Problem 16 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. Mixture Problem A solution of alcohol and water is \(80 \%\) alcohol. The solution is found to contain 32 milliliters of alcohol. How many milliliters total (both alcohol and water) are in the solution?

Step-by-Step Solution

Answer

The total solution is 40 milliliters.

Step 1: Identify the Components

The problem involves a mixture of alcohol and water where alcohol makes up 80% of the solution. We need to find the total volume of the solution (both alcohol and water).

Step 2: Restate as a Percent Problem

The problem can be restated as follows: If 80% of the solution equals 32 milliliters of alcohol, what is 100% of the solution? This translates to finding the total amount given a part and the percent it represents.

Step 3: Set Up the Equation

Let the total volume of the solution be denoted by \( x \). Since 80% of the solution is 32 milliliters, we set up the equation: \( 0.8x = 32 \).

Step 4: Solve for x

Solve the equation to find the total volume: \( 0.8x = 32 \). Divide both sides by 0.8: \[ x = \frac{32}{0.8} = 40. \]

Step 5: Interpret the Result

The total volume of the solution, including both alcohol and water, is 40 milliliters.

Key Concepts

Percent Equations

Percent equations are instrumental in solving problems related to percentages, especially when dealing with portions and parts of a whole. At its core, a percent equation helps us find unknown quantities by setting up a relationship between the part, whole, and the percentage. To do this, we often translate the given percent into its decimal form and use it in our equation.

For example, if we know that 80% of a solution is alcohol, and this constitutes 32 milliliters, it allows us to set up the equation using the formula:

• Part = Percent × Whole
• Given 80% as a decimal is 0.8, the equation becomes: 0.8 × Total = 32

Rearranging this formula can help us find the total quantity we're interested in. Percent equations are a handy way to translate real-world problems into mathematical expressions.

Mixture Problems

Mixture problems involve combining two or more substances into a single mixture, and then determining the proportions or amounts of certain components within it. A common task in mixture problems is finding out the total volume or concentration of a desired component, such as alcohol in an alcohol-water mixture.

• Identify the components and their respective percentages in the mixture.
• Restate the problem in terms of these percentages and quantities.

In the given exercise, the aim is to find the full amount of the mixture given the concentration of alcohol. This requires using the percent that represents alcohol and the known amount of alcohol to backtrack into the total mixture volume. Mixture problems are common in chemistry but appear in other fields where proportional relationships matter, such as cooking or solutions preparation.

Solving Equations

Solving equations is a fundamental skill in mathematics, crucial for finding unknown quantities. To solve equations, you need to isolate the variable, typically represented as 'x,' by using arithmetic operations. By manipulating the equation, you work towards expressing 'x' as a standalone quantity.In our example, the equation is:

• 0.8x = 32
• Further simplify by dividing both sides by 0.8, leading to: \( x = \frac{32}{0.8} \)
• This results in: \( x = 40 \)

A critical part of this process is understanding the balance between both sides of an equation. Operations performed on one side must be mirrored on the other to maintain equality. This principle ensures that as you simplify, the integrity of the equation remains intact.