Question

A 20-pound bag of Economy brand cement mix contains \(25 \%\) cement and \(75 \%\) sand. How much pure cement must be added to produce a cement mix that is \(40 \%\) cement?

Short answer

5 pounds

Step-by-step solution

Define Variables

Let the amount of pure cement to be added be denoted as \( x \) pounds.

Calculate Cement in Original Mix

The original 20-pound bag contains \( 25 \% \) cement. This means that the amount of cement in the original mix is \( 0.25 \times 20 = 5 \) pounds.

Create Equation for New Mix

After adding \( x \) pounds of pure cement, the total weight of the mix becomes \( 20 + x \) pounds. The total amount of cement in the new mix is \( 5 + x \) pounds.

Set Up Proportion

For the new mix to be \( 40 \, \% \) cement, the equation becomes: \[ \frac{5 + x}{20 + x} = 0.4 \]

Solve the Equation

Multiply both sides by \( 20 + x \) to clear the fraction: \[ 5 + x = 0.4(20 + x) \] Expand and simplify: \[ 5 + x = 8 + 0.4x \] Rearrange to isolate \( x \): \[ 0.6x = 3 \] Finally, solve for \( x \): \[ x = 5 \]

Key concepts

algebraic equations

In mixture problems like the one given, algebraic equations are used to represent mathematical relationships between the quantities involved. These equations are essential for modeling how different components of a mixture relate to each other.
For instance, in our cement mix problem, we used an equation to represent the relationship between the amount of cement and the total mixture. By defining variables, we can represent unknown quantities with symbols like \( x \), making it easier to manipulate and solve for them.
To connect the concepts, let's look at our equation: \( \frac{5 + x}{20 + x} = 0.4 \). This equation states that the ratio of cement to the total mix must equal 40%. Be sure to understand that forming the correct equation from the problem statement is crucial for solving it accurately.

percentages

Percentages are a key concept in mixture problems. They help us understand proportions and relationships in different parts of the mix. Percentages represent a part per hundred, making them ideal for showing ratios and comparisons.
In our exercise, the cement mix initially contains 25% cement and 75% sand, meaning that out of every 100 parts of the mix, 25 are cement and 75 are sand. We need to adjust these percentages to reach a mix that contains 40% cement.
This percentage change is crucial for setting up our equation. We then convert percentages into decimal form for easier calculations, such as converting 25% to 0.25 and 40% to 0.4.
Working through these conversions step-by-step ensures that we create accurate and solvable equations.

solving linear equations

Solving linear equations is a fundamental part of mixture problems. A linear equation is one in which each term is either a constant or the product of a constant and a single variable. The equation from our problem, \( \frac{5 + x}{20 + x} = 0.4 \), is a perfect example.
Here’s a quick breakdown of solving it:

• First, eliminate the fraction by multiplying both sides by \( 20 + x \).
• Next, simplify to form a linear equation: \( 5 + x = 8 + 0.4x \).
• Then, isolate the variable by rearranging terms: \( 0.6x = 3 \).
• Lastly, solve for \( x \) by dividing both sides by 0.6: \( x = 5 \).

Each step should be taken carefully to avoid errors, ensuring you correctly isolate and solve for the variable, providing the answer to the problem.

proportions in mixtures

Understanding proportions in mixtures involves calculating how different quantities relate to each other within the mix. These proportions are key to forming equations that reflect the desired mixture composition.
In the given problem, we start with a mix that has 25% cement and want to change it to one with 40% cement. This requires finding the right amount of pure cement to add.
We calculate proportions by:

• Identifying original components: 5 pounds of cement in a 20-pound mix.
• Determining the new total weight: \( 20 + x \) pounds after adding \( x \) pounds of pure cement.
• Ensuring the new cement content matches the desired proportion: \( \frac{5 + x}{20 + x} = 0.4 \).

By understanding and applying these proportions, we ensure the mixture meets the specified criteria, leading to a precise solution.