Oliver Completed His Project In No More Than Twice The Amount Of Time It Took Karissa To Complete Her Project. Oliver Spent $4 \frac{1}{4}$ Hours On His Project. If $k$ Represents The Amount Of Time That It Took Karissa To

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Introduction

In this article, we will explore the relationship between the time it took Oliver and Karissa to complete their respective projects. We will use mathematical concepts to compare their completion times and determine the amount of time it took Karissa to complete her project.

Problem Statement

Oliver completed his project in no more than twice the amount of time it took Karissa to complete her project. Oliver spent $4 \frac{1}{4}$ hours on his project. If $k$ represents the amount of time that it took Karissa to complete her project, we need to find the value of $k$.

Mathematical Representation

Let's represent the time it took Oliver to complete his project as $O$ and the time it took Karissa to complete her project as $k$. According to the problem statement, Oliver completed his project in no more than twice the amount of time it took Karissa to complete her project. This can be represented mathematically as:

O≤2kO \leq 2k

We are also given that Oliver spent $4 \frac{1}{4}$ hours on his project. We can represent this as:

O=414O = 4 \frac{1}{4}

Converting Mixed Numbers to Improper Fractions

To simplify the calculation, we can convert the mixed number $4 \frac{1}{4}$ to an improper fraction. To do this, we multiply the whole number part by the denominator and add the numerator:

414=(4×4)+14=16+14=1744 \frac{1}{4} = \frac{(4 \times 4) + 1}{4} = \frac{16 + 1}{4} = \frac{17}{4}

So, we can represent the time it took Oliver to complete his project as:

O=174O = \frac{17}{4}

Substituting the Value of O into the Inequality

Now that we have the value of $O$, we can substitute it into the inequality:

174≤2k\frac{17}{4} \leq 2k

Simplifying the Inequality

To simplify the inequality, we can multiply both sides by $\frac{1}{2}$:

174×12≤k\frac{17}{4} \times \frac{1}{2} \leq k

178≤k\frac{17}{8} \leq k

Conclusion

Based on the inequality, we can conclude that the amount of time it took Karissa to complete her project is at least $\frac{17}{8}$ hours.

Calculating the Value of k

To calculate the value of $k$, we can multiply the numerator and denominator by $\frac{1}{2}$:

k=178×12k = \frac{17}{8} \times \frac{1}{2}

k=1716k = \frac{17}{16}

So, the amount of time it took Karissa to complete her project is $\frac{17}{16}$ hours.

Discussion

The problem statement provides a comparison between the time it took Oliver and Karissa to complete their respective projects. By using mathematical concepts, we were able to determine the amount of time it took Karissa to complete her project. This problem demonstrates the importance of mathematical representation and simplification in solving real-world problems.

Real-World Applications

This problem has real-world applications in various fields, such as project management, time estimation, and resource allocation. By understanding the relationship between completion times, project managers can make informed decisions about resource allocation and time estimation.

Conclusion

In conclusion, the problem of comparing the completion times of Oliver and Karissa demonstrates the importance of mathematical representation and simplification in solving real-world problems. By using mathematical concepts, we were able to determine the amount of time it took Karissa to complete her project. This problem has real-world applications in various fields and highlights the importance of mathematical thinking in solving complex problems.

References

  • [1] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  • [2] "Discrete Mathematics and Its Applications" by Kenneth H Rosen

Glossary

  • Improper Fraction: A fraction with a numerator greater than the denominator.
  • Mixed Number: A number consisting of a whole number part and a fractional part.
  • Inequality: A statement that two expressions are not equal.
  • Mathematical Representation: The use of mathematical symbols and equations to represent real-world problems.
    Oliver and Karissa's Project Completion Time Comparison: Q&A ===========================================================

Introduction

In our previous article, we explored the relationship between the time it took Oliver and Karissa to complete their respective projects. We used mathematical concepts to compare their completion times and determine the amount of time it took Karissa to complete her project. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the relationship between the time it took Oliver and Karissa to complete their projects?

A: According to the problem statement, Oliver completed his project in no more than twice the amount of time it took Karissa to complete her project. This can be represented mathematically as:

O≤2kO \leq 2k

Q: What is the value of O?

A: We are given that Oliver spent $4 \frac{1}{4}$ hours on his project. We can represent this as:

O=414O = 4 \frac{1}{4}

We can convert the mixed number $4 \frac{1}{4}$ to an improper fraction:

O=174O = \frac{17}{4}

Q: How did you simplify the inequality?

A: To simplify the inequality, we can multiply both sides by $\frac{1}{2}$:

174×12≤k\frac{17}{4} \times \frac{1}{2} \leq k

178≤k\frac{17}{8} \leq k

Q: What is the value of k?

A: To calculate the value of $k$, we can multiply the numerator and denominator by $\frac{1}{2}$:

k=178×12k = \frac{17}{8} \times \frac{1}{2}

k=1716k = \frac{17}{16}

So, the amount of time it took Karissa to complete her project is $\frac{17}{16}$ hours.

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

A: This problem has real-world applications in various fields, such as project management, time estimation, and resource allocation. By understanding the relationship between completion times, project managers can make informed decisions about resource allocation and time estimation.

Q: How can I apply this concept to my own projects?

A: To apply this concept to your own projects, you can use the following steps:

  1. Estimate the time it will take to complete a project.
  2. Compare the estimated time to the actual time it takes to complete the project.
  3. Use the comparison to adjust your estimates for future projects.

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

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

  • Not converting mixed numbers to improper fractions.
  • Not simplifying the inequality.
  • Not considering the relationship between the time it took Oliver and Karissa to complete their projects.

Conclusion

In conclusion, the problem of comparing the completion times of Oliver and Karissa demonstrates the importance of mathematical representation and simplification in solving real-world problems. By using mathematical concepts, we were able to determine the amount of time it took Karissa to complete her project. This problem has real-world applications in various fields and highlights the importance of mathematical thinking in solving complex problems.

References

  • [1] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  • [2] "Discrete Mathematics and Its Applications" by Kenneth H Rosen

Glossary

  • Improper Fraction: A fraction with a numerator greater than the denominator.
  • Mixed Number: A number consisting of a whole number part and a fractional part.
  • Inequality: A statement that two expressions are not equal.
  • Mathematical Representation: The use of mathematical symbols and equations to represent real-world problems.