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GMAT Algebra (Data Sufficiency)
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Author:  questioner [ Thu Dec 09, 2010 9:07 pm ]
Post subject:  GMAT Algebra (Data Sufficiency)

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.

Author:  Gennadiy [ Thu Dec 09, 2010 9:14 pm ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

Quote:
"...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.

We should consider this sentence together with the previous one:
Quote:
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.

If the y pumpkins weighed less than 12, then the average weight of all the pumpkins would be less than 12 as well, because all the pumpkins would simply weigh less than 12.
So the y pumpkins weigh more than 12 and therefore more than 10, which means that they are heavier than the x pumpkins.

Author:  questioner [ Mon Jun 06, 2011 4:23 pm ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

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?

Author:  Gennadiy [ Mon Jun 06, 2011 4:35 pm ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

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.

Author:  questioner [ Sun Jan 29, 2012 7:12 pm ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

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.

Author:  Gennadiy [ Sun Jan 29, 2012 7:19 pm ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

questioner wrote:
It could be possible that the pumpkin with 10 pouns is heavier one.
If the heavier pumpkins weighed 10 pounds, then then the average weight would not be 12 pounds (It would be less than 10). This contradicts with the question statement.

Author:  questioner [ Thu Mar 29, 2012 4:06 am ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

Your answer is incorrect!

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.

Author:  Gennadiy [ Thu Mar 29, 2012 4:23 am ]
Post subject:  Re: math (test 3, question 21): data sufficiency, algebra

Quote:
10X + RY = 12X + 12Y => (R – 12)Y = 2X, since X, Y, R are positive, thus
=> R – 12 > 0 => R > 12
That is correct, but comes from the question statement alone.

Quote:
R > 12 => R = 10 + 5 = 15
This part is NOT correct. Statement (1) tells us
Quote:
(1)There are five more heavier pumpkins than lighter pumpkins.
It refers to quantities, not weights. It implies Y = X + 5. But it does NOT say anything about R.

Author:  questioner [ Mon Feb 04, 2013 12:57 pm ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

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!

Author:  Gennadiy [ Mon Feb 04, 2013 1:18 pm ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

Quote:
So the only possibility for r is to be 13!
That would be true if r had to be an integer. But the basic statement together with (1) do not imply that r has to be an integer.

Plug in some values for x to see what r can be.
For example,
x = 20 then y = 25 and r = 13.6

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