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

If i ≤ 36, then total pay=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,

If i > 36, then total pay=x*36 + 1.5x*(i-36)

So we have two equations, each with three variables (i, x, and total pay) ... 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=1 and x=480, so 480=480*1

or

i=32 and x=15, so 480=15*32

or

i=76 and x=5, so 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=4 and x=145, so 580=145*4

or

i=29 and x=20, so 580=20*29

or

i=41, and x=13 1/3, so 580=13 1/3*36 + 1.5(13 1/3)*(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, so 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, so 2.5x=30, so 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, so i=32

• x=10, 480=36(10)+1.5(10)(36-i), so 480=360+15(36-i), so 120=15(i-36), so 8=i-36, so 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.

We actually featured this problem recently on the PrepScholar GMAT blog as one of the 5 Hardest Data Sufficiency Questions. I recommend checking out the article for more strategies and trends we can take away from this and other 700+ level problems!