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Invisible Constraint. OG # 8 vs OG#123

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Invisible Constraint. OG # 8 vs OG#123

by enjoylife1788 » Sun Oct 30, 2011 5:48 am

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QUESTION 8

A citrus fruit grower receives $15 for each crate of
oranges shipped and $18 for each crate of grapefruit
shipped. How many crates of oranges did the grower
ship last week?

(1) Last week the number of crates of oranges that
the grower shipped was 20 more than twice the
number of crates of grapefruit shipped.

(2) Last week the grower received a total of
$38,700 from the crates of oranges and
grapefruit shipped.


If x represents the number of crates of oranges
and y represents the number of crates of
grapefruit, fi nd a unique value for x.
(1) Translating from words into symbols gives
x = 2y + 20, but there is no information
about y and no way to fi nd a unique value
for x from this equation. For example, if
y = 10, then x = 40, but if y = 100, then
x = 220; NOT sufficient.

(2) Translating from words to symbols gives
15x + 18y = 38,700, but there is no way to
fi nd a unique value for x from this equation.
For example, if y = 2,150, then x = 0 and if
y = 0, then x = 2,580; NOT suffi cient.
Taking (1) and (2) together gives a system of two
equations in two unknowns. Substituting the
equation from (1) into the equation from (2) gives
a single equation in the variable y. Th is equation
can be solved for a unique value of y from which a
unique value of x can be determined.
Th e correct answer is C;

both statements together are suffi cient.


QUESTION 123

Joanna bought only $0.15 stamps and $0.29 stamps.
How many $0.15 stamps did she buy?
(1) She bought $4.40 worth of stamps.
(2) She bought an equal number of $0.15 stamps
and $0.29 stamps.


Determine the value of x if x is the number of
$0.15 stamps and y is the number of $0.29
stamps.

(1) Given that 15x + 29y = 440, then
29y = 440 - 15x. Because x is an integer,
440 - 15x = 5(88 - 3x) is a multiple of 5.
Th erefore, 29y must be a multiple of 5, from
which it follows that y must be a multiple of
5. Hence, the value of y must be among the
numbers 0, 5, 10, 15, etc. To more effi ciently
test these values of y, note that
15x = 440 - 29y, and hence 440 - 29y
must be a multiple of 15, or equivalently,
440 - 29y must be a multiple of both 3 and
5. By computation, the values of 440 - 29y
for y equal to 0, 5, 10, and 15 are 440, 295,
150, and 5. Of these, only 150, which
corresponds to y = 10, is divisible by 3.
From 15x = 440 - 29y it follows that x = 10
when y = 10. Th erefore, x = 10 and y = 10;
SUFFICIENT.


(2) Although x = y, it is impossible to determine
the value of x because there is no information
on the total worth of the stamps Joanna
bought. For example, if the total worth, in
dollars, was 0.15 + 0.29, then x = 1, but if
the total worth was 2(0.15) + 2(0.29), then
x = 2; NOT suffi cient.

Th e correct answer is A;
statement 1 alone is suffi cient.
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by enjoylife1788 » Sun Oct 30, 2011 5:55 am

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My question regarding both the questions is


In question 123, they didnt have two equtions for two variables. they had only one equation. But, I get the point that the constraints are such that it would yield only one unique solution.

So my problem, How to determine when one equation is enough to solve for a unique solution and when its not.

I hope some one gets my question. Its too complicated to explain what m trying to say...
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by enjoylife1788 » Sun Oct 30, 2011 6:03 am

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For instance, what if I manipulated the 8th question to read as follows:

A citrus fruit grower ships only orange and grapefruit crates. He gets $0.15 for one orange crate and $0.20 for one grapefruit crate. How many orange crates did he ship?

1) he ships $4.40 worth of fruit.
2) he ships equal number of orange and grapefruit crates.

My purpose for this manipulation is to ask whether A is enough in this case and not regard the case where he could have shipped 0 orange crates and 4.4/0.2 grapefruit crates. Agreed, here the answer comes in decimal, which is not possible with number of crates. But, overall confusion remains as to when do we require two equation and when only one suffices.
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by Luke.Doolittle » Sun Oct 30, 2011 10:10 am

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In short, relevant to the GMAT:

If you have n unconstrained variables (unconstrained except for the fact that they are real, an assumption implicit in all GMAT problems) you will need n linear, independent equations to get the values for each variable.

When will less than n equations suffice? I can think of at least 2 common scenarios
(1) You don't need ALL the variable values, you only need one. Silly Example

If x, y and z are integers, what is the value of z?
(1) x + y + z = 5
(2) x + y = 4

This problem presents 3 variables but you can plainly see that you can get the value for z (and hence an answer to the question) with only 2 equations. This can also occur with composite value questions (like what is the value of x + y)

(2) When the values of the variables are constrained. The most typical of which is the integer/positive constraint you mentioned in your sample problem. If you have an equation Ax + By = C, where A, B and C are known constants and x and y are integers and positive, you MAY be able to arrive at a unique value of (x,y). However there is no way (that I know of anyway) to simply "look" at an equation of this type and infer that there is only one solution. You simply have to plug in numbers and check.

