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Ratio Question

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Ratio Question [#permalink] New post 01 Aug 2014, 17:46
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon
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Re: Ratio Question [#permalink] New post 01 Aug 2014, 17:59
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gordonf35 wrote:
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon


I use m to represent slope.

X = mY
Y = m(1/Z)

X = m(1/Z)

100 = m(1/50) --> m = 5000

X = 5000(1/100) --> X = 50
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Re: Ratio Question [#permalink] New post 03 Aug 2014, 04:56
LighthousePrep wrote:
gordonf35 wrote:
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon


I use m to represent slope.

X = mY
Y = m(1/Z)

X = m(1/Z)

100 = m(1/50) --> m = 5000

X = 5000(1/100) --> X = 50


Thanks for your reply. I don't think it is the correct answer as you assume the "m" is the same for both directly and inversely proportional case. I am just trying to figure out whether it is possible to obtain a solution given with one single example.
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Ratio Question [#permalink] New post 03 Aug 2014, 06:20
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gordonf35 wrote:
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon


x/y = k
y = x/k

y = m/z
yz = m

Therefore,
x*z/k = m or x*z = k*m.

Now when z=50, x = 100

k*m = 5000

Hence, when z = 100, substituting in x*z = m *k, we get

100 * x = 5000

x = 50.

Hope it is clear.

Last edited by distressedDamsel on 05 Aug 2014, 09:41, edited 1 time in total.
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Re: Ratio Question [#permalink] New post 03 Aug 2014, 18:22
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so x is directly proportional to y, and y is inversely proportional to z,

so 'x' is inversely proportional to 'z'. thats IT.

say,
x=k*y and
y=m/z

substitute the value of 'y' as x/k

x/k=m/z --> x = (m*k)/z , m*k is a constant and can be replaced with another constant 'c'. it does not matter what the relationship between 'x' and 'y' actually is..

finally its just x=c/z !!!!!

put in x=50,z=100, we get c=5000

so when z=100, x=50
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Re: Ratio Question [#permalink] New post 05 Aug 2014, 09:25
sshrivats wrote:
so x is directly proportional to y, and y is inversely proportional to z,

so 'x' is inversely proportional to 'z'. thats IT.

say,
x=k*y and
y=m/z

substitute the value of 'y' as x/k

x/k=m/z --> x = (m*k)/z , m*k is a constant and can be replaced with another constant 'c'. it does not matter what the relationship between 'x' and 'y' actually is..

finally its just x=c/z !!!!!

put in x=50,z=100, we get c=5000

so when z=100, x=50


THanks for everyone's input. I am really appreciate people who spent time on problem of the others , +1 kudo for those who contributes and seems correct to me. We are having a mixed answer here. :!:

Can someone point out which one is the correct answer please?
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Re: Ratio Question [#permalink] New post 05 Aug 2014, 09:45
gordonf35 wrote:
sshrivats wrote:
so x is directly proportional to y, and y is inversely proportional to z,

so 'x' is inversely proportional to 'z'. thats IT.

say,
x=k*y and
y=m/z

substitute the value of 'y' as x/k

x/k=m/z --> x = (m*k)/z , m*k is a constant and can be replaced with another constant 'c'. it does not matter what the relationship between 'x' and 'y' actually is..

finally its just x=c/z !!!!!

put in x=50,z=100, we get c=5000

so when z=100, x=50


THanks for everyone's input. I am really appreciate people who spent time on problem of the others , +1 kudo for those who contributes and seems correct to me. We are having a mixed answer here. :!:

Can someone point out which one is the correct answer please?


I've edited my earlier post as I had got the inverse relationship between y and z wrong. The previous post corrects my mistake and I believe that It is the correct solution.
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Re: Ratio Question [#permalink] New post 06 Aug 2014, 22:31
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gordonf35 wrote:
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon


Note that the question should explicitly mention "what is x when z is 100 keeping y constant"

"x is directly proportional to y"

x/y = k

"y is inversely proportional to z"

yz = m

Joint variation gives

x/(yz) = p
x/z = py

100/50 = py = 2

x/100 = 2
x = 200

Please see this post on Joint Variation: http://www.veritasprep.com/blog/2013/02 ... g-jointly/
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 21:02
VeritasPrepKarishma wrote:
gordonf35 wrote:
Hello guys, I saw similar question in GMAT OG but I think it would be much harder if I change it a bit. Please could you let me know your comment of this?

x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?

I am not sure how to build a relationship of this. Is it possible to obtain a solution on this question?

