Problem 55 - Algebra for College Students

Question

A mixture of alcohol and water contains a total of 36 oz of liquid. There are 9 oz of pure alcohol in the mixture. What percent of the mixture is water? What percent is alcohol?

Step-by-Step Solution

Answer

75% of the mixture is water, and 25% is alcohol.

Step 1: Identify the Total Volume

The total volume of the mixture is given as 36 oz. This will be used to calculate the percentage of water and alcohol in the mixture.

Step 2: Determine the Volume of Alcohol

It is given that the volume of pure alcohol in the mixture is 9 oz.

Step 3: Calculate the Volume of Water

Subtract the volume of alcohol from the total volume to find the volume of water: \( 36 \text{ oz} - 9 \text{ oz} = 27 \text{ oz} \).

Step 4: Calculate the Percentage of Water

To find the percent of the mixture that is water, use the formula: \[ \text{Percentage of Water} = \left( \frac{\text{Volume of Water}}{\text{Total Volume}} \right) \times 100 \% \]. Substituting the known values: \[ \text{Percentage of Water} = \left( \frac{27}{36} \right) \times 100 = 75\% \].

Step 5: Calculate the Percentage of Alcohol

To find the percent of the mixture that is alcohol, use the formula: \[ \text{Percentage of Alcohol} = \left( \frac{\text{Volume of Alcohol}}{\text{Total Volume}} \right) \times 100 \% \]. Substituting the known values: \[ \text{Percentage of Alcohol} = \left( \frac{9}{36} \right) \times 100 = 25\% \].

Key Concepts

Mixture Problems

Mixture problems often involve combining substances in different quantities and computing the resulting ratios. In this exercise, we have a mixture of alcohol and water. These types of problems usually require:

• Identifying individual components
• Determining their quantities
• Computing the overall ratio or percentage

For this example, understanding that the mixture has a total of 36 ounces and contains 9 ounces of alcohol is crucial.
This gives us the ability to know what portion is water by simple subtraction. Hence, we can determine that there are 27 ounces of water (36 oz total minus 9 oz alcohol).

Percentages

Percentages are a way to express a ratio as a fraction of 100. They are particularly useful for comparing and understanding the relative size of quantities. In this exercise, the percentage of a component within the mixture is found using the formula:

\[\text{Percentage} = \left( \frac{\text{Volume of Component}}{\text{Total Volume}} \right) \times 100 \]

With this formula, we can calculate:

• The percentage of water in the mixture
• The percentage of alcohol in the mixture

Let's apply this:
The volume of water is 27 oz, so
\[ \text{Percentage of Water} = \left( \frac{27}{36} \right) \times 100 \approx 75\% \]
And for the alcohol, the volume is 9 oz, so
\[ \text{Percentage of Alcohol} = \left( \frac{9}{36} \right) \times 100 = 25\% \]
This means that 75% of the mixture is water, and 25% is alcohol.

Algebra

Algebra involves using mathematical symbols to represent numbers and operations. It's a powerful tool for solving problems where values need to be found. In this exercise, algebra helps by formalizing the relationships between the quantities in the mixture.

For example:

• Let the total volume = 36 oz
• Let the volume of alcohol = 9 oz
• We can represent the volume of water as Total Volume - Volume of Alcohol

Mathematically, that's:
\[ \text{Volume of Water} = 36 - 9 = 27 \text{ oz} \]
By using the derived relationships, algebra further helps in converting these relationships into percentages, making it easier to understand the composition of mixtures in a straightforward way. Algebra also enhances the precision and flexibility of representing and solving such mixture problems.