If Bob produces 36 or fewer in a week, he is paid X dollars

If Bob produces 36 or fewer items in a week, he is paid X dollars per item. If Bob produces more than 36 items in a week, he is paid X dollars per item for the first 36 items and 3/2 times that amount for each additional item. How many items did Bob produce last week?

First let's set the equation for Bob's income:

When \(n\leq 36\), then \(I=nx\);

When \(n>36\), then \(I=36x+(n-36)1.5x=1.5nx-18x\).

The question asks to find the value of n.


(1) Last week Bob was paid total of $480 for the items that he produced that week.

So, \(I=480\).

Either \(I=480=nx\) OR \(I=480=1.5nx-18x\).

Not sufficient.


(2) This week Bob produced 2 items more than last week and was paid a total of $510 for the items that he produced this week.

So, \(I'=510\) and \(n'=n+2\).

Either: \(I'=510=(n+2)x\) OR \(I'=510=1.5(n+2)x-18x\).

Not sufficient.


(1)+(2) We can have three different systems of equations:

\(480=nx\) and \(510=(n+2)x\), meaning that \(n+2\leq 36\). In this case \(x=15\) and \(n=32\).

OR:

\(480=nx\) and \(510=1.5(n+2)x-18x\), meaning that \(n+1\leq 36\) and \(n+2>36\) (\(n=35\)). In this case n has no integer value, so this system doesn't work;

OR:

\(480=1.5nx-18x\) and \(510=1.5(n+2)x-18x\), meaning that \(n>36\). In this case \(x=10\) and \(n=44\).

So we can have two values for n: 32 and 44. Not sufficient.

Answer: E.

The last step can be done in another way:

We know that 2 more items resulted 30$ more.

If these two items were paid by 1.5x rate (n>=36) --> 1.5x+1.5x=30 --> x=10 and as n>=36, we should substitute this value in the second equation from (1), which gives n=44

If these two items were paid by x rate (n<=34) --> x+x=30 --> x=15 and as n<=34, we should substitute this value in the first equation from (1), which gives --> n=32

Already two different answers for n (no need to check for the third case when one item is paid by regular rate and another with overtime rate), hence insufficient.

Answer: E.
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Clearly each statement alone is NOT sufficient

Now, let's combine and see what happens

We have an extra US$30 revenue from the sale of two extra items. Now then, we could have two scenarios.

Original price 'x' equals to 15, then 2 extra items account for the US$30 difference (510 - 480). So we would have 480/15 = 32 items.

Or we could have that two items again represent US$30 but original price is in fact US$10 and US$15 are the 1.5x for each item > 36. Therefore, Maximum # with US$ 10 tag is 36=US$360. The rest is US$120 / 15 = 8 additional. So we would have a total of 36+8=44 items.

Two possible answers

Answer: E

Hope it clarifies
Cheers!
J

IMO E

we are having two variables no. of item is not known and price is x

with bth also we are not getting any thing

E,...
combining two st... $30 is price of either both at x(no<=36),one at x ,other at 1.5x. or both at 1.5x.....
so one sol x=15...items 32...
2nd sol x=10....items 360/10+120/15=42.....
3rd sit of one of x and second at 1.5x not possible
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and E

if you assume the no of items is less than 36, you get

x* (no of items) = 480
x* (no of items + 2) = 510
==> no of items = 15

if you assume the no of items is more than 36, you get

36* x + ( no of items - 36)*(1.5x) = 480
36*x + (no of items + 2 - 36)*(1.5x) = 510
==> no of items = 44

clearly the same set of information leads to 2 possible solutions
therefore additional data is required

ans : E

Its a pretty lengthy problem any other way to solve this under 2.5 mins?

Unfortunately not all question have "a silver bullet" solutions.
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I'm still not too clear on when it is appropriate to stop and be confident on the solution choice. I got to the part with the 2 equations combined, but then I selected "e", hoping there wasn't a constraint trick bc we clearly have too few equations and multiple unknowns. For some reason, I'm coming up on the 2 min mark after checking all of the previous out and it takes me another 2 mins (and a lot of energy) to find the exact #'s in Bunuel's solution. Is there a way to be confident without getting exact values for price and quantity in the combined statements case?

bump: MathRevolution, VeritasPrepKarishma, etc.

Let me try to answer.

You have not completely figured out the main crux of the problem. You need to be absolutely sure of whether the items were \(\leq\) 36 or > 36 . This will determine (for example) in statement 1 you need to use I =nx or I=1.5x(n-36)+36x (lets calls them equations 1.a and 1.b respectively). So, you get 2 distinct equations with no basis to eliminate either one. Thus this statement is NOT sufficient.

You will be able to use I=nx if x \(\leq\) 36 but if it is > 36, then you must use I=1.5x(n-36)+36x. This ambiguity makes this statement NOT sufficient.

You can now see that there might be a catch in this question based on analysis of statement 1 alone. Now, go onto statement 2 and you will again realize that there is no basis to eliminate 1 of the 2 possible equations (lets calls them equations 2.a and 2.b respectively) as you still have not been provided any information about the number of the items. They can be \(\leq\) 36 but at the same time can also be >36. We have no justification to choose 1 option over the other.

