Which Inequality Defines The Problem Below?In A Team Bike Race, Each Biker Has To Ride Exactly 50 Miles. If A Team Must Complete 560 Miles, At Least How Many Bikers Are On Each Team?A. 560 ≥ 50 B 560 \geq 50b 560 ≥ 50 B B. 560 B \textgreater 50 560b \ \textgreater \ 50 560 B \textgreater 50

Introduction

In a team bike race, each biker has to ride exactly 50 miles. If a team must complete 560 miles, at least how many bikers are on each team? This problem can be solved using inequalities, which are mathematical statements that compare two expressions. In this article, we will explore the two given inequalities and determine which one defines the problem.

Understanding the Problem

Let's break down the problem. Each biker has to ride exactly 50 miles, and the team must complete 560 miles. To find the minimum number of bikers required, we need to divide the total distance (560 miles) by the distance each biker rides (50 miles).

Inequality A: $560 \geq 50b$

The first inequality is $560 \geq 50b$. This inequality states that the total distance (560 miles) is greater than or equal to 50 times the number of bikers (b). In other words, the total distance is at least 50 times the number of bikers.

To understand this inequality, let's consider an example. Suppose there are 12 bikers on the team. Each biker rides 50 miles, so the total distance covered by the team is 12 x 50 = 600 miles. However, the team must complete only 560 miles. In this case, the inequality $560 \geq 50b$ is not satisfied, as 560 is less than 600.

Inequality B: $560b \ \textgreater \ 50$

The second inequality is $560b \ \textgreater \ 50$. This inequality states that 560 times the number of bikers (b) is greater than 50. In other words, the total distance covered by the team is more than 50 times the number of bikers.

To understand this inequality, let's consider an example. Suppose there are 12 bikers on the team. Each biker rides 50 miles, so the total distance covered by the team is 12 x 50 = 600 miles. In this case, the inequality $560b \ \textgreater \ 50$ is satisfied, as 560 x 12 = 6720 is greater than 50.

Which Inequality Defines the Problem?

Now that we have analyzed both inequalities, let's determine which one defines the problem. The problem states that the team must complete 560 miles, and each biker must ride exactly 50 miles. To find the minimum number of bikers required, we need to divide the total distance (560 miles) by the distance each biker rides (50 miles).

Using the first inequality $560 \geq 50b$, we can divide both sides by 50 to get b ≥ 11.2. This means that the minimum number of bikers required is 12, as we cannot have a fraction of a biker.

Using the second inequality $560b \ \textgreater \ 50$, we can divide both sides by 560 to get b > 0.0893. This means that the minimum number of bikers required is 1, as any positive number of bikers will satisfy this inequality.

However, the problem states that each biker must ride exactly 50 miles. Therefore, we cannot have a fraction of a biker, and the minimum number of bikers required is 12.

Conclusion

In conclusion, the inequality that defines the problem is $560 \geq 50b$. This inequality states that the total distance (560 miles) is greater than or equal to 50 times the number of bikers (b). Using this inequality, we can determine that the minimum number of bikers required is 12.

References

• [1] Khan Academy. (n.d.). Inequalities. Retrieved from https://www.khanacademy.org/math/algebra/x2f1d7d7/x2f1d7d7-inequalities/v/inequalities
• [2] Math Open Reference. (n.d.). Inequalities. Retrieved from https://www.mathopenref.com/inequalities.html

Additional Resources

• [1] Algebra.com. (n.d.). Inequalities. Retrieved from https://www.algebra.com/algebra/homework/inequalities/inequalities.faq
• [2] Purplemath. (n.d.). Inequalities. Retrieved from https://www.purplemath.com/modules/inequal.htm

Frequently Asked Questions

• Q: What is the minimum number of bikers required to complete 560 miles? A: The minimum number of bikers required is 12.
• Q: What is the inequality that defines the problem? A: The inequality that defines the problem is $560 \geq 50b$.
• Q: Can we have a fraction of a biker? A: No, we cannot have a fraction of a biker. Each biker must ride exactly 50 miles.
  Frequently Asked Questions (FAQs) =====================================

Q: What is the minimum number of bikers required to complete 560 miles?

A: The minimum number of bikers required is 12. This is calculated by dividing the total distance (560 miles) by the distance each biker rides (50 miles).

Q: What is the inequality that defines the problem?

A: The inequality that defines the problem is $560 \geq 50b$. This inequality states that the total distance (560 miles) is greater than or equal to 50 times the number of bikers (b).

Q: Can we have a fraction of a biker?

A: No, we cannot have a fraction of a biker. Each biker must ride exactly 50 miles. Therefore, the minimum number of bikers required is a whole number.

Q: How do we calculate the minimum number of bikers required?

A: To calculate the minimum number of bikers required, we divide the total distance (560 miles) by the distance each biker rides (50 miles). This gives us the minimum number of bikers required to complete the distance.

Q: What if the team wants to complete more than 560 miles?

A: If the team wants to complete more than 560 miles, we can use the inequality $560b \ \textgreater \ 50$. This inequality states that 560 times the number of bikers (b) is greater than 50. We can then calculate the minimum number of bikers required to complete the desired distance.

Q: Can we use a different inequality to solve the problem?

A: Yes, we can use a different inequality to solve the problem. However, the inequality $560 \geq 50b$ is the most straightforward and accurate way to solve the problem.

Q: What if the bikers ride different distances?

A: If the bikers ride different distances, we need to use a different approach to solve the problem. We can use the concept of weighted averages to calculate the minimum number of bikers required.

Q: Can we use algebraic expressions to solve the problem?

A: Yes, we can use algebraic expressions to solve the problem. We can use variables to represent the number of bikers and the distance each biker rides, and then use algebraic manipulations to solve for the minimum number of bikers required.

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

A: This problem has many real-world applications, such as:

• Calculating the minimum number of workers required to complete a project
• Determining the minimum number of resources required to complete a task
• Estimating the minimum number of people required to complete a task

Q: Can we use this problem to teach other mathematical concepts?

A: Yes, we can use this problem to teach other mathematical concepts, such as:

• Inequalities
• Algebraic expressions
• Weighted averages
• Real-world applications of mathematics

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

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

• Not considering the inequality $560 \geq 50b$
• Not using the correct algebraic manipulations
• Not considering the real-world applications of the problem
• Not using the correct units (e.g. miles, kilometers, etc.)

Q: Can we use technology to solve this problem?

A: Yes, we can use technology to solve this problem. We can use calculators, computers, or other digital tools to perform the calculations and solve the problem.

Q: What are some additional resources for learning more about this problem?

A: Some additional resources for learning more about this problem include:

• Online tutorials and videos
• Math textbooks and workbooks
• Online forums and discussion groups
• Math education websites and blogs