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If the price of a commodity is directly proportional to m^3

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If the price of a commodity is directly proportional to m^3 [#permalink]

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07 Apr 2013, 04:45
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If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?

A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72
[Reveal] Spoiler: OA

Last edited by Bunuel on 07 Apr 2013, 21:20, edited 1 time in total.
RENAMED THE TOPIC.
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Re: Price of a commodity [#permalink]

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07 Apr 2013, 04:56
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If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?
A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72

Let the price is P & constant is K
So P = $$\frac{K m^3}{q^2}$$

Now Just use the values in the options and put in the before mentioned equation.
A) P = $$\frac{K m^3}{q^2}$$ = $$\frac{K 3^3}{2^2}$$ = $$\frac{K 27}{4}$$ = 6.75 K
B) P = $$\frac{K m^3}{q^2}$$ = $$\frac{K 12^3}{12^2}$$ = 12 K
C) P = $$\frac{K m^3}{q^2}$$ = $$\frac{K 20^3}{20^2}$$ = 20 K
D) P = $$\frac{K m^3}{q^2}$$ = $$\frac{K 30^3}{36^2}$$ = 125/6 = >20K
E) P = $$\frac{K m^3}{q^2}$$ = $$\frac{K 36^3}{72^2}$$ = 9 K

So the answer is D. This is probably 600-700 level question.

Hope the explanation helps.
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Last edited by emmak on 07 Apr 2013, 05:00, edited 1 time in total.
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Re: Price of a commodity [#permalink]

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07 Apr 2013, 04:57
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Dipankar6435 wrote:
If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?
A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72

Function = $$\frac{m^3}{q^2}$$

A=$$\frac{3*3*3}{2*2}=\frac{27}{4}=7$$ almost
B=$$\frac{12*12*12}{12*12}=12$$
C=$$\frac{20*20*20}{20*20}=20$$
D=$$\frac{30*30*30}{36*36}=125/6$$
E=$$\frac{36*36*36}{72*72}=9$$

It's down to C or D, because $$\frac{120}{6}=20$$, $$\frac{125}{6}>20$$
D
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If the price of a commodity is directly proportional to m^3 [#permalink]

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07 Apr 2013, 21:22
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Dipankar6435 wrote:
If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?

A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72

Similar questions to practice:

DS:
the-amount-of-coal-a-train-burns-each-mile-is-directly-93667.html

PS:
a-is-directly-proportional-to-b-when-a-8-b-88971.html
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the-rate-of-a-chemical-reaction-is-directly-proportional-to-76921.html

Hope it helps.
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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09 Apr 2013, 03:56
Let price of the commodity be P = $$k *$$$$m^3$$ /$$Q^2$$

A. P = $$k * 27/4$$ = 6.75K
B. P = $$K * 12^3/12^2$$ = 12k
C. P = $$K * 20^3/20^2$$ = 20k
D. P = $$K * 30^3/36^2$$= $$k*5*5*30/6*6$$ = $$k * (125/6)$$ = 21K (approx)
E. P = $$k * 36^3/72^2$$ = $$K*(36/2*2)$$ = 9k

Ans : Option D.
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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10 Apr 2013, 17:09
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Could you please explain how you get $$\frac{m^3}{q^2}$$ or $$\frac{Km^3}{q^2}$$? Where am I wrong expressing this thing firstly as $$p=m^3k$$ and $$p=\frac{k}{q^2}$$?
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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10 Apr 2013, 22:19
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obs23 wrote:
Could you please explain how you get $$\frac{m^3}{q^2}$$ or $$\frac{Km^3}{q^2}$$? Where am I wrong expressing this thing firstly as $$p=m^3k$$ and $$p=\frac{k}{q^2}$$?

"directly proportional to m^3 and inversely proportional to q^2" is what the text says. So if m increases the price increases, if q increases the price decreases. The right formula here is $$\frac{m^3}{q^2}$$ ( or with k, it doesnt change anything), if the numerator grows, the price does the same; and as in every fraction, if the denominator grows, the price drops.
Your formulas are right but the price depends on both m and q, you have to include them in one equation.

Let me know if it is clear
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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11 Apr 2013, 02:51
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Zarrolou wrote:
obs23 wrote:
Could you please explain how you get $$\frac{m^3}{q^2}$$ or $$\frac{Km^3}{q^2}$$? Where am I wrong expressing this thing firstly as $$p=m^3k$$ and $$p=\frac{k}{q^2}$$?

"directly proportional to m^3 and inversely proportional to q^2" is what the text says. So if m increases the price increases, if q increases the price decreases. The right formula here is $$\frac{m^3}{q^2}$$ ( or with k, it doesnt change anything), if the numerator grows, the price does the same; and as in every fraction, if the denominator grows, the price drops.
Your formulas are right but the price depends on both m and q, you have to include them in one equation.

Let me know if it is clear

I see that makes sense. I guess I am making it more complicated than it really is. If you say that my formulas are correct, then $$\frac{m^3}{q^2}$$ should somehow be extracted from $$p=m^3k$$ and $$p=\frac{k}{q^2}$$, no? Technically speaking, I thought there was a way to combine them into $$\frac{m^3}{q^2}$$, just from pure algebraic manipulation (which I am very interested in here)... Or is it simply about common sense or the way you explained it?

P.S. Like your "kudos" tagline man!
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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11 Apr 2013, 03:05
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obs23 wrote:
I see that makes sense. I guess I am making it more complicated than it really is. If you say that my formulas are correct, then $$\frac{m^3}{q^2}$$ should somehow be extracted from $$p=m^3k$$ and $$p=\frac{k}{q^2}$$, no? I thought there was a way to combine them into $$\frac{m^3}{q^2}$$, just from pure algebraic manipulation (which I am very interested in here)... Or is it simply about common sense or the way you explained it?

To create the formula we should refer to the text:
"If the price of a commodity is directly proportional to m^3 [and at this point we write down Price= $$m^3$$] and inversely proportional to q^2 [and at this point we complete the foumula adding the Denominator so Price=$$\frac{m^3}{q^2}$$], which of the following values of m and q will result in the highest price for the commodity?"
$$\frac{m^3}{q^2}$$ is not extracted from $$p=m^3k$$ and $$p=\frac{k}{q^2}$$, and you cannot obtain it from pure algebraic manipulation.
The idea behind your formulas is correct p=m^3 expresses the direct correlation between p and m; and also p=1/q^2 expresses the inverse correlation between p and q. But the text uses "and" so those ideas must be expressed in one formula, so to obtain this final formula you "complete" one with the other =>m^3/p^2
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11 Apr 2013, 03:30
Got it! Thanks much for help.
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Re: If the price of a commodity is directly proportional to m^3 [#permalink]

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11 Apr 2013, 06:22
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Dipankar6435 wrote:
If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?

A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72

For more on direct, inverse and joint variation, check out these posts:

http://www.veritasprep.com/blog/2013/01 ... -directly/
http://www.veritasprep.com/blog/2013/02 ... inversely/
http://www.veritasprep.com/blog/2013/02 ... g-jointly/
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