Determine the Capacity of a Water Tank Using Math

Understanding the Problem

Problems like those involving water tanks and mathematical calculations are common in both academic and practical settings. The example provided deals with understanding the capacity of a water tank through a series of steps involving fractions and algebra. Such problems help build foundational skills in mathematics that are applicable in various scenarios.

Exploring the Scenario

Let's dive into a practical scenario involving a water tank. We are given a water tank that was initially half-filled. After using 500 liters, only a quarter of the tank's original capacity remains. Our task is to determine the total capacity of the water tank.

Solving the Problem Step-by-Step

Method 1: Using Proportional Relationships

Let's denote the total capacity of the water tank as C liters.

The problem states that 2/3 of the tank initially had water, and when 500 liters are removed, a quarter (1/4) of the tank's capacity remains.

Let's break this down further:

2/3 of the tank (2/3)C When 500 liters are removed, the remaining water is 1/4 of the tank's capacity: (1/4)C

Using the information that 500 liters corresponds to the difference between the initial 2/3 filled state and the remaining 1/4 filled state:

1/3 (2/3 - 1/4) of the tank corresponds to 500 liters since:

(1/3)C 500

Therefore, C 500 * 3

C 1500 * 3

C 5000 liters

Method 2: Using Fractional Subtraction

Another method involves directly working with the fractions involved in the problem:

When 500 liters are removed, the tank is left with 1/3 of its initial capacity, as 2/3 - 1/4 1/3. The remaining volume (1/3 of the tank) is 2500 liters, as it is given.

To find the full capacity:

Full capacity 2500 * 3

Full capacity 7500 * 1/3 (of the remaining quarter)

Full capacity 5000 liters

Method 3: Using Algebraic Equations

Let the total capacity of the water tank be x liters. Initially, the tank is half-filled, so it contains (x/2) liters of water.

After using 500 liters, the amount of water left in the tank is:

(x/2) - 500

According to the problem, this amount equals a quarter of the tank's capacity:

(x/2) - 500 x/4

Multiplying the entire equation by 4 to eliminate the fractions:

2x - 2000 x

Rearranging the equation:

2x - x 2000

x 2000

Therefore, the capacity of the water tank is 2000 liters.

Conclusion

Through these methods, we have successfully determined the capacity of the water tank to be 5000 liters. Understanding such problems not only aids in mathematical skill development but also in real-world applications, such as managing resources like water in tanks and reservoirs.

By practicing such problems, students and professionals can improve their problem-solving abilities and apply mathematical concepts to diverse situations effectively.