Data Sufficiency

Source: Veritas Prep

Steve works at an apple orchard and is paid by the bushel for the apples he harvests each day. For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel. For each additional bushel over 42, he is paid 1.5y. How many bushels of apples did Steve harvest yesterday?

(1) Yesterday, Steve was paid $180 for the apples that he harvested.
(2) Today, Steve was paid $240, and he harvested 10 more bushels of apples than he harvested yesterday.

Did you guys know it's really hard to pick plums to make vodka, by the way?

Here is the official explanation:

Correct answer: (E)

Solution: Because neither statement gives us information about y, neither statement alone is sufficient. For instance, in Statement (1) alone, Steve could have only harvested one bushel if y is $180, or he could have harvested 30 bushels if y is $6. Statement (1) is not sufficient, so eliminate answers (A) and (D). Similarly, because Statement (2) alone contains no information about y, it is also not sufficient. Eliminate answer (B). Choosing between (C) and (E) can be difficult. It appears that we can determine one value for y if we combine the two statements. If Statement (2) tells us that the additional 10 bushels added $60 to yesterday's wages, the value of y appears to be $6. Using this y, it appears that Steve harvested 30 bushels yesterday for $180 and 40 bushels today for $240. Before choosing answer (C), however, observe that the question stem gave us two potential scenarios. If those ten additional bushels that Steve harvested were already over the threshold of 42, the value of y would be $4 (10[1.5y]) = 60). Using y = $4 in Statement (1) suggests that Steve harvested 44 bushels yesterday. He would have earned $4 per bushel for the first 42 bushels ($168), and then the increased rate of $6 per bushel for the remaining 2 ($12), for a total earned of $180. Since, when we combine Statements (1) and (2), there are two possible scenarios for the number of bushels harvested yesterday, the answer is (E).


By the way, GMATmachoman, I see what you are saying about the official question with the "if." However, the Veritas question is worded differently and does not need the "if." The phrasing, "For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel" implies the "if" and, I think, does so a little more compactly than does the official question. If this was sentence correction the Veritas sentence wins since they convey the same information and this question does so more efficiently!

seems like E, my rationale. t- is the number of bushels.
(1)insufficient as we have two unknows, y and t
(2)insuff for the same reasons
together
if he picked fewer than <42 bushels,
ty=180, (t+10)y=240. solving we obtain y=6, and t=30

but on the other hand if t>42, then
42y+(t-42)1.5y=180
42y+(t+10-42)1.5y=240, solving for those
y=4. and t=some other number other than 30,
as we have two answers both also insuff

DanaJ wrote:Source: Veritas Prep

Steve works at an apple orchard and is paid by the bushel for the apples he harvests each day. For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel. For each additional bushel over 42, he is paid 1.5y. How many bushels of apples did Steve harvest yesterday?

(1) Yesterday, Steve was paid $180 for the apples that he harvested.
(2) Today, Steve was paid $240, and he harvested 10 more bushels of apples than he harvested yesterday.

Did you guys know it's really hard to pick plums to make vodka, by the way?

Statement 1:
If y=2, Steve was paid 2*42=84 for the first 42 bushels.
180-84 = 96.
The payrate for each additional bushel would be 1.5y = 1.5*2 = 3.
Thus, Steve would harvest 96/3 = 32 additional bushels.
Total bushels = 42+32 = 74.

If y=4, Steve was paid 4*42=168 for the first 42 bushels.
180-168 = 12.
The payrate for each additional bushel would be 1.5y = 1.5*4 = 6.
Thus, Steve would harvest 12/6 = 2 additional bushels.
Total bushels = 42+2 = 44.

Since the number of bushels could be 74 or 44, insufficient.

Statement 2:
If y=2, Steve was paid 2*42=84 for the first 42 bushels.
240-84 = 156.
The payrate for each additional bushel would be 1.5y = 1.5*2 = 3.
Thus, Steve would harvest 156/3 = 52 additional bushels.
Total bushels today = 42+52 = 94.
Total bushels yesterday = 94-10 = 84.

