Substituting the Known Values

Oct 13, 2025 by recoveryshake.com 8 views

Introduction

Horace is a skilled hairstylist who can give haircuts to both children and adults within a limited time frame of 7 hours. The problem is to determine the maximum number of adult haircuts Horace can give, given that he has already completed 5 child haircuts. To solve this, we need to use the given inequality and substitute the known values to find the solution.

The Inequality

The inequality given is:

0.75C + 1.25A ≤ 7

Where:

C represents the number of child haircuts
A represents the number of adult haircuts
0.75 is the rate at which Horace can give child haircuts (75% of the total time)
1.25 is the rate at which Horace can give adult haircuts (125% of the total time)
7 is the total time available (7 hours)

Substituting the Known Values

We are given that Horace has already completed 5 child haircuts. We can substitute this value into the inequality:

0.75(5) + 1.25A ≤ 7

Simplifying the Inequality

Now, let's simplify the inequality by multiplying 0.75 by 5:

3.75 + 1.25A ≤ 7

Isolating the Variable

Next, let's isolate the variable A by subtracting 3.75 from both sides of the inequality:

1.25A ≤ 7 - 3.75

Simplifying Further

Now, let's simplify the right-hand side of the inequality:

1.25A ≤ 3.25

Dividing by the Coefficient

Finally, let's divide both sides of the inequality by 1.25 to solve for A:

A ≤ 3.25 / 1.25

Solving for A

Now, let's calculate the value of A:

A ≤ 2.6

Conclusion

Therefore, Horace can give at most 2 adult haircuts within the 7-hour time frame, given that he has already completed 5 child haircuts.

Discussion

This problem is a classic example of a linear inequality, which is a fundamental concept in mathematics. The solution involves substituting the known values into the inequality, simplifying the expression, and isolating the variable. This type of problem is commonly encountered in real-world scenarios, such as scheduling, resource allocation, and optimization.

Real-World Applications

This problem has several real-world applications, including:

Scheduling: Horace needs to schedule his haircuts within the 7-hour time frame, taking into account the time required for each type of haircut.
Resource allocation: Horace needs to allocate his time and resources efficiently to maximize the number of haircuts he can give.
Optimization: Horace needs to optimize his schedule to minimize the time spent on each haircut and maximize the number of haircuts he can give.

Mathematical Concepts

This problem involves several mathematical concepts, including:

Linear inequalities: The problem involves solving a linear inequality to find the maximum number of adult haircuts Horace can give.
Substitution: The problem involves substituting the known values into the inequality to solve for the unknown variable.
Simplification: The problem involves simplifying the expression to isolate the variable.
Division: The problem involves dividing both sides of the inequality by the coefficient to solve for the variable.

Conclusion

Q: What is the main goal of Horace in this problem?

A: The main goal of Horace is to determine the maximum number of adult haircuts he can give within the 7-hour time frame, given that he has already completed 5 child haircuts.

Q: What is the given inequality in this problem?

A: The given inequality is:

0.75C + 1.25A ≤ 7

Where:

C represents the number of child haircuts
A represents the number of adult haircuts
0.75 is the rate at which Horace can give child haircuts (75% of the total time)
1.25 is the rate at which Horace can give adult haircuts (125% of the total time)
7 is the total time available (7 hours)

Q: How many child haircuts has Horace already completed?

A: Horace has already completed 5 child haircuts.

Q: What is the solution to the inequality?

A: The solution to the inequality is:

A ≤ 2.6

This means that Horace can give at most 2 adult haircuts within the 7-hour time frame.

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

A: Some real-world applications of this problem include:

Scheduling: Horace needs to schedule his haircuts within the 7-hour time frame, taking into account the time required for each type of haircut.
Resource allocation: Horace needs to allocate his time and resources efficiently to maximize the number of haircuts he can give.
Optimization: Horace needs to optimize his schedule to minimize the time spent on each haircut and maximize the number of haircuts he can give.

Q: What mathematical concepts are involved in this problem?

A: The mathematical concepts involved in this problem include:

Linear inequalities: The problem involves solving a linear inequality to find the maximum number of adult haircuts Horace can give.
Substitution: The problem involves substituting the known values into the inequality to solve for the unknown variable.
Simplification: The problem involves simplifying the expression to isolate the variable.
Division: The problem involves dividing both sides of the inequality by the coefficient to solve for the variable.

Q: Can Horace give more than 2 adult haircuts within the 7-hour time frame?

A: No, Horace cannot give more than 2 adult haircuts within the 7-hour time frame, given that he has already completed 5 child haircuts.

Q: What if Horace wants to give more adult haircuts? What would he need to do?

A: If Horace wants to give more adult haircuts, he would need to reduce the number of child haircuts he gives or increase the time available for haircuts.

Q: Can this problem be solved using other mathematical concepts?

A: Yes, this problem can be solved using other mathematical concepts, such as graphing or matrix operations. However, the solution using linear inequalities is the most straightforward and efficient method.

Q: Is this problem relevant to real-world scenarios?

A: Yes, this problem is relevant to real-world scenarios, such as scheduling, resource allocation, and optimization. It requires the application of mathematical concepts to solve a practical problem.