MIXTURE PROBLEM

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amitdgr: Master | Next Rank: 500 Posts; Posts: 228; Joined: Sun Jun 29, 2008 8:47 pm; Thanked: 15 times; Followed by:1 members

MIXTURE PROBLEM

by amitdgr » Fri Aug 01, 2008 6:48 pm

Q) In what proportion can 3 varieties of sugar priced at $10, $12 and $18 per kg, be mixed so that the price of the resultant mixture is $14 per kg?

A) 2:2:5
B) 2:3:4
C) 1:3:4
D) 3:4:5
E) 1:2:8

Please help how to approach. I will post the correct answer afterwards.

Thanks
Amit

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Source: Beat The GMAT — Problem Solving | Fri Aug 01, 2008 6:48 pm

pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Sat Aug 02, 2008 12:33 am

Once again that is not an academic one method. Here is how I did it:

x: quantity of sugar at $10 in kg
y: quantity of sugar at $12 in kg
z: quantity of sugar at $18 in kg

(10x+12y+18z)/(x+y+z)=14
0=4x+2y-4z
2z=2x+y

And you try each of the possible answers to find which of them respects the equation.

When you are on D) it gives you 2*5=2*3+4
Then D)
Takes you less than one minute.

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egybs: Master | Next Rank: 500 Posts; Posts: 178; Joined: Wed May 14, 2008 3:51 pm; Thanked: 16 times; Followed by:1 members; GMAT Score:99%+

by egybs » Sat Aug 02, 2008 2:24 am

agreed with pepeprepa on this... just do some trial and error.

But there is one thing you could do which would narrow your choices down to a couple to test:

Looking at 10, 12 and 18.. and knowing that we need 14 as an average. You can basically see that if there were no $12s you'd need an equal number of 10s and 18s to get an average of 14 (since (10+18)/2 = 14). From there, if you were to substitute some of the 10s for 12s, you'll see that you'll need a smaller number of 18s to get that same 14 average.

So what does this mean? The number of 10s and 12s should, together, be more than the number of 18s.

Another way to look at it is that if you were combine only the 10s and 12s, the lowest the average could be is 10 and the highest 12. If it was 10, there would be a 1:1 ration with the 18s.. and if the average was 12, there would be a 2:1 ration with 18s. Of course we're looking for something in between.

In any case, this can eliminate choices A, C, and E... So you'd only need to test B and D.

Of course, if you were to realize this, you'd only really save 20-30 seconds.. so it's not the end of the world if you don't.