mixture problems with more than 1 variable

Expert replies

topspin360: Master | Next Rank: 500 Posts; Posts: 103; Joined: Sat Jun 02, 2012 9:46 pm; Thanked: 1 times

by topspin360 » Sat Jul 07, 2012 7:45 am

is there a standard framework we can use to solve mixture problems with more than 1 variable such as follows:

A coin made of alloy of aluminum and silver measures 2 x 15 mm (it is 2 mm thick and its diameter is 15 mm). If the weight of the coin is 30 grams and the volume of aluminum in the alloy equals that of silver, what will be the weight of a coin measuring 1 x 30 mm made of pure aluminum if silver is twice as heavy as aluminum?
36 grams
40 grams
42 grams
48 grams
50 grams

Thanks.

Like this post

Source: Beat The GMAT — Problem Solving | Sat Jul 07, 2012 7:45 am

eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sat Jul 07, 2012 8:31 am

I am not sure about whether there is a standard way of doing 2 or more variable mixture problems. I would just try to evaluate each on a case by case basis.

Here are two ways of solving the question. One is algebraic and second is reasoning-based.

Algebraic:
Volume of first coin = V1= pi*2*(15/2)^2
Volume of second coin = V2 = pi*1*(30/2)^2
V2/V1 = (pi*1*(30/2)^2) / (pi*2*(15/2)^2) = 30*30*1/(15*15*2) = 2
V2=2V1
In the first coin, volume of aluminum = volume of silver = V1/2
Weight of silver / Weight of Aluminum = 2:1

Hence for the first coin, Weight of Aluminum in the coin = 1/3*30 = 10 grams.

So V1/2 weighs = 10 grams.
Second coin is made entirely of aluminum.
Hence V2 = 2*V1 = 2*(2*V1/2) = 4*(V1/2) must weigh 4*10 = 40 grams.

Here's the non-standard reasoning approach that I used to solve the question without doing any pen paper calculation:

Volume of first one comes from 2*15. Second one is from 1*30. Volume is proportional to height and proportional to square of the diameter. Since 30 is twice 15, and height is halved, volume of second one must be 2^2*1/2 = twice that of first one.
Now first one has equal volume of silver and aluminum, and silver is twice as heavy as aluminum. Since they have equal volumes, silver is twice as heavy as aluminum, and total weight is 30, weight of aluminum must be 10 and that of silver must be 20.

So, aluminum's weight in half of first coin = 10 grams. If the whole of first coin were made of aluminum, it would weigh = 20 grams. Since the volume of second coin is double that of the first, it must weigh twice of 20 = 40 grams.

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Jul 07, 2012 2:10 pm

topspin360 wrote:is there a standard framework we can use to solve mixture problems with more than 1 variable such as follows:

A coin made of alloy of aluminum and silver measures 2 x 15 mm (it is 2 mm thick and its diameter is 15 mm). If the weight of the coin is 30 grams and the volume of aluminum in the alloy equals that of silver, what will be the weight of a coin measuring 1 x 30 mm made of pure aluminum if silver is twice as heavy as aluminum?
36 grams
40 grams
42 grams
48 grams
50 grams

Thanks.

This is more of a PROPORTION problem than a mixture problem.

Coin made of alloy:
Volume = âˆ�rÂ²h = âˆ�(15Â²)(2) = 450âˆ�.
The total weight = 30 grams.
Since silver weighs twice as much as aluminum, silver = 20 grams and aluminum = 10 grams.

Since the volume of each metal is the same, if the silver -- with a weight of 20 grams -- is replaced with aluminum -- with a weight of 10 grams -- the total weight of the resulting coin will be 20 grams.
Thus, the weight of an aluminum coin with a volume of 450âˆ� = 20 grams.

Pure aluminum coin:
V = âˆ�rÂ²h = âˆ�(30Â²)(1) = 900âˆ�.
Since the weight of an aluminum coin with a volume of 450âˆ� = 20 grams, the weight of an aluminum coin that is twice as big (with a volume of 900âˆ�) = 40 grams.

The correct answer is B.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review