Understanding the Mathematics Behind the Problem

Oct 13, 2025 by miodeya.com 76 views

Understanding Manuel's Savings

Manuel's Savings Problem: A Mathematical Approach

Manuel has been saving a certain amount of money each week, and we are given some information about his savings. He has been saving $\$6$ each week, and today, he spent $\$90$ of his savings, leaving him with $\$48$ left. We need to find out how many weeks he has been saving.

Formulating the Problem

Let's denote the number of weeks Manuel has been saving as $x$ . Since he saves $\$6$ each week, the total amount of money he has saved is $6x$ . However, he spent $\$90$ of his savings, so the amount of money he has left is $6x - 90$ . We are given that he has $\$48$ left, so we can set up the equation:

6x - 90 = 48

Solving the Equation

To solve for $x$ , we need to isolate the variable. First, let's add $90$ to both sides of the equation:

6x = 48 + 90

6x = 138

Next, let's divide both sides of the equation by $6$ :

x = \frac{138}{6}

x = 23

Interpreting the Result

So, Manuel has been saving for $23$ weeks. This means that he has been saving $\$6$ each week for $23$ weeks, and he has a total of $\$138$ saved.

Conclusion

In this problem, we used a simple algebraic equation to solve for the number of weeks Manuel has been saving. By setting up the equation and solving for the variable, we were able to find the answer to the problem.

Understanding the Mathematics Behind the Problem

The mathematics behind this problem involves basic algebraic concepts, such as solving linear equations. The equation we set up is a linear equation in one variable, and we solved it using basic algebraic techniques.

Real-World Applications

This problem has real-world applications in finance and economics. Understanding how to calculate savings and expenses is an important skill for individuals and businesses alike. By applying mathematical concepts to real-world problems, we can make informed decisions and manage our finances effectively.

Additional Examples and Exercises

Here are some additional examples and exercises to help you practice solving linear equations:

If Manuel saves $\$8$ each week and has $\$56$ left after spending $\$120$ , how many weeks has he been saving?
If a company saves $\$10$ each week and has $\$240$ left after spending $\$300$ , how many weeks has the company been saving?
If a person saves $\$12$ each week and has $\$72$ left after spending $\$180$ , how many weeks has the person been saving?

Solutions to Additional Examples and Exercises

If Manuel saves $\$8$ each week and has $\$56$ left after spending $\$120$ , we can set up the equation:

8x - 120 = 56

Solving for $x$ , we get:

8x = 56 + 120

8x = 176

x = \frac{176}{8}

x = 22

So, Manuel has been saving for $22$ weeks.

If a company saves $\$10$ each week and has $\$240$ left after spending $\$300$ , we can set up the equation:

10x - 300 = 240

Solving for $x$ , we get:

10x = 240 + 300

10x = 540

x = \frac{540}{10}

x = 54

So, the company has been saving for $54$ weeks.

If a person saves $\$12$ each week and has $\$72$ left after spending $\$180$ , we can set up the equation:

12x - 180 = 72

Solving for $x$ , we get:

12x = 72 + 180

12x = 252

x = \frac{252}{12}

x = 21

So, the person has been saving for $21$ weeks.

Conclusion

In this article, we used algebraic equations to solve for the number of weeks Manuel has been saving. We also provided additional examples and exercises to help you practice solving linear equations. By applying mathematical concepts to real-world problems, we can make informed decisions and manage our finances effectively.
Manuel's Savings Problem: A Q&A Guide

Frequently Asked Questions

In this article, we will answer some frequently asked questions about Manuel's savings problem.

Q: What is the main goal of the problem?

A: The main goal of the problem is to find out how many weeks Manuel has been saving.

Q: What information do we have about Manuel's savings?

A: We know that Manuel saves $\$6$ each week and has $\$48$ left after spending $\$90$ .

Q: How do we set up the equation to solve for the number of weeks?

A: We set up the equation $6x - 90 = 48$ , where $x$ is the number of weeks Manuel has been saving.

Q: How do we solve for the number of weeks?

A: We add $90$ to both sides of the equation to get $6x = 138$ , and then we divide both sides by $6$ to get $x = 23$ .

Q: What does the solution mean?

A: The solution means that Manuel has been saving for $23$ weeks.

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

A: This problem has real-world applications in finance and economics. Understanding how to calculate savings and expenses is an important skill for individuals and businesses alike.

Q: Can we use this problem to solve other similar problems?

A: Yes, we can use this problem as a model to solve other similar problems. For example, if we know how much someone saves each week and how much they have left after spending a certain amount, we can set up a similar equation to solve for the number of weeks they have been saving.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Not setting up the equation correctly
Not solving for the variable correctly
Not checking the solution to make sure it makes sense in the context of the problem

Q: How can we use this problem to practice solving linear equations?

A: We can use this problem to practice solving linear equations by setting up the equation and solving for the variable. We can also try solving the problem with different values for the savings and expenses.

Q: What are some additional examples and exercises that we can use to practice solving linear equations?

A: Here are some additional examples and exercises that we can use to practice solving linear equations:

If Manuel saves $\$8$ each week and has $\$56$ left after spending $\$120$ , how many weeks has he been saving?
If a company saves $\$10$ each week and has $\$240$ left after spending $\$300$ , how many weeks has the company been saving?
If a person saves $\$12$ each week and has $\$72$ left after spending $\$180$ , how many weeks has the person been saving?

Solutions to Additional Examples and Exercises

If Manuel saves $\$8$ each week and has $\$56$ left after spending $\$120$ , we can set up the equation:

8x - 120 = 56

Solving for $x$ , we get:

8x = 56 + 120

8x = 176

x = \frac{176}{8}

x = 22

So, Manuel has been saving for $22$ weeks.

If a company saves $\$10$ each week and has $\$240$ left after spending $\$300$ , we can set up the equation:

10x - 300 = 240

Solving for $x$ , we get:

10x = 240 + 300

10x = 540

x = \frac{540}{10}

x = 54

So, the company has been saving for $54$ weeks.

If a person saves $\$12$ each week and has $\$72$ left after spending $\$180$ , we can set up the equation:

12x - 180 = 72

Solving for $x$ , we get:

12x = 72 + 180

12x = 252

x = \frac{252}{12}

x = 21

So, the person has been saving for $21$ weeks.

Conclusion

In this article, we answered some frequently asked questions about Manuel's savings problem. We also provided additional examples and exercises to help you practice solving linear equations. By applying mathematical concepts to real-world problems, we can make informed decisions and manage our finances effectively.