Question

solution problems

1. how many liters of a 14 percent alcohol solution must be mixed with 20 liters of a 50 percent alcohol solution to get a 20 percent alcohol solution?
2. a chemist has 6 liters of a 25% alcohol solution. how much pure alcohol must he add so that the resulting solution contains 50% alcohol?
3. milk that has 5% butterfat is mixed with milk that has 2% butterfat. how much of each is needed to obtain 60 gallons of milk that has 3% butterfat?
4. chris has 3 gallons of a solution that is 30 percent antifreeze, which he wants to use to winterize his car. how much pure antifreeze should he add to this solution so that the new solution will be 65 percent antifreeze?
5. suppose 30 liters of a solution with an unknown percentage of alcohol is mixed with 5 liters of a 90% alcohol solution. if the resulting mixture is a 62% alcohol solution, what is the percentage of alcohol in the first solution?
6. sterling silver is 92.5% pure silver. how many grams of pure silver and sterling silver must be mixed to obtain 100g of a 94% silver alloy?

Explanation:

1.

Step1: Set up the equation

Let $x$ be the number of liters of the 14 - percent alcohol solution. The amount of alcohol in the 14 - percent solution is $0.14x$, the amount of alcohol in the 20 - liter 50 - percent solution is $0.5\times20$, and the amount of alcohol in the final 20 - percent solution is $0.2(x + 20)$. So the equation is $0.14x+0.5\times20=0.2(x + 20)$.

Step2: Expand and simplify the equation

$0.14x + 10=0.2x+4$.

Step3: Solve for $x$

$10 - 4=0.2x-0.14x$, $6 = 0.06x$, $x=\frac{6}{0.06}=100$.

Step1: Set up the equation

Let $x$ be the amount of pure alcohol (100% alcohol, so 1 in decimal form) to be added. The amount of alcohol in the initial 6 - liter 25% solution is $0.25\times6$, and the amount of alcohol in the final 50% solution is $0.5(6 + x)$. So the equation is $0.25\times6+x=0.5(6 + x)$.

Step2: Expand the equation

$1.5+x = 3+0.5x$.

Step3: Solve for $x$

$x-0.5x=3 - 1.5$, $0.5x=1.5$, $x = 3$.

Step1: Let variables

Let $x$ be the number of gallons of 5% butter - fat milk and $y$ be the number of gallons of 2% butter - fat milk. We have two equations: $x + y=60$ (total volume) and $0.05x+0.02y=0.03\times60$ (butter - fat content). From $x + y=60$, we get $y = 60 - x$.

Step2: Substitute $y$ into the second equation

$0.05x+0.02(60 - x)=1.8$.

Step3: Expand and simplify

$0.05x+1.2-0.02x=1.8$, $0.03x=1.8 - 1.2$, $0.03x=0.6$, $x = 20$.

Step4: Find $y$

Since $y=60 - x$, $y = 40$.

Answer:

100 liters

2.