Questions: A chemist needs 30 mL of a 12% acid solution for an experiment. The lab has available a 10% and a 25% solution. How many milliliters of the 10% and how many milliliters of the 25% solutions should the chemist mix to make the 12% solution?

A chemist needs 30 mL of a 12% acid solution for an experiment. The lab has available a 10% and a 25% solution. How many milliliters of the 10% and how many milliliters of the 25% solutions should the chemist mix to make the 12% solution?

Transcript text: A chemist needs 30 mL of a $12 \%$ acid solution for an experiment. The lab has available a $10 \%$ and a $25 \%$ solution. How many milliliters of the $10 \%$ and how many milliliters of the $25 \%$ solutions should the chemist mix to make the $12 \%$ solution?

Solution

Solution Steps

Step 1: Define Variables

Let $ x $ be the amount of the $ 10\% $ solution in milliliters, and $ y $ be the amount of the $ 25\% $ solution in milliliters. We need to find the values of $ x $ and $ y $.

Step 2: Set Up the System of Equations

We have two conditions to satisfy:

The total volume of the mixture should be 30 mL: \[ x + y = 30 \]
The total amount of acid in the mixture should be $ 12\% $ of 30 mL: \[ 0.10x + 0.25y = 0.12 \times 30 \]

Step 3: Simplify the Equations

First, simplify the second equation: \[ 0.10x + 0.25y = 3.6 \]

Step 4: Solve the System of Equations

From the first equation, express $ y $ in terms of $ x $: \[ y = 30 - x \]

Substitute $ y = 30 - x $ into the second equation: \[ 0.10x + 0.25(30 - x) = 3.6 \]

Simplify and solve for $ x $: \[ 0.10x + 7.5 - 0.25x = 3.6 \] \[ -0.15x + 7.5 = 3.6 \] \[ -0.15x = 3.6 - 7.5 \] \[ -0.15x = -3.9 \] \[ x = \frac{-3.9}{-0.15} = 26 \]

Substitute $ x = 26 $ back into the equation for $ y $: \[ y = 30 - 26 = 4 \]