Question

1. a chemist has 70 ml of a 50% methane solution. how much of a 80% solution must she add so the final solution is 60% methane? 5. pure acid (100%) is to be added to a 10% acid solution to obtain 54l of a 20% acid solution. what amounts of each should be used? 6. a coffee mix is to be made that sells for $2.50 by mixing two types of coffee. the cafe usually packages has 40 ml of guatemalan coffee for a price of $3.00. they want to blend it with the arabica coffee that costs $1.50. how much of the arabica should the cafe mix into the blend?

Explanation:

4.

Step1: Set up the equation based on the amount of methane

Let $x$ be the volume (in mL) of the 80% methane solution to be added. The amount of methane in the 50% solution is $0.5\times70$, the amount of methane in the 80% solution is $0.8x$, and the amount of methane in the final 60% solution is $0.6\times(70 + x)$. So the equation is $0.5\times70+0.8x=0.6\times(70 + x)$.

Step2: Expand and simplify the equation

$35 + 0.8x=42+0.6x$.
Subtract $0.6x$ from both sides: $35 + 0.8x-0.6x=42+0.6x - 0.6x$, which gives $35 + 0.2x=42$.

Step3: Solve for $x$

Subtract 35 from both sides: $0.2x=42 - 35=7$.
Divide both sides by 0.2: $x=\frac{7}{0.2}=35$ mL.

Step1: Let variables

Let $x$ be the volume (in L) of pure acid (100% acid) and $y$ be the volume (in L) of the 10% acid - solution. We know two equations: $x + y=54$ (total volume) and $1.0x+0.1y = 0.2\times54$ (amount of acid).

Step2: Rewrite the second - equation

The second equation $x + 0.1y=10.8$.
From the first equation $x = 54 - y$. Substitute $x$ into the second equation: $(54 - y)+0.1y=10.8$.

Step3: Expand and solve for $y$

$54 - y+0.1y=10.8$, which simplifies to $54-0.9y=10.8$.
Subtract 54 from both sides: $-0.9y=10.8 - 54=-43.2$.
Divide both sides by - 0.9: $y=\frac{-43.2}{-0.9}=48$ L.

Step4: Solve for $x$

Since $x + y=54$ and $y = 48$, then $x=54 - 48 = 6$ L.

Step1: Set up the cost - based equation

Let $x$ be the volume (in mL) of Arabica coffee. The total cost of the blend is the sum of the costs of each type of coffee. The cost of Guatemalan coffee is $3.00\times40$, the cost of Arabica coffee is $1.50x$, and the cost of the blend is $2.50\times(40 + x)$. So the equation is $3.00\times40+1.50x=2.50\times(40 + x)$.

Step2: Expand the equation

$120+1.50x=100 + 2.50x$.

Step3: Solve for $x$

Subtract $1.50x$ from both sides: $120=100 + 2.50x-1.50x$, which gives $120=100 + x$.
Subtract 100 from both sides: $x = 20$ mL.

Answer:

35 mL

5.