Problem 16 - Prealgebra

Question

Use the table that shows the average amount of time students ages 14 to 18 spend on homework per week. $$\begin{array}{|l|c|}\hline \text { Group } & \text { Average Hours per Week } \\\\\hline \text { Male } & 5.4 \\\\\hline \text { Female } & 6.8 \\\\\hline\end{array}$$ Inali spends at least an hour more than the average time spent by boys on homework each week. Write an inequality for Inali's homework time.

Step-by-Step Solution

Answer

Inali's homework time is \( x \geq 6.4 \) hours per week.

Step 1: Understand the Problem

We have a table showing the average weekly homework times for male and female students. Inali, whose gender is not specified, spends at least an hour more than the average time spent by boys on homework each week. We need to represent this situation as an inequality.

Step 2: Identify the Average Time for Boys

From the table, the average weekly homework time for male students (boys) is 5.4 hours. This information will be used to establish the minimum time Inali spends on homework.

Step 3: Formulate the Inequality

Inali spends at least one hour more than the average time (5.4 hours) spent by boys. This can be expressed as: if Inali's time is represented by \( x \), then the inequality is \( x \geq 5.4 + 1 \).

Step 4: Simplify the Inequality

Simplify the expression \( 5.4 + 1 \) to get \( 6.4 \). Thus, the inequality becomes \( x \geq 6.4 \). This means Inali spends 6.4 hours or more on homework each week.

Key Concepts

Understanding Average Homework Time

Average homework time is a critical measure that helps us understand how much students are engaging with their academic work outside of school hours. In this exercise, we have data for two groups: male and female students. - Male students, on average, spend 5.4 hours per week on homework. - Female students, on average, spend 6.8 hours per week on homework. These figures represent a typical week for each gender and provide a baseline for analyzing individual student behaviors. It's essential to remember that these averages are just that—averages. Not every student will spend exactly this amount of time. Some will spend less, and others more. Understanding these averages helps educators, parents, and even students themselves gauge where they stand in comparison to typical expectations.

Mathematical Representation of Scenarios

In mathematics, it's common to use numbers and symbols to express real-world situations. This is known as mathematical representation. In this case, we are tasked with expressing how much time Inali spends on homework in comparison to an average.The table tells us the average homework time, but we need to create an equation that expresses Inali's unique situation. Understanding how to use inequalities is key here because the problem asks us to define a situation where Inali spends 'at least' a certain amount of time.To represent this:- We assign a variable, such as \( x \), to the unknown quantity (Inali's homework time).- Since Inali spends at least one hour more than the average of boys (5.4 hours), the situation can be expressed by the inequality \( x \geq 5.4 + 1 \). - Simplifying, we find \( x \geq 6.4 \).This inequality serves as a concise and clear representation of the problem statement.

Effective Problem Solving Steps

Solving mathematical problems often calls for a systematic approach. Here, let's walk through the problem-solving steps used in this exercise:1. **Understand the Problem** - We begin by identifying key details from the table and the scenario described. For this problem, it's important to focus on the average homework time for boys and the condition given for Inali.2. **Identify Key Numbers** - Extract critical information such as the average time for males, which is 5.4 hours. This becomes foundational to our calculations.3. **Formulate the Inequality** - Transition the problem's words into a mathematical statement. For Inali, spending at least one hour more means adding 1 hour to the average 5.4 hours, resulting in the expression \( x \geq 6.4 \).4. **Simplify and Conclude** - Ensure the inequality is in its simplest form. Here, it's already simplified, making \( x \geq 6.4 \) the final inequality.By breaking down the problem into manageable steps, we can systematically approach and resolve complex scenarios mathematically. This method not only helps in solving the exercise but also builds confidence in tackling similar future problems.