HANDY TRICK
Remember that all GMAT quant problems can be solved in 2 minutes or under. Most of these "plug and chug" problems will only have 4-5 different possible x,y pairs that come close to satisfying the equation. However sometimes it would appear that the quantity of pairs you have to check is huge (al la your second problem in the OP). When that is the case the method that you used in statement (1) is excellent for compressing those possibilities.
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by parul9 » Sun Oct 30, 2011 10:23 am

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enjoylife1788 wrote:My question regarding both the questions is


In question 123, they didnt have two equtions for two variables. they had only one equation. But, I get the point that the constraints are such that it would yield only one unique solution.

So my problem, How to determine when one equation is enough to solve for a unique solution and when its not.

I hope some one gets my question. Its too complicated to explain what m trying to say...
I am not an expert. But the little that I have seen so far about GMAT, it is not a test of rules or point blank formulae. Every question is different and should be treated differently.
A DS qsn will have answer A if it can be answered using statement 1 alone.
Can we answer the first qsn by using only statement 1? No. So the answer is not A.

On the other hand, in the second qsn, 0.15 + 0.29 = 0.44.
So, if we consider statement one, the only possible values of the respective stamps can be 10 each.
So, we can answer using statement 1 alone. Statement 2 makes a statement that coincides with what we derived from 1, but in isolation from 1, it is incapable of solving this qsn. So the answer for the second qsn is A.
enjoylife1788 wrote:For instance, what if I manipulated the 8th question to read as follows:

A citrus fruit grower ships only orange and grapefruit crates. He gets $0.15 for one orange crate and $0.20 for one grapefruit crate. How many orange crates did he ship?

1) he ships $4.40 worth of fruit.
2) he ships equal number of orange and grapefruit crates.

My purpose for this manipulation is to ask whether A is enough in this case and not regard the case where he could have shipped 0 orange crates and 4.4/0.2 grapefruit crates. Agreed, here the answer comes in decimal, which is not possible with number of crates. But, overall confusion remains as to when do we require two equation and when only one suffices.
Assuming u mean 0.29$ for grapefruit, yes, the answer for this DS qsn would be A.

Hope this helps! :)

Cheers!
Parul
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by enjoylife1788 » Sun Oct 30, 2011 8:33 pm

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Thank you all for your replies. There is no direct way of knowing such equations. So basically, as parul said, we just have to consider each question individually and try to solve it.

Also, I dont if its gonna help anyone. But I read somewhere that if u come across a weird prime number such as 29 in question # 123, then there is gotta more to the problem than it seems at first. :)
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by ceilidh.erickson » Tue Nov 13, 2018 11:32 am

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enjoylife1788 wrote:My question regarding both the questions is


In question 123, they didnt have two equtions for two variables. they had only one equation. But, I get the point that the constraints are such that it would yield only one unique solution.

So my problem, How to determine when one equation is enough to solve for a unique solution and when its not.

I hope some one gets my question. Its too complicated to explain what m trying to say...
Your question is a very understandable one. If you want to be absolutely sure whether there is only one pair of values that work or multiple pairs of values, you should create a table / chart. Start by charting the maximum amount of the variable with the higher coefficient, and see if there's a corresponding value for the 2nd variable that would add to the total.

For example, on the oranges and grapefruits question, you could create a chart for statement 2:

Image

As soon as you see 2 pairs of values that fit, it's insufficient.

For statement 1 of Joanna's stamps, here is the chart I'd use. Note: multiples of 15 have to end in a units digit of 5 or 0, so to add up to a sum with a units digit of 0, the 29 cent stamps would have to be in a multiple of 5 or 10. Those are the only values we have to test.

Image

In this chart, we have only 1 possibility that works.
For more on charting, see: https://www.beatthegmat.com/if-a-certai ... tml#822857

Now that said... that's my answer to your question about how to be SURE if there's 1 possibility or multiple. Realistically, though, you won't have time to go through all of that on test day. The arithmetic is too time-consuming! So here is the general rule that I follow:

1 Equation or 2 for Integer Constraints?
If the coefficients do not share any factors (e.g. 15 and 29), it's most likely that a given total will only yield 1 set of values that work. So bet on 1 equation being sufficient. If they do share factors (e.g. 15 and 18), it's likely that there will be overlap, and there will be several possibilities.**

**Note that there *could* always be exceptions to this rule. Consider:
I bought $2 apples and $3 oranges and spent a total of $20.
Well, I could have bought 4 apples and 4 oranges for $8 and $12 respectively, or I could have bought 1 apple and 6 oranges for $2 and $18 respectively.

But... it's highly unlikely that the GMAT would do this to you, especially with large numbers. What the GMAT is really testing is whether you'll fall for the assumption that "I always need 2 equations for 2 variables." They want to trick the student who blindly follows the rules and doesn't think. For that reason, it's highly unlikely that they'd ever make you do all of that work to get to the same answer that someone who blindly followed the rule would have gotten to.

More on how the GMAT messes with 2 equations / 2 variables rules here: https://www.manhattanprep.com/gmat/blog ... ons-rules/
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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