Initial Capacity of the Bank

Oct 13, 2025 by talynt.org 77 views

Introduction

Jessica's penny bank is a container used to store a large number of pennies. The problem at hand is to determine the maximum capacity of Jessica's penny bank. We are given that the bank is initially $\frac{3}{8}$ full and after adding 240 pennies, it becomes $\frac{3}{4}$ full. Our goal is to find out how many pennies the bank can hold.

Initial Capacity of the Bank

Let's assume that the bank can hold $x$ pennies. Since the bank is initially $\frac{3}{8}$ full, the number of pennies in the bank at this point is $\frac{3}{8}x$ . We can represent this as an equation:

\frac{3}{8}x = \text{Initial number of pennies}

Adding Pennies to the Bank

After adding 240 pennies, the bank becomes $\frac{3}{4}$ full. This means that the total number of pennies in the bank is now $\frac{3}{4}x$ . We can represent this as an equation:

\frac{3}{4}x = \text{Initial number of pennies} + 240

Setting Up the Equation

We can now set up an equation using the two expressions we have derived. We know that the initial number of pennies is $\frac{3}{8}x$ , and after adding 240 pennies, the total number of pennies is $\frac{3}{4}x$ . We can write the equation as:

\frac{3}{4}x = \frac{3}{8}x + 240

Solving the Equation

To solve for $x$ , we can start by multiplying both sides of the equation by 8 to eliminate the fractions:

6x = 3x + 1920

Next, we can subtract $3x$ from both sides of the equation to get:

3x = 1920

Finally, we can divide both sides of the equation by 3 to solve for $x$ :

x = \frac{1920}{3}

Calculating the Maximum Capacity

Now that we have solved for $x$ , we can calculate the maximum capacity of Jessica's penny bank. Plugging in the value of $x$ , we get:

x = \frac{1920}{3} = 640

Therefore, Jessica's penny bank can hold a maximum of 640 pennies.

Conclusion

In this problem, we were given that Jessica's penny bank is initially $\frac{3}{8}$ full and after adding 240 pennies, it becomes $\frac{3}{4}$ full. We used algebraic equations to solve for the maximum capacity of the bank, which turned out to be 640 pennies. This problem demonstrates the importance of using mathematical equations to solve real-world problems.

Key Takeaways

We can use algebraic equations to solve real-world problems.
We can represent the initial and final number of pennies in the bank as equations.
We can solve for the maximum capacity of the bank by setting up and solving an equation.

Real-World Applications

This problem has real-world applications in various fields, such as:

Business: Understanding the capacity of a container is crucial in business, especially when it comes to storing and managing inventory.
Engineering: Engineers need to calculate the capacity of containers to design and build efficient systems.
Science: Scientists use mathematical equations to model and analyze real-world phenomena.

Final Thoughts

Frequently Asked Questions

Q: What is the initial capacity of Jessica's penny bank?

A: The initial capacity of Jessica's penny bank is not explicitly given. However, we can represent it as an equation: $\frac{3}{8}x = \text{Initial number of pennies}$ , where $x$ is the maximum capacity of the bank.

Q: How many pennies are added to the bank?

A: 240 pennies are added to the bank.

Q: What is the final capacity of Jessica's penny bank?

A: The final capacity of Jessica's penny bank is $\frac{3}{4}x$ , where $x$ is the maximum capacity of the bank.

Q: How do we solve for the maximum capacity of the bank?

A: We can solve for the maximum capacity of the bank by setting up and solving an equation. We know that the initial number of pennies is $\frac{3}{8}x$ , and after adding 240 pennies, the total number of pennies is $\frac{3}{4}x$ . We can write the equation as: $\frac{3}{4}x = \frac{3}{8}x + 240$ .

Q: What is the solution to the equation?

A: To solve for $x$ , we can start by multiplying both sides of the equation by 8 to eliminate the fractions: $6x = 3x + 1920$ . Next, we can subtract $3x$ from both sides of the equation to get: $3x = 1920$ . Finally, we can divide both sides of the equation by 3 to solve for $x$ : $x = \frac{1920}{3}$ .

Q: What is the maximum capacity of Jessica's penny bank?

A: The maximum capacity of Jessica's penny bank is 640 pennies.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as business, engineering, and science. Understanding the capacity of a container is crucial in these fields, especially when it comes to storing and managing inventory, designing and building efficient systems, and modeling and analyzing real-world phenomena.

Q: Why is it important to use mathematical equations to solve real-world problems?

A: Using mathematical equations to solve real-world problems is important because it allows us to make informed decisions and design efficient systems. Mathematical equations provide a precise and accurate way to model and analyze real-world phenomena, which is essential in various fields.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

We can use algebraic equations to solve real-world problems.
We can represent the initial and final number of pennies in the bank as equations.
We can solve for the maximum capacity of the bank by setting up and solving an equation.

Q: How can this problem be applied to real-world scenarios?

A: This problem can be applied to real-world scenarios in various fields, such as business, engineering, and science. For example, understanding the capacity of a container is crucial in business, especially when it comes to storing and managing inventory. Engineers need to calculate the capacity of containers to design and build efficient systems. Scientists use mathematical equations to model and analyze real-world phenomena.

Q: What are some final thoughts on this problem?

A: In conclusion, this problem demonstrates the importance of using mathematical equations to solve real-world problems. By understanding the capacity of a container, we can make informed decisions and design efficient systems. We hope that this problem has provided you with a deeper understanding of algebraic equations and their applications in real-world scenarios.