Many thanks,

Gordon


Note that the question should explicitly mention "what is x when z is 100 keeping y constant"

"x is directly proportional to y"

x/y = k

"y is inversely proportional to z"

yz = m

Joint variation gives

x/(yz) = p
x/z = py

100/50 = py = 2

x/100 = 2
x = 200

Please see this post on Joint Variation: http://www.veritasprep.com/blog/2013/02 ... g-jointly/


x is directly proportional to y and y is inversely proportional to z. If x is 100 when z is 50, what is x when z is 100?


x is directly proportional to y and y is inversely proportional to z -> this implies x and z are inversely proportional
or product of xz = constant. In other words, when x increases, then z decreases and when x decreases, then z increases.
Given that, when x equals 100, z is 50.
Therefore, when z increases from 50 to 100, then x must decrease from 100 to 50. (I have assumed that variations are linear)

Am I missing something?
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 22:07
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arunspanda wrote:

x is directly proportional to y and y is inversely proportional to z -> this implies x and z are inversely proportional
or product of xz = constant. In other words, when x increases, then z decreases and when x decreases, then z increases.
Given that, when x equals 100, z is 50.
Therefore, when z increases from 50 to 100, then x must decrease from 100 to 50. (I have assumed that variations are linear)

Am I missing something?


In joint variation, when you establish relation between three or more variables, you have to ensure that independent relation between any two variables remains the same when the third variable is constant.

So
x/y = k
yz = m
gives

x/yz = p (constant)

Now if z is constant, x/y is constant
If x is constant, yz is constant

So if y is constant, x is directly proportional to z - that is how x and z are related.

Please check the link mentioned above.
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 22:27
VeritasPrepKarishma wrote:
arunspanda wrote:

x is directly proportional to y and y is inversely proportional to z -> this implies x and z are inversely proportional
or product of xz = constant. In other words, when x increases, then z decreases and when x decreases, then z increases.
Given that, when x equals 100, z is 50.
Therefore, when z increases from 50 to 100, then x must decrease from 100 to 50. (I have assumed that variations are linear)

Am I missing something?


In joint variation, when you establish relation between three or more variables, you have to ensure that independent relation between any two variables remains the same when the third variable is constant.

So
x/y = k
yz = m
gives

x/yz = p (constant)

Now if z is constant, x/y is constant
If x is constant, yz is constant

So if y is constant, x is directly proportional to z - that is how x and z are related.

Please check the link mentioned above.


The following text is reproduced from the blog.

1. x varies directly with y and directly with z.
x/y = k
x/z = k
Joint variation: x/yz = k
2. x varies directly with y and y varies inversely with z.
x/y = k
yz = k
Joint variation: x/yz = k

Both cases above are shown with the same equation.
I am confused. Kindly clarify.
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 22:38
Expert's post
arunspanda wrote:
The following text is reproduced from the blog.

1. x varies directly with y and directly with z.
x/y = k
x/z = k
Joint variation: x/yz = k
2. x varies directly with y and y varies inversely with z.
x/y = k
yz = k
Joint variation: x/yz = k

Both cases above are shown with the same equation.
I am confused. Kindly clarify.


Yes, they are the same.

x varies directly with y (z constant) and x varies directly with z (y constant). -> implies y varies inversely with z (if x is constant)

x varies directly with y (z constant) and y varies inversely with z (x constant) -> implies x varies directly with z (if y is constant)

Hence they both give the same expression: x/yz = constant
Try each case. Put a finger on the variable you want constant. See if the relation between the other two variables holds.
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Ratio Question [#permalink] New post 07 Aug 2014, 23:01
VeritasPrepKarishma wrote:
arunspanda wrote:
The following text is reproduced from the blog.

1. x varies directly with y and directly with z.
x/y = k
x/z = k
Joint variation: x/yz = k
2. x varies directly with y and y varies inversely with z.
x/y = k
yz = k
Joint variation: x/yz = k

Both cases above are shown with the same equation.
I am confused. Kindly clarify.


Yes, they are the same.

x varies directly with y (z constant) and x varies directly with z (y constant). -> implies y varies inversely with z (if x is constant)

x varies directly with y (z constant) and y varies inversely with z (x constant) -> implies x varies directly with z (if y is constant)

Hence they both give the same expression: x/yz = constant
Try each case. Put a finger on the variable you want constant. See if the relation between the other two variables holds.


These variations appear similar to gas law in Chemistry.
P (Pressure) is proportional to T (Temperature), keeping volume constant
P (Pressure) is inversely proportional to V (Volume), keeping Temperature constant.
Combining both variations, \frac{PV}{T} = constant

Assuming x, y and z as T (Temperature), P (Pressure) and V (Volume) respectively, we get\frac{yz}{x} = constant.

Is the above analogy presented wrong?
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 23:20
Expert's post
arunspanda wrote:
VeritasPrepKarishma wrote:
arunspanda wrote:
The following text is reproduced from the blog.

1. x varies directly with y and directly with z.
x/y = k
x/z = k
Joint variation: x/yz = k
2. x varies directly with y and y varies inversely with z.
x/y = k
yz = k
Joint variation: x/yz = k

Both cases above are shown with the same equation.
I am confused. Kindly clarify.


Yes, they are the same.

x varies directly with y (z constant) and x varies directly with z (y constant). -> implies y varies inversely with z (if x is constant)

x varies directly with y (z constant) and y varies inversely with z (x constant) -> implies x varies directly with z (if y is constant)

Hence they both give the same expression: x/yz = constant
Try each case. Put a finger on the variable you want constant. See if the relation between the other two variables holds.


