A company makes and sells two products, P and Q. The costs per unit of

From the stem we get:

Profit of P = 10 – 8 = 2
Profit of Q = 13 – 9.5 = 3.5

P + Q = 834 => 2P = 2(834 – Q) => 2P = 1668 – 2Q

Question : 2P + 3.5Q > 2000 ?
Substitute 2P in the question: (1668 – 2Q) + 3.5Q > 2000?

The question becomes: is Q > 221.3?

1) P > Q = > 417 > Q > 0, INSUFFICIENT
2) Q > 100 => INSUFFICIENT
1 and 2 => 417 > Q> 100 => INSUFFICIENT. If Q is 300 the answer is YES; if Q is 200 the answer is NO.

Answer is E

p+q = 834
we need to know whether 2p + 3.5q > 2,000
(1) During the month, more units of P than units of Q were sold.
=> p>q
=> 2p + 3.5q >5.5q
But we do not know the value of q => insufficient
(2) During the month, at least 100 units of Q were sold.
=> q>=100, p <=734
=> 2p + 3.5q >= 2p + 350
Insufficient
(1) + (2): p>834/2 = 417 => 2p + 3.5q >= 2p + 350 >834+350

Do not know the valute of 2p + 3.5q

Insufficient

Ans: E

I think the answer is 'A,' I am unsure why 'E' is the answer. The stem basically asks is "2P + 3.5Q > 2,000" and the stem also states 834 unites were sold. Stmt A states that "p > q" so the lowest values to make that statement true is 418 (p) and 416 (q) which gives us 836 + 1,456 which is greater than 2,000.

Where is the hole in my thinking?

AS7x ,

Statement1 : p> q
p can take any value from 418 to 834 .Let's consider the below 2 cases-


Case 1.If p = 834
then 2p = 1668 < 2000

Case 2.If p = 418 , q = 416
2* 418 + 416 * 3.5 , which will be greater than 2000

Hope it helps!!

Let's talk strategy here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full "GMAT Jujitsu" for this question:

The target of this question is "total profit", and we know that profit is equal to the selling price minus the cost. Thus, the profit per unit for product P would be \($_P = 10-8 = 2\), and the profit per unit for product Q would be \($_Q = 13-9.5 = 3.5\). The total profit would therefore be (where \(P\) and \(Q\) represent the number of units of each product):

\(T=P($_P )+Q($_Q)\)
\(T=P(2)+Q(3.5)\)

Now that we have set up the basic equation, we need to pause in our discussion here. Many people spend too much time on Data Sufficiency questions because they think they need to get to the bitter end. The question asks “Was the total profit on these items more than $2,000.00?” This is a “Yes/No” question – a very common structure for Data Sufficiency problems. The fundamental trap for problems like these is to bait you into thinking that you actually need to solve for every value. You don’t. As soon as you have enough information to conclude whether you can come up with only one answer to the question, you can move on. For “Yes/No” questions, if you can think of two situations (or two variable inputs) that are consistent with all of the problem’s constraints but come up with different answers to the question, you know a statement is insufficient. In my classes, I call this strategy “Play Both Sides.”

Let’s analyze each statement, and you will see what I mean. Statement #1 tells us that there are more units of \(P\) than there are units of \(Q\). Now, look at the shape of the equations. From the question stem, we know that \(P+Q=834\). “834” is a bit of a messy number, so approximating this would make our analysis a lot easier. First, let’s assume that \(P\) is roughly equal to \(Q\), making both values around \(400\). Thus,

\(T=P(2)+Q(3.5)≈400(2)+400(3.5)≈800+1400≈2200\)

So, it is possible to exceed \($2,000\), and we know we can answer “Yes” to the question. The next question we ask ourselves is: Can we play both sides? Is it possible to have a solution where the total is under \($2,000\)? If \(P\) dominated the relationship (in other words, if \(P\) were around \(800\), then we would have less than \($2,000\). After all, \(800*2\) is only equal to \($1,600\). We are not even close to the boundary of \($2,000\), so approximating is sufficient here. Since one scenario would clearly give us a combined profit of less than \($2,000\) while the other would give a total profit of more than \($2,000\), Statement #1 is insufficient by itself.

