Problem 9 - Algebra for College Students

Question

A tank contains 5832 gallons of water. Each day onethird of the water in the tank is removed and not replaced. How much water remains in the tank at the end of 6 days? 512 gallons

Step-by-Step Solution

Answer

After 6 days, 512 gallons of water remain.

Step 1: Understand the Problem

We have a tank with 5832 gallons of water, and each day one-third of the water is removed. We need to find out how much water remains after 6 days.

Step 2: Express Reduction Mathematically

Every day, one-third of the water in the tank is removed, which means two-thirds of the water remains. Therefore, we need to multiply the remaining water by \( \frac{2}{3} \) each day.

Step 3: Calculate Day-Wise Reduction

Let's compute the amount of water remaining at the end of each day:- Day 1: \( 5832 \times \frac{2}{3} = 3888 \) gallons- Day 2: \( 3888 \times \frac{2}{3} = 2592 \) gallons- Day 3: \( 2592 \times \frac{2}{3} = 1728 \) gallons- Day 4: \( 1728 \times \frac{2}{3} = 1152 \) gallons- Day 5: \( 1152 \times \frac{2}{3} = 768 \) gallons- Day 6: \( 768 \times \frac{2}{3} = 512 \) gallons

Step 4: Conclusion

After 6 days, the amount of water remaining in the tank is 512 gallons.

Key Concepts

Fraction Operations

Understanding fraction operations is key when dealing with situations like the water tank problem. Here, we see a real-world scenario involving repeated fractional reduction of a quantity. Fraction operations allow us to systematically reduce amounts, in this case, the water in the tank is reduced by one-third each day.
Every day, removing one-third of the water translates into keeping two-thirds. Mathematically, this is represented by the fraction \( \frac{2}{3} \). To calculate how much remains each day, we multiply the current amount of water by \( \frac{2}{3} \). Performing these operations correctly is crucial for problem-solving, as it helps in breaking down larger problems into manageable parts.

• Start by identifying the operation needed, like removing or keeping a fraction of an amount.
• Translate verbal descriptions into mathematical expressions.
• Perform operations systematically to achieve consistent results.

Mastering fraction operations can simplify many complex problems, making it easier to predict outcomes and make calculations efficiently.

Problem-Solving

Problem-solving in mathematics often involves breaking down a problem into smaller, more manageable pieces. With the water tank problem, we start by understanding the scenario: 5832 gallons of water with a daily reduction of one-third.
To effectively tackle such problems, follow these steps:

• Identify the problem: What are we trying to find? In this problem, it is the remaining quantity of water after six days.
• Understand the processes involved: Each day, a specific fraction of water is removed. Recognize how this operation affects the total each day.
• Apply logical sequencing: It involves sequentially applying the same operation over different steps (or days, in our case). This can often be represented through iterations that help to build the complete solution over time.

By employing these problem-solving tactics, even complex problems can be deciphered smoothly. It is about applying known methods to work through the steps necessary to uncover the solution.

Mathematical Modeling

Mathematical modeling translates real-world scenarios into mathematical language and expressions. In the tank problem, we model the daily reduction of water using fractions and repeated multiplications.
The given problem is an excellent example of iterative modeling, where the same mathematical operation \( \left( \times \frac{2}{3} \right) \) is applied repeatedly. Each day's water amount becomes the model for the next, demonstrating how iterative calculations can accurately represent continual processes.

• Identify real-world variables: Here, the total amount of water is a variable that changes daily.
• Create a mathematical framework: Use fractions and multiplication to represent the problem.
• Adjust as needed to reflect real changes over time.

Using models helps to visualize and solve problems that might otherwise seem complex. These techniques are invaluable in predicting outcomes based on set parameters or rules.