Mixture Problem with Refresher in answers: GMAT, GRE, SAT

Four-ninths of a drink mix is dye and the rest is pure ugar. When the mix is added to water a gallon of the drink is made that is 7.5% sugar. What percentage of the gallon is dye?

(A) 6
(B) 6.5
(C) 7.5
(D) 8
(E) 24

If we take the problem literally, we can make an equation quickly. 4/9 of the total is dye and the remaining 5/9 is sugar. When we're done, the total mix will be 7.5% sugar. 

49/x100=59/7,5100  $\frac{4}{9}/\frac{x}{100}=\frac{5}{9}/\frac{7,5}{100}$

x=6

OR

Let's make a proportion of dye to sugar

Dye/Sugar = (4/9)/(5/9) = x/7.5

4/5 = x/7.5 

x = 4*7.5/5

6

Refresher on Mixture Problems 

We need a 15% acid solution for a certain project, but our supplier only ships a 10% solution and a 30% solution. We need to mix 10% solution with 30% solution, to make your own 15% solution required for the project. We'll need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should we use?

Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30% solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at all suggestive of what they stand for. While the two amounts could be the same using different variables allows them to be different amounts.) Let's plot this into a grid to see each part of the equation: 

	liters sol'n	percent acid	total liters acid
10% sol	x	0.10	0.10x
30% sol	y	0.30	0.30y
mixture	x + y = 10	0.15	(0.15)(10) = 1.5

Since x + y = 10, then x = 10 − y. 

We can substitute for x in our grid, and eliminate one of the variables:   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

	liters sol'n	percent acid	liters acid
10% sol	10 − y	0.10	0.10(10 − y)
30% sol	y	0.30	0.30y
mixture	x + y = 10	0.15	(0.15)(10) = 1.5

When the problem is set up like this, you can usually use the last column to write your equation: The liters of acid from the 10% solution, plus the liters of acid in the 30% solution, add up to the liters of acid in the 15% solution. :

0.10(10 − y) + 0.30y = 1.5
1 − 0.10y + 0.30y = 1.5
1 + 0.20y = 1.5
0.20y = 0.5
y = 0.5/0.20 = 2.5