Statement #2 tells us that \(Q\) is “at least 100 units.” We have already shown by our analysis of Statement #1 that if \(Q\) is around \(400\), then the total will exceed \($2,000\). There is no reason to redo this. As we “Play Both Sides”, we already know that there is a solution where \(Q\) is “at least 100 units” that allows us to answer [color=#00a651]“Yes”[/color] to the original question. So, we need to know if we can show that the total may be less than \($2,000\). The phrase “at least” is massive leverage. This is a minimum amount, so let’s see what a minimum amount would do – in other words, if \(Q=100\). Again, approximating here is more enough. If \(Q=100\), then \(P≈700\). Thus:

\(T≈700(2)+100(3.5)≈1400+350≈1750\)

Since we can also show that the total can be less than \($2,000\), we have “played both sides” and Statement #2 is insufficient.

We now need to look at combining them to see if they are sufficient. But we have already done all the analysis we need. The hypothetical situations we created for Statements #1 and #2 show two possibilities where (1) P can be greater than Q and (2) Q is at least 100. Since we can “play both sides” of this Yes/No question even when the statements are simultaneously true, the answer is “E”.

Now, let’s look back at this problem from the perspective of strategy. This problem can teach us several patterns seen throughout the GMAT. First, minimize your math. You only need to do enough math to prove the answer to the question. With Yes/No questions, this often means “playing both sides” – coming up with two possible solutions that follow the rules of the problem, but give different answers to the question. Second, notice how approximation can often cut through otherwise messy math. This saves time and energy, allowing you to focus on other, potentially harder, questions. And that is how you think like the GMAT.

VeritasKarishma Bunuel chetan2u MathRevolution
What does this language mean :
costs per unit of making and selling P and Q are $8.00 and $9.50

cost of making and selling P is 8
this is Cost Price, i understand but what does this mean?
should not it be the cost of making only?
sellling cost is sp ,right ?

Cost of making and selling would mean two items - one the actual cost of material and labour AND second the cost of selling incurred in transporting or advertisement or packing or whatever it may be etc

Solution:

We see that product P has a profit of $2 per unit and product Q has a profit of $3.50 per unit. If we let p = the number of units of product P sold and q = the number of units of product Q sold, we can create the equations:

p + q = 834

and

2p + 3.5q = total profit

We need to determine whether 2p + 3.5q > 2000.

Statement One Only:

During the month, more units of P than units of Q were sold.

From statement one, we see that at least 834/2 + 1 = 418 units of P were sold, and at most 834/2 - 1 = 416 units of Q were sold. If exactly 418 units of P and 416 units of Q were sold, then the total profit is 2(418) + 3.5(416) = $2292, which is more than $2000. However, if 834 units of P and no units of Q were sold, then the total profit is 2(834) = $1668, which is less than $2000.

Statement one alone is not sufficient to answer the question.

Statement Two Only:

During the month, at least 100 units of Q were sold.

From statement two, we see that at least 100 units of Q were sold and at most 834 - 100 = 734 units of P were sold. If exactly 734 units of P and 100 units of Q were sold, then the total profit is 2(734) + 3.5(100) = $1818, which is less than $2000. However, if 418 units of P and 416 units of Q were sold, then the total profit is $2292 (see the analysis for Statement One Only), which is more than $2000.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

With the two statements, it’s possible for the total profit to be more than $2000 (for example, 418 units of P and 416 units of Q were sold, for a total profit is $2292) or less than $2000 (for example, 734 units of P and 100 units of Q were sold, for a total profit is $1818).

The two statements together are still not sufficient to answer the question.

Answer: E

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Answer: Option E

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Given: A company makes and sells two products, P and Q. The costs per unit of making and selling P and Q are $8.00 and $9.50, respectively, and the selling prices per unit of P and Q are $10.00 and $13.00, respectively. In one month the company sold a total of 834 units of these products.