Again, you get 2 more distinct equations , making statement 2 not sufficient alone.

For combining the 2 statements, you can now clearly see that you can have the following 2 sets of distinct equations:

1. 1.a and 2.a or
2. 1.b and 2.b

Either way, you will not be getting the same answer ---> E is thus the correct answer. But be careful about this step as if you do get the same result for 'x' or 'n' from the 2 systems of equations, then it will be C instead of E.

In totality, it took close to 1-1.5 minutes to analyse both statements individually with another 45-60 seconds to analyse both statements together and mark E as the answer.
Remember that in GMAT Quant, you need to spend an AVERAGE of 2 minutes per question and not more. This does not mean that all questions will take you full 2 minutes. Some of the questions are bound to take less than 2 minutes while some of them will take you >2 minutes, bringing the average close to 2 minutes per question.

As this is a 'difficult' question as categorized by the GMATCLUB timer results, it is fine if you spent 2-3 minutes on this question. In GMAT, you should be able to recover this time if you know how to pick your battles and move on.

Hope this helps.


P.S.:

1. Having 2 equations for 2 variables may or may not sufficient to give you a sufficient answer.

Example, 2a+3b=6 and 4a+6b=12 are although 2 equations but they are NOT distinct and you will not get a unique value for a,b. Thus you need to check for DISTINCT equations and not just any equations while solving for variables.

2. Lets say you ended up getting 2 systems of equations as

a) 2x+3y=4 and x+y=4 ,you get x=3 and y=1

b) 3x+4y=13 and 2x+y=7, you again get x=3 and y=1

Thus in this case, you must mark C as you are getting the same unique values for x and y.
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Engr2012, that makes sense and thank you for your response. However, I'm wondering how much calculation we have in this specific scenario since there are actually 3 cases:

1) 1a & 2a
2) 1a & 2b
3) 2b & 3b

Without putting out the numbers is there anyway we can pick "e" without finding the exact prices and quantities for x & y given that the total number is the only constraint and prices can be decimal quantities?

You are correct that we will have 3 systems of equations as Bunuel had mentioned in his post but in effect you only have 2 systems as the 2nd one mentioned below does not work as '' can only take integer values.

\(480=nx\) and \(510=(n+2)x\), meaning that \(n+2<=36\). In this case \(x=15\) and \(n=32\).

OR:
\(480=nx\) and \(510=1.5(n+2)x-18x\), meaning that \(n+1<=36\) and \(n+2>36\) (\(n=35\)). In this case n has no integer value, so this system doesn't work;

OR:
\(480=1.5nx-18x\) and \(510=1.5(n+2)x-18x\), meaning that \(n>36\). In this case \(x=10\) and \(n=44\).

But your reason for choosing E is not correct. We are marking 'E' as you will end up getting more than 1 sets of values for x and n, hence even when you combine the statements, you will not get unique values for x and n.

Hope this helps.
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Engr2012, thanks for entertaining my questions. Maybe I'm dumb here, but I can't solve these three simultaneous situations as quickly as you all seemingly do. Would my thinking be incorrect to think, at a high level and through inspection:

1) We have two equations where we know x will be (510 - 480)/2 = 15. N must be 480/15 = 960/30 = 32. Ok this holds.
2) Maybe this is right, I can't tell n can't be an integer...
3) Maybe this is right, I've tapped out on my mental capacity for this question and I need to finish this exam...

E seems more likely to be correct because we only need one more solution that works and for c to be right 2 & 3 must either give the same answer a (1), not be possible bc n must be an integer, or a combination of those. I just can't see myself reasonably throwing any more firepower at a question like this so that's why I'm asking for a shortcut.

You are not dumb to ask these questions. Asking questions only goes to show that you are actually thinking about this question along the right path. As for us figuring out the systems, it all comes down to practicing similar DS questions. Once you have solved 50-100 questions, you will start seeing patterns. GMAT is all about pattern recognition and time management. These 2 things come only after you have practiced a lot.

As mentioned in my post above if you see a difficult question such as this and you have already spent 2-2.5 or even 3 minutes to reach the last combined step, then yes, your high level assumptions might work. This will atleast help you to cut your losses further timewise and move on to the next question. Your explanation for (1) is correct and the same goes for (2). For (3) if you see running short on time, mark E and move on.

1 way I see between 1 and 3 is to recognise that the equations are very different in these 2 systems and your best bet will be that you will end up getting 2 different values for x and n . This in itself will be sufficient for you to mark E and move on. You dont even need to check (2) as you already have 2 different answers.

But to close the loop on this discussion, for (3) in order to look at the values, you will have to solve the equations to get x=10 and n=44. There is no other way I see to circumvent this requirement and to be absolutely sure that you will not end up getting the same numbers as those you got from (1). But this route is only recommended when you are still within the 2-2.5 minute mark.

Hope this helps.

