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 Post subject: GMAT Algebra (Data Sufficiency) Posted: Thu Dec 09, 2010 9:07 pm Joined: Sun May 30, 2010 3:15 am
Posts: 424
A pumpkin patch contains x pumpkins that weigh 10 pounds each and y pumpkins that weigh r pounds each. If the average (arithmetic mean) weight of the pumpkins is 12 pounds, what is the value of r?
(1) There are five more heavier pumpkins than lighter pumpkins.
(2) The weight in pounds of each of the heavier pumpkins is 3 more than their number.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) The original statement says the average weight of a pumpkin is 12 pounds. Since x pumpkins weigh 10 pounds each, the y pumpkins must be heavier.
Writing an equation for the average weight of the pumpkins is (10x + ry) / (x + y) = 12. This equation has three unknowns.
Solve for r to get:
r = 12 + (2x/y)

Statement (1) defines the relationship between x and y as y = x + 5 or y = x – 5. Plug in the formula for r to get
r = 12 + (2(y – 5)/y)
r = 14 – 10/y
Plug in couple values of y (greater than 5) to see that they yield different results for r.
Statement (1) is not sufficient.

Statement (2) gives the relationship between y and r as r = y + 3. This isn't enough information as you can plug it into the original equation and see that too many variables remain. Plug in couple values for y to see that we get different possible valuee of r and x.

Using the given with the equations from (1) and (2), the system is
(10x + ry) / (x + y) = 12
y = x + 5
r = y + 3

Rewrite r = y + 3 as r – 3 = y.
Substitute r – 3 = y into y = x + 5 and get r – 3 = x + 5 so r = x + 8.
This gives an equation for y in terms of x, and an equation for r in terms of x.
Substitute these equations into the original equation.
(10x + ry) / (x + y) = 12. Substitute y = x + 5 and r = x + 8.
(10x + (x + 8)(x + 5)) / (x + (x + 5)) = 12
10x + (x² + 13x + 40) = 12(2x + 5)
x² + 23x + 40 = 24x + 60
x² – x – 20 = 0
(x – 5)(x + 4) = 0
x = 5 or x = -4 = 0
So there are 5 of the smaller pumpkins.
Using y = x + 5, y = 5 + 5 = 10, so there are 10 of the larger pumpkins.
And using r = y + 3, r = 13, the larger pumpkins weigh 13 pounds.
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"...Since x pumpkins weigh 10 pounds each, the y pumpkins must be heavier..." is not an objective step. Please explain how you decide y to be heavier and not x.

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Thu Dec 09, 2010 9:14 pm Joined: Sun May 30, 2010 2:23 am
Posts: 498

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Mon Jun 06, 2011 4:23 pm Joined: Sun May 30, 2010 3:15 am
Posts: 424
Is it required to solve the entire question? What do we have to do: just to form the equations or just to find out how many unknow variables the equations have? From this information, can we comment as to whether we can get the answer or not?

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Mon Jun 06, 2011 4:35 pm Joined: Sun May 30, 2010 2:23 am
Posts: 498
Our main goal is to find which information is sufficient to answer the question. But we do NOT have to calculate the actual answer.

It takes at least 3 linear equations to solve for ALL three variables. However, three equations might NOT be enough, if those equations are proportional, e.g.:
x + y + z = 1
2x + 2y + 2z = 2
3x + 3y + 3z = 3

On the other hand, if you do NOT need to find ALL the variables, but just one, two equations might be enough, e.g.:
x + y + z = 1
x + y = 0
So z = 1.

Therefore you should NOT just calculate the number of equations and variables you have, BUT to analyze:
- are these linear equations?
- what variable(s) do you need to solve for?
- are any of the equations proportional?

So you do NOT need to do all the calculations, but to be sure that calculations will give you the answer.

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Sun Jan 29, 2012 7:12 pm Joined: Sun May 30, 2010 3:15 am
Posts: 424
This question does not tell us which type of pumpkin is the heavier one. In that case how can we take the validtity of the statement that x = y + 5. It could be possible that the pumpkin with 10 pouns is heavier one.

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Sun Jan 29, 2012 7:19 pm Joined: Sun May 30, 2010 2:23 am
Posts: 498

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Thu Mar 29, 2012 4:06 am Joined: Sun May 30, 2010 3:15 am
Posts: 424

From (1) => 10X + RY = 12X + 12Y => (R – 12)Y = 2X, since X, Y, R are positive, thus
=> R – 12 > 0 => R > 12 => R = 10 + 5 = 15
Therefore, (1) by itself is sufficient. => A is the right answer.

Top Post subject: Re: math (test 3, question 21): data sufficiency, algebra Posted: Thu Mar 29, 2012 4:23 am Joined: Sun May 30, 2010 2:23 am
Posts: 498

Top Post subject: Re: GMAT Algebra (Data Sufficiency) Posted: Mon Feb 04, 2013 12:57 pm Joined: Sun May 30, 2010 3:15 am
Posts: 424
I think that statement 1 is sufficient to answer the question and here is how:
(10x + ry) / (x + y) = 12 => (r – 12)y = 2x
Statement 1: y = 5 + x ; if we use this in the previous equation we have:
(r – 12)(5 + x) = 2x => 5(r – 12) = (14 – r)x

We know that r > 12, otherwise the average of the pumpkins would not be 12. So the left term is positive. Hence the right term needs to be positive as well: x is a positive quantity => 14 - r > 0. So the only possibility for is to be 13! Solved only with equation 1!

Top Post subject: Re: GMAT Algebra (Data Sufficiency) Posted: Mon Feb 04, 2013 1:18 pm Joined: Sun May 30, 2010 2:23 am
Posts: 498

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