Problem 29 - Prealgebra

Question

Andrea swims 40 meters per minute, and she wants to swim at least 2000 meters this morning. Write and solve an inequality to find how long she should swim.

Step-by-Step Solution

Answer

Andrea should swim for at least 50 minutes.

Step 1: Identify the Variables

First, identify what you are trying to find and the information given. Let \( t \) represent the time in minutes Andrea needs to swim. We are informed that she swims 40 meters per minute and needs to swim at least 2000 meters.

Step 2: Set Up the Inequality

Using the information, set up the inequality to represent the situation. The inequality is based on the distance condition: \( 40t \geq 2000 \), which means the distance covered by swimming for \( t \) minutes must be at least 2000 meters.

Step 3: Solve the Inequality

Solve the inequality \( 40t \geq 2000 \) by isolating \( t \). Divide both sides of the inequality by 40: \( t \geq \frac{2000}{40} \). This simplifies to \( t \geq 50 \).

Step 4: Interpret the Solution

The solution to the inequality tells us Andrea needs to swim for at least 50 minutes to meet her goal of at least 2000 meters.

Key Concepts

Identifying Variables

In mathematics, identifying variables is an essential first step in solving problems, particularly those related to inequalities or equations. A variable is essentially a symbol, often denoted by letters like \( t \) or \( x \), representing an unknown quantity we aim to determine. In our exercise, the variable is \( t \), which stands for the time Andrea will spend swimming. Knowing what each variable represents helps us set up the correct equation or inequality efficiently.

Here are some simple tips to help identify variables in a problem:

• Look for phrases like "how long," "how much," or "how many," indicating what needs to be found.
• Take note of units associated with the variable, such as minutes in this exercise.
• Use the given information to define what the variable stands for in the context of the problem.

Recognizing variables helps translate a word problem into a mathematical form, which can then be solved.

Understanding Inequalities in Mathematics

Inequalities in mathematics express a relationship where one value is greater than or less than another. They are like equations, but instead of an equals sign \( = \), inequalities utilize symbols such as \( >, <, \geq, \) or \( \leq \). In the exercise, the inequality is \( 40t \geq 2000 \).

This statement means the distance Andrea swims (which is \( 40t \) meters per minute multiplied by \( t \) minutes) must be equal to, or greater than, 2000 meters. Inequalities can have:

• One variable, which is what we often see in simple word problems.
• Strict inequality with \( > \) or \( < \) or a non-strict one with \( \geq \) or \( \leq \).

Solving inequalities involves similar steps as solving equations, like isolating the variable, but it's important to follow specific rules like flipping the inequality sign when multiplying or dividing by a negative number.

Exploring the Distance and Time Relationship

Distance and time have a simple and straightforward relationship often represented in problems like this one. Here, Andrea's swimming speed is given as 40 meters per minute. This means every minute, she covers 40 meters.

To find out how long Andrea needs to swim to cover 2000 meters, a relationship formula is used:
\[ \text{Distance} = \text{Speed} \times \text{Time} \] This relationship can be re-arranged depending on what you are solving for. In Andrea’s situation, solving for time (\( t \)) involves setting the inequality:

• With Distance as 2000 meters
• Speed as 40 meters per minute
• Time as the unknown variable \( t \) we need to find out

Understanding distance and time relationships can help break down more complex problems later on and is a fundamental skill in both algebra and physics.