Problem 59 - Algebra A Combined Function

Question

Answer each question. If a tank is filled in 3 hours, what part of the tank is filled in 1 hour?

Step-by-Step Solution

Answer

In 1 hour, \( \frac{1}{3} \) of the tank is filled.

Step 1: Understanding the Problem

We need to determine what fraction of a tank is filled in 1 hour when it takes 3 hours to fill the entire tank. This means we need to understand the rate at which the tank is filled.

Step 2: Defining the Rate of Filling

Since the tank is filled completely in 3 hours, the rate of filling is that the entire tank, or 1 tank, is filled in 3 hours. This can be represented mathematically as the rate being 1 tank per 3 hours.

Step 3: Calculate the Fraction Filled in 1 Hour

To find out how much of the tank is filled in 1 hour, we divide the total work (1 full tank) by the number of hours (3 hours). The math operation is 1 tank divided by 3, or \( \frac{1}{3} \).

Step 4: Conclusion

In 1 hour, \( \frac{1}{3} \) of the tank is filled. This makes sense because 3 hours are needed for the whole tank, so each hour contributes one third towards filling the tank.

Key Concepts

Fraction

A fraction represents a part of a whole. In this exercise, we are interested in what part of the tank is filled in one hour if the whole tank is filled in three hours. This question involves dividing the whole into equal parts. When dealing with fractions, you often refer to the numerator and the denominator.

• The numerator indicates how many parts are being considered.
• The denominator shows the total number of equal parts the whole is divided into.

In our context, the tank can be seen as being divided into three equal segments. Each hour corresponds to filling one of these segments, or one of three parts. Therefore, the fraction of the tank filled in one hour is represented by the fraction \( \frac{1}{3} \). This simple division allows us to break down and understand how much of the task is completed at a specific time.

Problem Solving

Problem solving in mathematics is about breaking down a situation to find a solution effectively. For rate problems like this, it's crucial to understand the relationship between the given information and what is being asked. Start by identifying what you know and what you need to find out.

In this problem, we know:

• The tank fills completely in 3 hours.
• The problem asks what fraction of the tank is filled in just one hour.

By understanding that you need to find how one hour compares to the whole process (3 hours), a single mathematical division gives the answer. Setting up a problem this way helps in organizing the information and using it effectively. Applying logical steps ensures that you can navigate through to finding a solution accurately.

Algebra

Algebra involves using symbols and letters to represent numbers and relationships between them. Although this problem might seem like a simple fraction, it's fundamentally rooted in algebra.

For instance, you can represent the problem algebraically as \( x = \frac{1}{3} \), where \( x \) is the amount of tank filled in one hour. Algebra helps in generalizing solutions. Even simple equations illustrate the concept clearly, allowing you to understand and apply them to similar real-world problems.
Understanding algebra can further help in expressing the solution in more complex rate problems, where other variables or unknowns might be involved. It acts as a solid foundation when learning and applying mathematical concepts.

Time Management

In mathematical problems, especially those involving rates, understanding time management is key. This involves knowing how to distribute work over time effectively. For this exercise, recognizing that the tank filling divides evenly across the total time gives you insight into efficient time allocation.

To apply this:

• Determine the total time available or required (3 hours here).
• Calculate the contribution every single unit of time (1 hour) makes towards the completion (\( \frac{1}{3} \) of the tank).

Effective time management means segmenting tasks into manageable parts and understanding their cumulative effect. In math, this could mean breaking problems into smaller, solvable segments, thereby achieving accuracy step by step. Such skills are not only applicable to textbook exercises but also to various real-life scenarios, like project planning or resource management.