If y=4, Steve was paid 4*42=168 for the first 42 bushels.
240-168 = 72.
The payrate for each additional bushel would be 1.5y = 1.5*4 = 6.
Thus, Steve would harvest 72/6 = 12 additional bushels.
Total bushels today = 42+12 = 54.
Total bushels yesterday = 54-10 = 44.

Since the number of bushels yesterday could be 84 or 44, insufficient.

Statements 1 and 2 together:
Total payment today = 240, total payment yesterday = 180.
The 10 additional bushels harvested today yielded 240-180 = 60 more dollars.
Thus, the payrate for each of the 10 additional bushels = 60/10 = 6.
This means either that y=6 or that 1.5y=6 and y=4. (Tricky!)
If y=6, Steve harvested today 240/6 = 40 bushels and yesterday 180/6 = 30 bushels.
If y=4, we saw in statement 2 that Steve harvested today 54 bushels and yesterday 44 bushels.
Since the number of bushels harvested yesterday could be 30 or 44, insufficient.

The correct answer is E.

I dont find any issues pick E as the correct answer.

But recently i noticed same kind of problem in a official GMAT question...

If [/spoiler]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 1.5 times that amount for each additional item. How many items did
Bob produce last week?
(1) Last week Bob was paid a 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

At the outset, people may tend to solve for y and the for total number of bushels assuming that " farmer has harvested more than 42 bushels"... Thats the trick here!!

but i feel "If" should have been added in that "veritas Prep" question.

I'm not good with these things yet so I apologize if this sounds silly but don't the 2 statements contradict each other? Again, I'm not sure because I haven't covered this quant subject yet so don't get all nasty on my butt!:P

Nobody is going to mean to you!

I think you might be referring to the fact that on GMAT questions the two statements must each be true at the same time and so are never allowed to contradict each other. Official questions will never contradict so if you think they do then you are wrong and need to go back and find away for each to be true at the same time. Unofficial questions also should not have the statements contradict, if they do it is a major problem and the question would need to be edited again.

In this case they do not contradict.

Statement 1 gives you the amount that Steve was paid yesterday for x number of bushels in this case 180 dollars. So x the price he was paid per bushel is 180/ x. X could be 3 and he gets sixty dollars per bushel or 18 and he gets ten dollars per bushel, etc. So alone statement one does not tell us what we are looking for which is the number of bushels harvested yesterday.

Statement 2 fast-forwards to today and gives you a different amount paid, $240, for x + 10 bushels (that is ten more bushels than yesterday). So this statement really does not work because it is talking about the number of bushels today and we need the number for yesterday!

This is common in data sufficiency that the question quickly comes down to two options. In this case C (both together) vs E (not enough info to solve).

Taking both together we really have two clear possibilities for the number of bushels yesterday. The question stem says that when you get over 42 bushels Steve is paid 50% more. So he could have been paid $4 per bushel for the first 42 bushels. That would be $168 dollars (4 x 42) and then another 2 bushels at $6 gets you to $180. Where did I get the $4 - well we already know that it is $6 per bushel for the extra 10 bushels today, so if that is 150% of the base pay then the base is $4.

Another option is that the base price is $6 and this means 30 bushels x $6 = $180 yesterday. Today could still be $6 for each bushel including the extra 10 bushels as this would be 40 bushels at $6 or $240 and none of these at the increased price because he is not over 42 bushels!

So no contradiction.

Good luck in you studies!

Thanks David!:) Your post helped me a lot!:)

HI:
I follows the simple rule that in order to solve 2 variables in linear equations, you need two equations.
So given that there are 2 variables x (no of bushels) and y (amount paid), we need both equations.

Now assume x >42 and solve it for both x and y, then do the same for x <=32 and you get another answer. do the same for 32<x<=42 and again you get different answer.

So you can't solve it with these two conditions....

DanaJ wrote:Source: Veritas Prep

Steve works at an apple orchard and is paid by the bushel for the apples he harvests each day. For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel. For each additional bushel over 42, he is paid 1.5y. How many bushels of apples did Steve harvest yesterday?

(1) Yesterday, Steve was paid $180 for the apples that he harvested.
(2) Today, Steve was paid $240, and he harvested 10 more bushels of apples than he harvested yesterday.