The more straightforward way to solve this system is to substitute for 'n' in order to get x=10. After this you can plug in this value of x to get n.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If Bob produces 36 or fewer items in a week, he is paid X dollars per item. If Bob produces more than 36 items in a week, he is paid X dollars per item for the first 36 items and 3/2 times that amount for each additional item. How many items did Bob produce last week?

(1) Last week Bob was paid total of $480 for the items that he produced that week.

(2) This week Bob produced 2 items more than last week and was paid a total of $510 for the items that he produced this week.


When you modify the original condition and the question, you need to figure out the number of item he produced(L), the number of item he produced this week(T), and x. So there are 3 variables, which should match with the number of equations. So you need 3 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer.
When 1) & 2), they become 36x+(L-36)(3x/2)=480, 36x+(L+2-36)(3x/2)=510 or Lx=480 and (L+2)x=510, which is not unique and not sufficient. Therefore, the answer is E.


 For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

Statement 1:
Clearly not sufficient

Statement 2:
Clearly not sufficient

1 + 2

Total paid last week = 480
Total paid this week = 510

510 - 480 = 30/2 = 15 each when assuming items are > 36

15/1.5 = 10 each when assuming items are <= 36

I believe the confusion begins when people are forgetting that statements 1 and 2 are not providing enough details about total items.
if you take 480/10 = 48 items then you have >36 items
if you take 480/15 = 32 which are < 36

The question is... do you know with certainty how many items?

because of this ambiguity of two possible answers, then statements 1 and 2 are not sufficient. Therefore, answer is E

Whenever we have a word problem, like this one, we want to translate the words into math. Scanning over the problem, we see the phrases “36 or fewer” and “more than 36” — these are classic signs that we’re dealing with inequalities. This particular problem gives us two scenarios for calculating how much Bob is paid based on how many total items he produces in a given week (one for 36 or fewer items, one for more than 36 items), so we want to create two equations: one for each scenario. Letting i = the number of items Bob makes in a given week, we can translate our first scenario as

Ifi≤36thentotalpay=x×i

Our second sentence is a little more complicated. If Bob produces more than Bob is paid x for the first 36 items (or 36x). Then for all of the items after 36 (or i−36), he is paid 1.5x (or 1.5x×(i−36)). Putting that together,

Ifi>36thentotalpay=x×36+1.5x×(i−36)

So we have two equations, each with three variables (i, x, and totalpay) … which means we need a bunch of information to figure out an answer. To figure out a value for i, we need information about

which of the two equations to use
the value of x
the total pay

Statement 1

This statement tells us how much Bob was paid last week, but it doesn’t tell us anything about the specific value of x or which of the two equations we should use. So we could have:

i=1andx=480→480=480×1

or

i=32andx=15→480=15×32

or

i=76 andx=5→480=5×36+1.5(5)×(40)

and so on. Statement 1 is insufficient.

Statement 2

This one tells us how much Bob was paid this week, and it compares the number of items he produced this week to the number he produced last week. Well, we don’t know anything about how many items Bob produced last week, so the last piece of information doesn’t tell us much about x — he could have produced 1 item last week and 3 this week or 100 items last week and 102 this week. And, like in Statement 1, we don’t know whether or not i is greater than 36, so we don’t know which statement to use. So we could have:

i=4andx=145→580=145×4

or

i=29andx=20→580=20×29

or

i=41andx=1313→580=1313×36+1.5(1313)×(5)

and so on. Statement 2 is insufficient.


BOTH

What if we put the two statements together? Well, now we know something: the additional two items Bob produced this week earned him $30 more than he earned last week. This means that Bob earned an extra /$15 per item. But we’re still missing a key piece of information: which scenario are we dealing with?

Did Bob produce 36 or fewer items this week? If so, then both items were produced at a rate of x, so that x=15.
Did Bob produce at least 38 items this week? If so, then both items were produced at a rate of 1.5x, so that 1.5x=15 → x=10?
OR did Bob produce exactly 35 items last week and 37 items this week? If so, then the first item was produced at a rate of x and the second item was produced at a rate of 1.5x, so that x+1.5x=30 → 2.5x=30 → x=12.

We’ve got a few options here, so let’s try each individually. Remember, we want to solve for the number of items Bob produced last week, so we’ll use that equation:

x=15, 480=15i → i=32
x=10, 480=36(10)+1.5(10)(36−i) → 480=360+15(36−i) → 120=15(i−36) → 8=i−36 → i=44

We already have two possible solutions, so we don’t need to look at our third, more complicated option. We cannot determine whether Bob made 32 or 44 items last week, so we cannot solve the problem with both statements. The correct answer is E: Statements 1 and 2 TOGETHER are NOT sufficient to answer the question.

Video solution from Quant Reasoning:
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Hi Bunuel please advise once we know that their can be three different possibilities for a situation and hence we will be getting three different values do we actually need to compute them.

Or we can straightforward rule it out insufficient together

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n represents the number of items, so it has to be an integer. As you can see from three cases for (1)+(2) only two give integer values of n but if only one of the cases gave integer value of n, then the answer would have been C, not E. So, unfortunately we have to solve to check how many of the cases give valid solutions of n.

Hope it's clear.
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