One fourth

lheiannie07 wrote:One fourth of a solution that was 10% sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was what percent sugar by weight?

A. 34%
B. 24%
C. 22%
D. 18%
E. 8.5%

We can let the weight of the original solution = n.

Thus, the original solution had 0.1n sugar.

If 1/4 of the original solution was removed, the new solution now has:

0.1n x 3/4 = 0.1n x 0.75 = 0.075n sugar

Some amount of solution was added back so that the overall weight was 16% sugar. We can let x = the amount of sugar in the second solution and create the following equation:

(0.075n + x)/n = 0.16

0.075n + x = 0.16n

x = 0.085n

We see that the amount of sugar in the second solution, in terms of n, is 0.085n. However, the total weight of the second solution is (¼)n or 0.25n since that is the amount being replaced. Thus, the percent of the second solution that is sugar is:

0.085n/0.25n = 85/250 = 340/1000 = 34/100 = 34%

Alternate Solution:

We start with x liters of a 10% solution. From it, we take out x/4 liters (which is also 10%). We then add back in x/4 liters of z% solution, resulting in x liters of a 16% solution.

Let’s express this in an equation and solve for z:

x(0.10) – (x/4)(0.10) + (x/4)z = x(0.16)

Multiplying both sides by 4, we obtain:

0.40x – 0.10x + zx = 0.64x

0.30x + zx = 0.64x

zx = 0.34x

z = 0.34, or 34%.

Answer: A