Did you guys know it's really hard to pick plums to make vodka, by the way?

the general formula for Steve's pay is 42y + 1.5y(x-42)

S1: 42y+1.5y(a-42) = 180, where a = yesterday's harvest
S2: 42y+1.5y(b-42) = 240, where b = today's harvest; b=a+10

combining S1 and S2, we have

1.5yb - 1.5ya = 60 ....... (1)
b - a = 10 .................... (2)

these are similar equations and will never give us either b or a.

Answer E.

yes the answer should be E because there is not sufficient information given combining both statements

E is the answer.

Will not be able to successfully isolate 2 equations and 2 unknowns.

S1.
Case 1 x = no of bushels <42
180 = xy
Case 2. x > 42
1.5 xy = 180 + 21y
Not Sufficient
S2.
Case 1. x < 42 and x+10 < 42
y(x+10) = 240
Case 2.
x < 42 and x+10 > 42
42y + 1.5y*(x+10-42) = 240
Case 3.
x > 42
42y + 1.5y*(x+10-42) = 240
Not Sufficient
Combining,
Case 1. x < 42 and x+10 < 42
No solution exists
Case 2. x<42 and x+10 > 42
x = 36
Case 3. x>42
x = 44
Hence two different solutions.
(E) is answer.

Thanks David, your post was superb, it really opened the eyes as to what GMAT can ask...

Thank You once again...

GMATGuruNY wrote:

DanaJ wrote:Source: Veritas Prep

Steve works at an apple orchard and is paid by the bushel for the apples he harvests each day. For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel. For each additional bushel over 42, he is paid 1.5y. How many bushels of apples did Steve harvest yesterday?

(1) Yesterday, Steve was paid $180 for the apples that he harvested.
(2) Today, Steve was paid $240, and he harvested 10 more bushels of apples than he harvested yesterday.

Did you guys know it's really hard to pick plums to make vodka, by the way?

Statement 1:
If y=2, Steve was paid 2*42=84 for the first 42 bushels.
180-84 = 96.
The payrate for each additional bushel would be 1.5y = 1.5*2 = 3.
Thus, Steve would harvest 96/3 = 32 additional bushels.
Total bushels = 42+32 = 74.

If y=4, Steve was paid 4*42=168 for the first 42 bushels.
180-168 = 12.
The payrate for each additional bushel would be 1.5y = 1.5*4 = 6.
Thus, Steve would harvest 12/6 = 2 additional bushels.
Total bushels = 42+2 = 44.

Since the number of bushels could be 74 or 44, insufficient.

Statement 2:
If y=2, Steve was paid 2*42=84 for the first 42 bushels.
240-84 = 156.
The payrate for each additional bushel would be 1.5y = 1.5*2 = 3.
Thus, Steve would harvest 156/3 = 52 additional bushels.
Total bushels today = 42+52 = 94.
Total bushels yesterday = 94-10 = 84.

If y=4, Steve was paid 4*42=168 for the first 42 bushels.
240-168 = 72.
The payrate for each additional bushel would be 1.5y = 1.5*4 = 6.
Thus, Steve would harvest 72/6 = 12 additional bushels.
Total bushels today = 42+12 = 54.
Total bushels yesterday = 54-10 = 44.

Since the number of bushels yesterday could be 84 or 44, insufficient.

Statements 1 and 2 together:
Total payment today = 240, total payment yesterday = 180.
The 10 additional bushels harvested today yielded 240-180 = 60 more dollars.
Thus, the payrate for each of the 10 additional bushels = 60/10 = 6.
This means either that y=6 or that 1.5y=6 and y=4. (Tricky!)
If y=6, Steve harvested today 240/6 = 40 bushels and yesterday 180/6 = 30 bushels.
If y=4, we saw in statement 2 that Steve harvested today 54 bushels and yesterday 44 bushels.
Since the number of bushels harvested yesterday could be 30 or 44, insufficient.

The correct answer is E.

i see how your explanation gets to answer E. But when preparing for the GMAT, we learn that if you have 2 variables, and 2 equations, you can find the answer. What is the rule here as to why we cannot find the 2 variables?