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 MathematicallyObtuse January 1st, 2011 04:13 PM

Ratios/fractions

$\text{In an 8-gram solution of water and alcohol, the ratio by mass of water\\to alcohol is 3 to 1.\\If 12 grams of a solution consisting of 2 grams water for each gram of\\alcohol is added to the 8-gram solution, what fraction by mass of the\\new solution is alcohol?}$

I understand the first sentence perfectly well. It is the second sentence I'm not understanding.

Water : Alcohol = 3 : 1 in the 8-gram solution.
Water : Alcohol = 2 : x in the 12-gram solution.
Water : Alcohol = 5 : x in the 20-gram solution ( 12 + 8 )

 MarkFL January 1st, 2011 04:26 PM

Re: Ratios/fractions

....................fraction of alcohol
8 gram solution: 1/(3+1) = 1/4
12 gram solution: 1/(2+1) = 1/3
20 gram total: $\frac{\frac{1}{4}\cdot8+\frac{1}{3}\cdot12}{20}=\f rac{3}{10}$

 MathematicallyObtuse January 1st, 2011 04:37 PM

Re: Ratios/fractions

Quote:
 Originally Posted by MarkFL ....................fraction of alcohol 8 gram solution: 1/(3+1) = 1/4 12 gram solution: 1/(2+1) = 1/3 20 gram total: $\frac{\frac{1}{4}\cdot8+\frac{1}{3}\cdot12}{20}=\f rac{3}{10}$
Hmm...You simply added the amount of alcohol in the 8 and 12 gram solutions; the rest followed.
Darn, what a way to shine the light on my mathematical vacuity!

Anyway, thanks :-)

 soroban January 1st, 2011 04:55 PM

Re: Ratios/fractions

Hello, MathematicallyObtuse!

Quote:
 In an 8-gram solution of water and alcohol, the ratio by mass of water to alcohol is 3 to 1. If 12 grams of a solution consisting of 2 grams water for each gram of alcohol is added to the 8-gram solution, what fraction by mass of the new solution is alcohol?

We can baby-step our way through this one . . .

[color=beige]. [/color] We start with 8 grams which has: 6 grams of water and 2 grams of alcohol.
[color=beige]. . . . [/color]We add 12 grams which has: 8 grams of water and 4 grams of alcohol.
And we have 20 grams which has: 14 grams of water and 6 grams of alcohol.

Therefore . . .

 MarkFL January 1st, 2011 05:04 PM

Re: Ratios/fractions

That seemed to be the easiest way to answer the question. :mrgreen:

If you want to add ratios, you need to make sure the two numbers x:y add up to the total grams in the solution.

3:1 (3 + 1 = 4, 8/4 = 2, thus we use 6:2)
2:1 (2 + 1 = 3, 12/3 = 4, thus we use 8:4)

6:2
8:4
-----
14:6

6/(14+6) = 3/10

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