Select The Correct Answer.Susan Is Planting Marigolds And Impatiens In Her Garden. Each Marigold Costs $\$9$, And Each Impatien Costs $\$7$. Susan Wants The Number Of Marigolds To Be More Than Twice The Number Of
Introduction
Mathematical problem solving is an essential skill that involves applying mathematical concepts and techniques to solve real-world problems. In this article, we will focus on a specific problem involving the selection of marigolds and impatiens for Susan's garden. The problem requires mathematical reasoning and calculation to determine the correct answer.
The Problem
Susan is planting marigolds and impatiens in her garden. Each marigold costs $9, and each impatien costs $7. Susan wants the number of marigolds to be more than twice the number of impatiens. If she has a total budget of $300, how many marigolds and impatiens can she buy?
Mathematical Representation
Let's represent the number of marigolds as M and the number of impatiens as I. We know that each marigold costs $9 and each impatien costs $7. Therefore, the total cost of marigolds and impatiens can be represented as:
9M + 7I ≤ 300
We also know that Susan wants the number of marigolds to be more than twice the number of impatiens. This can be represented as:
M > 2I
Solving the Inequality
To solve the inequality, we can start by isolating the variable M. We can do this by subtracting 7I from both sides of the inequality:
9M ≤ 300 - 7I
Next, we can divide both sides of the inequality by 9:
M ≤ (300 - 7I) / 9
Finding the Minimum Value of I
To find the minimum value of I, we can set M to its minimum value, which is 2I + 1 (since M > 2I). Substituting this value into the inequality, we get:
2I + 1 ≤ (300 - 7I) / 9
Multiplying both sides of the inequality by 9, we get:
18I + 9 ≤ 300 - 7I
Adding 7I to both sides of the inequality, we get:
25I + 9 ≤ 300
Subtracting 9 from both sides of the inequality, we get:
25I ≤ 291
Dividing both sides of the inequality by 25, we get:
I ≤ 11.64
Since I must be an integer, the minimum value of I is 11.
Finding the Maximum Value of M
To find the maximum value of M, we can substitute the minimum value of I into the inequality:
M ≤ (300 - 7(11)) / 9
Simplifying the expression, we get:
M ≤ (300 - 77) / 9
M ≤ 223 / 9
M ≤ 24.78
Since M must be an integer, the maximum value of M is 24.
Conclusion
In conclusion, Susan can buy a maximum of 24 marigolds and 11 impatiens with a total budget of $300. This solution satisfies the condition that the number of marigolds is more than twice the number of impatiens.
Final Answer
The final answer is: 24 marigolds and 11 impatiens.
Discussion
This problem requires mathematical reasoning and calculation to determine the correct answer. The solution involves solving an inequality and finding the minimum and maximum values of the variables. The problem also requires an understanding of the relationship between the number of marigolds and impatiens.
Real-World Applications
This problem has real-world applications in various fields, such as:
- Gardening: The problem can be applied to determine the optimal number of marigolds and impatiens to plant in a garden, given a limited budget.
- Business: The problem can be applied to determine the optimal number of products to produce, given a limited budget and a set of constraints.
- Science: The problem can be applied to determine the optimal number of experiments to conduct, given a limited budget and a set of constraints.
Future Research Directions
Future research directions in this area include:
- Developing more efficient algorithms for solving inequalities and finding the minimum and maximum values of variables.
- Applying mathematical problem solving to real-world problems in various fields, such as gardening, business, and science.
- Developing new mathematical models to represent real-world problems and solve them using mathematical techniques.
Introduction
In our previous article, we solved a mathematical problem involving the selection of marigolds and impatiens for Susan's garden. In this article, we will provide a Q&A section to help readers understand the problem and its solution better.
Q: What is the problem about?
A: The problem is about selecting the number of marigolds and impatiens that Susan can buy with a total budget of $300, given that each marigold costs $9 and each impatien costs $7.
Q: What is the condition that Susan wants to satisfy?
A: Susan wants the number of marigolds to be more than twice the number of impatiens.
Q: How do we represent the problem mathematically?
A: We represent the number of marigolds as M and the number of impatiens as I. The total cost of marigolds and impatiens can be represented as:
9M + 7I ≤ 300
We also know that Susan wants the number of marigolds to be more than twice the number of impatiens, which can be represented as:
M > 2I
Q: How do we solve the inequality?
A: We start by isolating the variable M. We can do this by subtracting 7I from both sides of the inequality:
9M ≤ 300 - 7I
Next, we can divide both sides of the inequality by 9:
M ≤ (300 - 7I) / 9
Q: How do we find the minimum value of I?
A: To find the minimum value of I, we can set M to its minimum value, which is 2I + 1 (since M > 2I). Substituting this value into the inequality, we get:
2I + 1 ≤ (300 - 7I) / 9
Multiplying both sides of the inequality by 9, we get:
18I + 9 ≤ 300 - 7I
Adding 7I to both sides of the inequality, we get:
25I + 9 ≤ 300
Subtracting 9 from both sides of the inequality, we get:
25I ≤ 291
Dividing both sides of the inequality by 25, we get:
I ≤ 11.64
Since I must be an integer, the minimum value of I is 11.
Q: How do we find the maximum value of M?
A: To find the maximum value of M, we can substitute the minimum value of I into the inequality:
M ≤ (300 - 7(11)) / 9
Simplifying the expression, we get:
M ≤ (300 - 77) / 9
M ≤ 223 / 9
M ≤ 24.78
Since M must be an integer, the maximum value of M is 24.
Q: What is the final answer?
A: The final answer is: 24 marigolds and 11 impatiens.
Q: What are the real-world applications of this problem?
A: This problem has real-world applications in various fields, such as:
- Gardening: The problem can be applied to determine the optimal number of marigolds and impatiens to plant in a garden, given a limited budget.
- Business: The problem can be applied to determine the optimal number of products to produce, given a limited budget and a set of constraints.
- Science: The problem can be applied to determine the optimal number of experiments to conduct, given a limited budget and a set of constraints.
Q: What are the future research directions in this area?
A: Future research directions in this area include:
- Developing more efficient algorithms for solving inequalities and finding the minimum and maximum values of variables.
- Applying mathematical problem solving to real-world problems in various fields, such as gardening, business, and science.
- Developing new mathematical models to represent real-world problems and solve them using mathematical techniques.
Conclusion
In conclusion, this Q&A section provides a detailed explanation of the mathematical problem and its solution. We hope that this article has helped readers understand the problem and its solution better.