If the price of a commodity is directly proportional to m3 a

[rkafc81]

If the price of a commodity is directly proportional to m3 and inversely proportional to q2, 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


ans = d

in the MGMAT answer, it says
"We know that when m^3 gets bigger, the price of our commodity gets bigger. However, when q^2 gets bigger, that same price gets smaller. We can express this as: Price = km^3 / q^2"

... i understand most of direct and indirect proportionality, however I don't get how the question's author has come to this equation?

please elaborate for me!

thanks

[jnelson0612]

If the price of a commodity is directly proportional to m3 and inversely proportional to q2, 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


ans = d

in the MGMAT answer, it says
"We know that when m^3 gets bigger, the price of our commodity gets bigger. However, when q^2 gets bigger, that same price gets smaller. We can express this as: Price = km^3 / q^2"

... i understand most of direct and indirect proportionality, however I don't get how the question's author has come to this equation?

please elaborate for me!

thanks

Sure! Let's think for a second about what direct proportionality means: if one variables grows another grows. They have a direct relationship: the resulting value = (some constant) * the original value.

For example, maybe we learn that the more I study the GMAT the higher my score is. The formula for my score could be
GMAT score = 1.05 (number of hours studying)^3. Thus, as my number of hours studying go up my score also goes up. They are directly proportionate.

An inverse proportion would indicate that the less of something I have the more of something else I have. For example, maybe I can rate my test anxiety on a scale, and the less test anxiety I have the greater my GMAT score is. Now, it doesn't make sense to set these directly equal to each other like I did in the previous paragraph. As I increase my test anxiety, my score becomes lower. What's the solution? To set up this kind of a formula:
GMAT score = constant/text anxiety

Let's cross multiply: my GMAT score * my level of test anxiety = some constant value. Thus, if I can decrease my test anxiety my GMAT score will rise, and vice versa.

Thus, in this question, to make this formula we're just combining these two equations:

Direct:
Price = k * m^3 (k is the constant value)
Indirect:
Price = k/q^2

So how do we combine them? They both have price on the left and k in the numerator on the right. Now we just continue to use m^3 on the top and q^2 on the bottom. Test out the answers and determine which one gives you the best value.

Please let us know if we can help you further. :-)

[ToThePromiseLand]

Hi Jamie,

Very helpful reply, but something still isn't clicking for me. All is good until this point:

"So how do we combine them? They both have price on the left and k in the numerator on the right. Now we just continue to use m^3 on the top and q^2 on the bottom."

What are the algebraic steps taken to combine the two? Could someone else throw some light on this for me?

Thanks!

[tim]

Just multiply them together. That's the rule for combining multiple proportions in this way.

[griffin.811]

Why is it ok to assume that the constant is the same under both the directly and the inversely proportional situations?

I was thinking directly proportional: p=k1(m^3) and inversely proportional: p=k2/(q^2)

where k1 and k2 are different constants.

Thanks

[RonPurewal]

Why is it ok to assume that the constant is the same under both the directly and the inversely proportional situations?

I was thinking directly proportional: p=k1(m^3) and inversely proportional: p=k2/(q^2)

where k1 and k2 are different constants.

Thanks

your first formula excludes q, so it works only if q never changes (i.e., if q is held constant).

similarly, your second formula won't work unless m is constant.

[RonPurewal]

just think of an easy real-world example: let's say you hire some movers. each person charges $20 per hour.

• if both the number of movers ("m") and the number of hours for the job ("h") are variables, then the formula is
total cost = 20mh
(directly proportional to both)

• if the number of movers is known, then we can write a formula with just "h". for instance, if there must be exactly 4 movers, then the total price is 80h. (it's $80 per hour for the four-person crew)

• similarly, if the job has a fixed length (which would be weird here, since one would generally expect more people to finish faster... but let's disregard that), then we can write a formula using only "m". for instance, if the job is exactly 5 hours long, then the total price is 100m. (it's $100 per mover for a five-hour job)

on the other hand, if both the number of movers and the length of the job are variable, it should be clear that it's impossible to write a formula using just one or the other.

[griffin.811]

This makes sense, thanks. Also, above Tim says just multiply the two proportions (P=km^3 and P=K/(q^2) to get (Km^3)/q^2), but wouldn't this result in a k^2 term in the numerator?

For me I think the biggest issue was not associating the "and" with creating one equation. Now its pretty clear that it's one constant (k) multiplied by the whole relationship (m^3)/(q^2).

Thanks!

[RonPurewal]

an unknown constant times another unknown constant is still ... an unknown constant.

in my examples above, the "constants" are 20, 80, and 100.
the reason they're different is that the constant includes everything that isn't a variable.
for instance, in the middle example in which the constant is 80, that value of 80 takes into account not only the hourly rate (20$ per person per hour) but also the number of people (4 people).

[RonPurewal]

^^ so, if you wanted to be very formally correct, you could change the name of the constant every time you did a new step. C', C'', etc.

there isn't usually any good reason to do so. well, not on a test of short problems (like the gmat), anyway.
(if you were doing a long derivation with many different C's, and were going to return later to one of the earlier steps--e.g., to substitute a value somewhere--THEN it would be crucial to keep track of C, C', C'', and so on. that's definitely not going to happen here.)

[griffin.811]

Thanks Ron!

[RonPurewal]

you're welcome.

[vkt219]

Hi,

Was trying to formulate a "Back solving" technique for this example. I found this problem to be a good candidate for back solving when under time pressure, however have few doubts on the process:

- Started out with C and quickly derived the value as 20. Checked B as it is on the same lines as C. Derived the value as 12.
Skipped A as the values are ascending order ( B=12 , C=20 ) and we need the max.
- Evaluated D as 20+ and E as 9. Select D as the final answer.

Is the above process buggy or is this a bad candidate for quick backsolving?

Can the values on the real GMAT answer choices change the All ascending or All descending pattern midway (E decreases in value after D) ?

Any way to speed up eliminate answer choice E ?

[RonPurewal]

ok, so, basically, you are just using the formula that's given in the first post (km^3 / q^2), but you've chosen k = 1.
this is fine, of course.

...but, how are you getting C as an answer?

C gives 20.
D gives MORE than 20.
you want the biggest answer.

huh?

[RonPurewal]

also, this isn't really "backsolving", because it's the only way to do the problem!

"backsolving" is a thing for problems that can be solved without answer choices, but on which you can "cheat" and circumvent having to do some of the work by using the choices.
that's not what is happening here -- this problem wouldn't even mean anything without the answer choices.