Question

1. a student finishes the first half of an exam in $\frac{2}{3}$ the time it takes him to finish the second half. if the entire exam takes him an hour, how many minutes does he spend on the first half of the exam? (a) 20 (b) 24 (c) 27 (d) 36 11. a 25 ounce solution is 20% alcohol. if 50 ounces of water are added to it, what percent of the new solution is alcohol? (a) $6\frac{2}{3}%$ (b) $7\frac{1}{2}%$ (c) 10% (d) $13\frac{1}{3}%$ 12. marty has exactly 5 blue pens, 6 black pens, and 4 red pens in his backpack. if he pulls out one pen at random from his backpack, what is the probability that the pen is either red or black? (a) $\frac{2}{3}$ (b) $\frac{3}{5}$ (c) $\frac{2}{5}$ (d) $\frac{1}{3}$

Explanation:

Step1: Set up an equation for problem 10

Let the time spent on the second - half be $x$ minutes. The time spent on the first - half is $\frac{2}{3}x$ minutes. The total time is 60 minutes. So, $\frac{2}{3}x + x=60$.

Step2: Combine like terms

$\frac{2}{3}x+\frac{3}{3}x = 60$, which simplifies to $\frac{2x + 3x}{3}=60$, or $\frac{5x}{3}=60$.

Step3: Solve for $x$

Multiply both sides by $\frac{3}{5}$: $x = 60\times\frac{3}{5}=36$ minutes. The time spent on the first - half is $\frac{2}{3}\times36 = 24$ minutes.

Step4: Solve problem 11

The amount of alcohol in the original 25 - ounce solution is $0.2\times25 = 5$ ounces. The new volume of the solution is $25 + 50=75$ ounces. The percentage of alcohol in the new solution is $\frac{5}{75}\times100=\frac{100}{15}=6\frac{2}{3}\%$.

Step5: Solve problem 12

The total number of pens is $5 + 6+4 = 15$ pens. The number of red or black pens is $4 + 6=10$ pens. The probability of picking a red or black pen is $\frac{4 + 6}{15}=\frac{10}{15}=\frac{2}{3}$.

Answer:

1. B. 24
2. A. $6\frac{2}{3}\%$
3. A. $\frac{2}{3}$