Calculating the Total Bill

Oct 13, 2025 by vinooalluri.com 47 views

Understanding the Problem

Alonso paid for repairs on his car, and $\frac{3}{5}$ of the bill was for labor costs. The cost of the labor was $79.50. We need to find the total bill amount, denoted by $b$ . To solve this problem, we will use algebraic equations and proportions.

Setting Up the Equation

Let's assume the total bill is $b$ . Since $\frac{3}{5}$ of the bill was for labor costs, the amount spent on labor is $\frac{3}{5}b$ . We are given that the cost of the labor was $79.50. Therefore, we can set up the following equation:

\frac{3}{5}b = 79.50

Solving the Equation

To solve for $b$ , we can multiply both sides of the equation by $\frac{5}{3}$ to isolate $b$ :

b = 79.50 \times \frac{5}{3}

Calculating the Total Bill

Now, let's calculate the total bill amount:

b = 79.50 \times \frac{5}{3}

b = 132.50

Therefore, the total bill amount is $132.50.

Conclusion

In this problem, we used algebraic equations and proportions to find the total bill amount. We set up an equation based on the given information and solved for the unknown variable $b$ . The correct equation and solution are:

\frac{3}{5}b = 79.50

b = 79.50 \times \frac{5}{3}

b = 132.50

This problem demonstrates the importance of using algebraic equations and proportions to solve real-world problems involving labor costs and total bill amounts.

Additional Tips and Variations

To check the solution, we can substitute the value of $b$ back into the original equation to ensure that it is true.
We can also use a calculator to check the solution.
If the problem had multiple labor costs, we could use a system of equations to solve for the total bill amount.
We can also use proportions to solve for the total bill amount if we know the ratio of labor costs to total bill costs.

Common Mistakes to Avoid

Not setting up the equation correctly
Not solving for the unknown variable
Not checking the solution
Not using proportions to solve for the total bill amount

Real-World Applications

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

Business: Labor costs and total bill amounts are important considerations in business decision-making.
Finance: Labor costs and total bill amounts are important factors in financial planning and budgeting.
Engineering: Labor costs and total bill amounts are important considerations in engineering projects and construction.

Conclusion

Q: What is the relationship between labor costs and total bill amounts?

A: Labor costs are a portion of the total bill amount. In this problem, $\frac{3}{5}$ of the bill was for labor costs, which means that the labor costs are 3/5 of the total bill amount.

Q: How do I set up the equation to find the total bill amount?

A: To set up the equation, you need to know the amount spent on labor costs and the ratio of labor costs to total bill costs. In this problem, we know that the cost of the labor was $79.50 and that $\frac{3}{5}$ of the bill was for labor costs. We can set up the equation as follows:

\frac{3}{5}b = 79.50

Q: How do I solve for the total bill amount?

A: To solve for the total bill amount, you need to isolate the variable $b$ . You can do this by multiplying both sides of the equation by the reciprocal of the coefficient of $b$ . In this problem, the coefficient of $b$ is $\frac{3}{5}$ , so we can multiply both sides of the equation by $\frac{5}{3}$ :

b = 79.50 \times \frac{5}{3}

Q: What if I have multiple labor costs? How do I solve for the total bill amount?

A: If you have multiple labor costs, you can use a system of equations to solve for the total bill amount. For example, if you have two labor costs, $x$ and $y$ , and the total bill amount is $b$ , you can set up the following system of equations:

\frac{3}{5}b = x

\frac{2}{5}b = y

You can solve this system of equations using substitution or elimination.

Q: How do I check the solution?

A: To check the solution, you can substitute the value of $b$ back into the original equation to ensure that it is true. For example, if we substitute $b = 132.50$ back into the original equation, we get:

\frac{3}{5}(132.50) = 79.50

This equation is true, so we can be confident that our solution is correct.

Q: What are some common mistakes to avoid when solving for the total bill amount?

A: Some common mistakes to avoid when solving for the total bill amount include:

Not setting up the equation correctly
Not solving for the unknown variable
Not checking the solution
Not using proportions to solve for the total bill amount

Q: What are some real-world applications of labor costs and total bill amounts?

A: Labor costs and total bill amounts have real-world applications in various fields, such as:

Business: Labor costs and total bill amounts are important considerations in business decision-making.
Finance: Labor costs and total bill amounts are important factors in financial planning and budgeting.
Engineering: Labor costs and total bill amounts are important considerations in engineering projects and construction.

Q: How can I use proportions to solve for the total bill amount?

A: You can use proportions to solve for the total bill amount by setting up a proportion based on the ratio of labor costs to total bill costs. For example, if you know that $\frac{3}{5}$ of the bill was for labor costs and the cost of the labor was $79.50, you can set up the following proportion:

\frac{3}{5} = \frac{79.50}{b}

You can solve this proportion by cross-multiplying and solving for $b$ .

Conclusion

In conclusion, this Q&A guide provides a comprehensive overview of labor costs and total bill amounts. We have discussed how to set up the equation, solve for the total bill amount, and check the solution. We have also covered common mistakes to avoid and real-world applications of labor costs and total bill amounts. By following these guidelines, you can confidently solve problems involving labor costs and total bill amounts.