Question

Identify an unknown and rewrite each expression as an algebraic expression. The number of gallons of antifreeze in a radiator containing \(x\) gallons of a mixture that is \(11 \%\) antifreeze.

Short answer

The algebraic expression for the number of gallons of antifreeze is 0.11x.

Step-by-step solution

Define the Variable

Let the variable x represent the total number of gallons of the mixture in the radiator.

Determine the Percentage of Antifreeze

Since the mixture is 11% antifreeze, we convert this percentage to a decimal by dividing by 100. The decimal equivalent of 11% is 0.11.

Calculate the Amount of Antifreeze

To find the amount of antifreeze, multiply the total amount of the mixture (x gallons) by the decimal equivalent of the antifreeze percentage (0.11). The algebraic expression for the number of gallons of antifreeze is given by 0.11x.

Key concepts

Antifreeze Percentage Problem

When dealing with real-life situations such as figuring out the amount of antifreeze in a car's radiator, algebraic expressions come in handy. An 'antifreeze percentage problem' typically involves determining what fraction of a mixture consists of a certain substance—in this case, antifreeze.

The exercise presents such a scenario by asking for the amount of antifreeze when given the total amount of mixture in the radiator as well as the percentage of the mixture that is antifreeze. To solve similar problems, one needs to understand how percentages represent parts of a whole. Here, the percentage describes how many gallons of antifreeze are present per 100 gallons of mixture. Hence, we translate this percentage into a part of the variable representing the total mixture.

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is essential for solving algebraic expressions involving percentages. A percentage represents a number out of 100. To convert it to a decimal, which is necessary for mathematical operations, simply divide the percentage by 100.

For instance, the step by step solution reveals that 11% is converted to 0.11 by dividing 11 by 100. This seemingly simple act transforms the percentage into a multiplier that can be applied to any number to find 11% of it. Therefore, it is crucial to be comfortable with this conversion as it unlocks the ability to work with percentages in algebraic expressions and equations effectively.

Variable Definition in Algebra

In algebra, a variable is a symbol that represents an unknown value that can change or vary. It acts as a placeholder for a (typically) numeral that is not yet known or that can take on different values in an equation or expression.

In the exercise, the variable 'x' is used to represent the total amount of the mixture in the radiator. By defining 'x' as the total mixture, you can establish a relationship between the total volume and the amount of antifreeze. The algebraic expression '0.11x' then describes the amount of antifreeze as a function of the total volume of the mixture. This utilization of variables allows for a general solution to be found that can apply to many different values of 'x', showcasing the power and universality of algebra in problem-solving.