Problem 49 - Algebra 1

Question

The average 12 -to- 17 -year-old spends 645 minutes per month on a personal computer. This is 732 fewer minutes per month than the average 18 -to- 24 -year-old spends. How many minutes per month does the average 18 -to- 24 -year-old spend on a personal computer? D Source: Media Matrix

Step-by-Step Solution

Answer

The average 18 to 24-year-old spends 1377 minutes per month on a personal computer.

Step 1: Identify the known variables

The average 12 to 17-year-old spends 645 minutes per month on a personal computer. This is 732 fewer minutes per month than the average 18 to 24-year-old spends on a personal computer.

Step 2: Write an equation based on the problem

Let \(x\) be the number of minutes the average 18 to 24-year-old spends on a personal computer. The problem states that 'This (645 minutes) is 732 fewer minutes per month than the average 18 to 24 -year-old spends' which can be translated to \(x = 645 + 732\) minutes.

Step 3: Solve the equation

Solving the equation gives \(x = 1377\) minutes.

Key Concepts

Variables in Equations

When solving equations, understanding variables is crucial. A variable is a symbol, often represented by letters like \(x\), \(y\), or \(z\), that stands in for an unknown number. In equations, variables play the role of placeholders that help us figure out what the unknown number is by using the relationship given in a mathematical problem.
In our initial exercise, the variable \(x\) represents the unknown amount of time the average 18 to 24-year-old spends on a personal computer each month.
This helps us create a mathematical expression that translates words into numbers. By identifying what the variable should represent, we set the foundation for solving the equation.

Algebraic Expressions

An algebraic expression like \(x = 645 + 732\) is a form of communication in mathematics. It includes numbers, operators, and variables to succinctly express a calculation or idea. In essence, algebraic expressions are tools that allow us to model real-world situations using mathematical language.

• The expression we used in the exercise provided a clear and direct way to translate the word problem into math terms.
• This ensured that the relationship among the different quantities was preserved accurately.

By constructing the algebraic expression \(x = 645 + 732\), we depicted the relationship between the time spent by the two age groups in a straightforward manner. Having a well-defined expression guides us through the problem-solving process.

Mathematical Problem Solving

Solving mathematical problems involves a methodical step-by-step process. In our example, it began with understanding the problem by identifying known quantities and determining what we needed to find.
With the correct algebraic expression set up, solving the equation step involved performing arithmetic calculations. This gave us the exact number of minutes: 1377, which the average 18 to 24-year-old spends on a personal computer each month.

• This solution technique can be applied broadly and is valuable beyond the exercise since it emphasizes clarity in defining what you know and systematically working towards what you want to discover.
• Finally, confirming our solution was reasonable and matched the problem's context was crucial—ensuring that it made sense was the last step in effective problem solving.

Understanding each stage ensures you can tackle similar problems confidently, using logical thinking and structured methods.