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.
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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

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

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.

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?

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.

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.
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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.
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