Question

8.0 cubic feet of cement is contained in a concrete mixture that is \(12 \%\) cement by volume. What is the volume of the total mixture?

Short answer

The volume of the total mixture is approximately 66.67 cubic feet.

Step-by-step solution

Understand the Percentage of Cement in the Mixture

The problem states that the concrete mixture contains 8.0 cubic feet of cement which makes up 12% of the total volume of the mixture.

Calculate the Total Volume

Since 12% of the total volume is cement, you can represent the total volume of the mixture with a variable, let's say V. Therefore, the volume of cement is 0.12V. You know the volume of cement is 8.0 cubic feet, so set up the equation 0.12V = 8 cubic feet.

Solve for Total Volume V

Divide both sides of the equation by 0.12 to solve for V: V = 8 / 0.12.

Calculate and Find the Answer

Perform the division to find the value of V: V = 8 / 0.12 = 66.67 cubic feet. This is the total volume of the concrete mixture.

Key concepts

Volume Calculation

Understanding volume calculation is essential for solving various real-world problems, especially in construction and manufacturing. Volume refers to the amount of space occupied by a substance or object, and it's typically measured in cubic units, such as cubic feet, cubic meters, or liters. When we speak of the volume of a mixture, it's a case of combining different materials and calculating the total space they occupy together.

In our exercise, we calculated the volume of a concrete mixture. To visualize this, imagine a container large enough to hold the entire mixture. The amount of space inside this container that the mixture takes up is what we're calculating. Always ensure the units used for volume are consistent throughout the problem to avoid any confusion. For example, had the mixture been measured in cubic meters but we are given a volume in cubic feet, a conversion would be necessary before solving the problem.

Mathematical Problem-Solving

Mathematical problem-solving is a critical skill that students must develop, as it is not just about understanding mathematical procedures but also about applying logical thinking to reach a solution. The first step in problem-solving is to understand the question and identify what is known and what needs to be found.

In our exercise, the problem clearly outlined that the volume of cement is a certain percentage of the total mixture's volume. Identifying that percentage relationship is key to setting up the correct equation to find the total volume. The problem-solving process often involves writing down what you know in an equation or formula and then manipulating that equation to find the unknown value. Critical thinking and a systematic approach to solving equations, like isolating the variable, can lead to the correct answer, as demonstrated in the solution provided.

Percentage by Volume

Percentage by volume is a concept used to express the concentration of a component in a mixture or solution. It is calculated as the ratio of the volume of the component of interest to the total volume of the mixture, multiplied by 100%. For instance, if you have a bottle of a drink with 5% of the volume being sugar syrup, this means that out of 100 units of total volume, 5 units are sugar syrup.

In the context of our concrete mixture problem, the percentage by volume tells us how much of the mixture is made up of cement. To find the total volume of the mixture, we used the given percentage and the volume of cement. This concept extends well beyond the math classroom, as it is widely used in chemistry for solutions, in cooking recipes, and in determining the strength of various materials and products. Mastering how to calculate percentage by volume is thus a valuable skill across many fields.