Questions: A scientist mixes water (containing no salt) with a solution that contains 45% salt. She wants to obtain 225 ounces of a mixture that is 15% salt. How many ounces of water and how many ounces of the 45% salt solution should she use?

Solution Steps

To solve this problem, we need to set up a system of linear equations based on the given information. Let \( x \) be the ounces of water and \( y \) be the ounces of the 45% salt solution. We have two equations: one for the total volume of the mixture and another for the concentration of salt in the mixture. The first equation is \( x + y = 225 \) (total ounces), and the second equation is \( 0.45y = 0.15 \times 225 \) (total salt content). Solving this system will give us the values of \( x \) and \( y \).

To solve the problem, we need to set up a system of linear equations. Let \( x \) be the ounces of water and \( y \) be the ounces of the 45% salt solution. We have two equations based on the problem statement:

1. The total volume of the mixture is 225 ounces: \[ x + y = 225 \]

2. The total salt content in the mixture should be 15% of 225 ounces: \[ 0.45y = 0.15 \times 225 \]

First, calculate the right-hand side of the second equation: \[ 0.15 \times 225 = 33.75 \] Thus, the second equation becomes: \[ 0.45y = 33.75 \]

Now, solve the system of equations:

1. From the second equation, solve for \( y \): \[ y = \frac{33.75}{0.45} = 75 \]

2. Substitute \( y = 75 \) into the first equation to find \( x \): \[ x + 75 = 225 \implies x = 225 - 75 = 150 \]

Final Answer