Profit made by selling one unit of product P = $10.00 - $8.00 = $2.00
Profit made by selling one unit of product Q = $13.00 - $9.50 = $3.50

Let P = the number of units of product P sold.
Let Q = the number of units of product Q sold.
So, the TOTAL profit = 2P + 3.5Q

Also, since the company sold a total of 834 units, we can write: P + Q = 834

Target question: Was the total profit on these items more than $2,000?

Statement 1: During the month, more units of P than units of Q were sold.
In other words: P > Q
Let's examine two extreme cases:
Case a: P = 833 and Q = 1 (since we must satisfy the condition that P + Q = 834). In this case, the profit = 2(833) + 3.5(1) = $1669.50. So, the answer to the target question is NO, the total profit was not greater than $2000
Case b: P = 418 and Q = 416. In this case, the profit = 2(418) + 3.5(416) = $2292. So, the answer to the target question is YES, the total profit was greater than $2000
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: During the month, at least 100 units of Q were sold.
There are many scenarios that satisfy statement 2 (and the given information). Here are two:
Case a: P = 724 and Q = 100. In this case, the profit = 2(724) + 3.5(100) = $1798. So, the answer to the target question is NO, the total profit was not greater than $2000
Case b: P = 418 and Q = 416. In this case, the profit = 2(418) + 3.5(416) = $2292. So, the answer to the target question is YES, the total profit was greater than $2000
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that the counter-examples I used for statement 2 satisfy BOTH statements. So, those same counter-examples will show that the two statements COMBINED are not sufficient.
In other words,
Case a: P = 724 and Q = 100. In this case, the profit = 2(724) + 3.5(100) = $1798. So, the answer to the target question is NO, the total profit was not greater than $2000
Case b: P = 418 and Q = 416. In this case, the profit = 2(418) + 3.5(416) = $2292. So, the answer to the target question is YES, the total profit was greater than $2000
Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

Video solution from Quant Reasoning:
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­Is 2p + 3.5q > 2000
ie,
4p +7q > 4000 ?
Given: p+q = 834

0) If p=q, then
           p = 417, q = 417
1) SAY p > q
                        A) p = 418, q = 416…………..yes (for mind calculations, take p = 400, q=400…if ans is yes then stop, if ans is NO                                                                              then try for actual p & q)
                                                                             
                        B) p = 834, q = 0……………….no (for mind calculation, take p = 800 u get ans as 4(800), now add 4*30=120 and                                                                              still it’s not near 4k)
                                                                                
                         INSUFFICIENT
2) SAY q>=100
                         A) p = 734, q = 100……………no

                         B) check if we have done q > 100 where answer is a yes,
                              Oh yea, it was done in 1A case
                              p = 418, q = 416…………..yes
                              else u could have simply done p = 0, q=834………….yes
                          INSUFFICIENT

Now we need 1 & 2 both together,
for this select from 1 where 2 is valid(ie, where q>=100)
1A) p = 418, q = 416…………..yes
Now select from 2 where 1 is valid(ie, where p>q)
2A) p = 734, q = 100……………no
INSUFFICIENT

Therefore the Answer is option E



 ­­­­

Bunuel KarishmaB

If the question was changed to Was the total profit on these items more than $1,500? Instead of $2,000, statement 2 would be sufficient and hence the answer is B. Is that correct?

Posted from my mobile device

­Profit for each Q sold = (selling price) - (cost per unit) = 13 - 9.5 = 3.5
Profit for each P sold = (selling price) - (cost per unit) =10 - 8 = 2

When analyzing the statements, test EXTREMES.

Case 1: 100 units of Q are sold, 734 units of P are sold, for a total of 834 units
Total profit = (100*3.5) + (734*2) = 350 + 1468 = 1818

Case 2: 416 units of Q are sold, 418 units of P are sold, for a total of 834 units
Total profit = (416*3.5) + (418*2) = 1456 + 836 = 2292

Each case satisfies the two statements.
Since the answer to the question stem is NO in Case 1 but YES in Case 2, the two statements combined are INSUFFICIENT.

­