These variations appear similar to gas law in Chemistry.
P (Pressure) is proportional to T (Temperature), keeping volume constant
P (Pressure) is inversely proportional to V (Volume), keeping Temperature constant.
Combining both variations, \frac{PV}{T} = constant

Assuming x, y and z as T (Temperature), P (Pressure) and V (Volume) respectively, we get\frac{yz}{x} = constant.

Is the above analogy presented wrong?


No, it's correct. You get yz/x = constant which is same as saying x/yz = constant because 1/constant = New constant.
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Re: Ratio Question [#permalink] New post 07 Aug 2014, 23:23
Expert's post
arunspanda wrote:
These variations appear similar to gas law in Chemistry.
P (Pressure) is proportional to T (Temperature), keeping volume constant
P (Pressure) is inversely proportional to V (Volume), keeping Temperature constant.
Combining both variations, \frac{PV}{T} = constant

Assuming x, y and z as T (Temperature), P (Pressure) and V (Volume) respectively, we get\frac{yz}{x} = constant.

Is the above analogy presented wrong?


Note here:
P directly proportional to T (P increases, T increases)
P inversely proportional to V (P increases, V decreases)
So can we say that when P increases, T increases and V decreases so T and V are inversely proportional? No. V and T are directly proportional - we see that from our expression PV/T = constant.

Note that the third variable is assumed to be a constant.
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Re: Ratio Question [#permalink] New post 08 Aug 2014, 00:03
VeritasPrepKarishma wrote:
arunspanda wrote:
These variations appear similar to gas law in Chemistry.
P (Pressure) is proportional to T (Temperature), keeping volume constant
P (Pressure) is inversely proportional to V (Volume), keeping Temperature constant.
Combining both variations, \frac{PV}{T} = constant

Assuming x, y and z as T (Temperature), P (Pressure) and V (Volume) respectively, we get\frac{yz}{x} = constant.

Is the above analogy presented wrong?


Note here:
P directly proportional to T (P increases, T increases)
P inversely proportional to V (P increases, V decreases)
So can we say that when P increases, T increases and V decreases so T and V are inversely proportional? No. V and T are directly proportional - we see that from our expression PV/T = constant.

Note that the third variable is assumed to be a constant.



1. x varies directly with y and directly with z.
x/y = k
x/z = k
Joint variation: x/yz = k
2. x varies directly with y and y varies inversely with z.
x/y = k
yz = k
Joint variation: x/yz = k

From 1. and 2., it can be concluded that both expressions "x varies directly with y and y varies inversely with z" and x varies directly with y and directly with z" are same.

Let us consider 2 which states that x varies directly with y and y varies inversely with z.
x/y = k....(A)
yz = k......(B)

Multiplying (A) and (B) (assuming k is not equal to zero and y is same in both (A) and (B)
(x/y)(yz) = constant
or, xz = constant
This leads to wrong conclusion that x is inversely proportional to z.

Kindly elaborate.
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Ratio Question [#permalink] New post 08 Aug 2014, 00:54
Expert's post
arunspanda wrote:
Let us consider 2 which states that x varies directly with y and y varies inversely with z.
x/y = k....(A)
yz = k......(B)

Multiplying (A) and (B) (assuming k is not equal to zero and y is same in both (A) and (B)
(x/y)(yz) = constant
or, xz = constant
This leads to wrong conclusion that x is inversely proportional to z.

Kindly elaborate.


You are focusing on equations while forgetting the constraints under which they are true. note that the third variable is a constant.

x/y = k.... only when z is a constant
yz = k......Here z is not a constant. This equation holds only when x is a constant.

How are you multiplying these equations? They are not always true. They hold in different cases.

Please read up on Joint Variation from a Math book.
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Ratio Question [#permalink] New post 08 Aug 2014, 02:23
VeritasPrepKarishma wrote:
arunspanda wrote:
Let us consider 2 which states that x varies directly with y and y varies inversely with z.
x/y = k....(A)
yz = k......(B)

Multiplying (A) and (B) (assuming k is not equal to zero and y is same in both (A) and (B)
(x/y)(yz) = constant
or, xz = constant
This leads to wrong conclusion that x is inversely proportional to z.

Kindly elaborate.


You are focusing on equations while forgetting the constraints under which they are true. note that the third variable is a constant.

x/y = k.... only when z is a constant
yz = k......Here z is not a constant. This equation holds only when x is a constant.

How are you multiplying these equations? They are not always true. They hold in different cases.

Please read up on Joint Variation from a Math book.


Let us consider following two cases.

Case 1.
x is directly proportional to y, z is held constant -> x/y=constant for all x and y keeping z as constant.
x is inversely proportional to z, y is held constant-> xz=constant for all x and z keeping y as constant.
Then, xz/y = constant.

Case 2.
x is directly proportional to y, z is held constant ->x/y=constant for all x and y keeping z as constant.
y is inversely proportional to z, x is held constant -> yz=constant for all y and z keeping x as constant.
Then, x/yz =constant or x is proportional to z.

Are these joint relations correct?-In particular, the case 2, in which x is not directly related to z.
I am requesting you to kindly explain.
Ratio Question   [#permalink] 08 Aug 2014, 02:23
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