options for this question?
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Welcome to GMAT Club :cool:

Data Sufficiency questions (all questions) share the same answer choices.
Even the Official Guide does not post them as you are expected to know them by heart. They are below for reference.

  1. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
  2. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are not sufficient.


txu5008
I’m so confused. Where are multiple choice options for this question?
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M24-04 28 Jul 2025, 08:41
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Bunuel,

General question, for (2): You have taken two sub cases involving x or y=0 -> shouldn't we take some other cases to solve - because since question said x gallons and y gallons is bought, each should be bought in SOME quantity greater than 0, right?

(Although I understand that using other cases, we can still prove it:
e.g:
- case 1: x=3, y=5 => NO
- case 2: x=2, y=1 => YES)
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Bunuel,

General question, for (2): You have taken two sub cases involving x or y=0 -> shouldn't we take some other cases to solve - because since question said x gallons and y gallons is bought, each should be bought in SOME quantity greater than 0, right?

(Although I understand that using other cases, we can still prove it:
e.g:
- case 1: x=3, y=5 => NO
- case 2: x=2, y=1 => YES)
Show more
First of all, the question does not say that any amount was bought. It asks whether $9 is enough to buy x gallons of Mixture A and y gallons of Mixture B. So there's no requirement that x or y be greater than 0.

Therefore, values like x = 0 or y = 0 are completely valid to test. There's nothing in the question preventing that.
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I like the solution - it’s helpful.
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Basically 2x+5y <= 9?


Statement 1....x:y = 2:1

Case 1. X = 2, y=1
Inputting ..2x + 5y = 9..Yes...

Case 2...x = 4, Y =2 ...
Inputting 2x + 5y = 18...No....

Not Sufficient
4x + 5y/2 > 20
8x +5y > 20 .......
x = 2, y=1...Satisfies statement 2 and is equal to 9..Yes Case....
x=4, y = 2...Satisfies statement 2 and is greater than 9....No Case..

Combining...Same cases we can use....and still not get the answer.......

Answer E....









Bunuel
The price of Mixture A is $2 per gallon and the price of Mixture B is $5 per gallon. Is $9 enough to buy \(x\) gallons of Mixture A and \(y\) gallons of Mixture B?



(1) The ratio of \(x\) to \(y\) is 2 to 1

(2) $10 is NOT enough to buy \(2x\) gallons of Mixture A and \(\frac{y}{2}\) gallons of Mixture B
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