The New Proportion of Yellow Pins

Oct 13, 2025 by miodeya.com 113 views

Introduction

In this article, we will delve into a fascinating mathematical problem involving Yaqub and his bag of green and yellow pins. The problem requires us to apply our understanding of fractions and proportions to determine the number of pins in Yaqub's bag. We will break down the problem step by step, using mathematical concepts to arrive at a solution.

The Initial Situation

Yaqub has a bag containing only green and yellow pins. Initially, $$\frac{3}{7}$$ of the pins are green. This means that the remaining $$\frac{4}{7}$$ of the pins are yellow. Let's assume that the total number of pins in the bag is 7x, where x is a positive integer. This assumption will allow us to work with fractions and proportions.

Yaqub Picks a Green Pin

Yaqub picks out a green pin from his bag and gives it to his sister. This action reduces the number of green pins in the bag by 1. The new proportion of green pins in the bag is $$\frac{3}{7} - \frac{1}{7} = \frac{2}{7}$$. This means that $$\frac{2}{7}$$ of the remaining pins in the bag are green.

The New Proportion of Yellow Pins

After Yaqub picks a green pin, the new proportion of yellow pins in the bag is $$\frac{4}{7} - \frac{1}{7} = \frac{3}{7}$$. This means that $$\frac{3}{7}$$ of the remaining pins in the bag are yellow.

The Remaining Pins in the Bag

Let's assume that the total number of pins in the bag is 7x. After Yaqub picks a green pin, the number of remaining pins in the bag is 7x - 1. The number of green pins remaining in the bag is $$\frac{2}{7}$$ of 7x - 1, which is $$\frac{2}{7} \times (7x - 1)$$.

Solving for x

We are given that $$\frac{2}{5}$$ of the remaining pins in the bag are green. This means that $$\frac{2}{5}$$ of 7x - 1 is equal to $$\frac{2}{7} \times (7x - 1)$$. We can set up an equation to solve for x:

\frac{2}{5} \times (7x - 1) = \frac{2}{7} \times (7x - 1)

Simplifying the equation, we get:

\frac{14x - 2}{5} = \frac{14x - 2}{7}

Cross-multiplying, we get:

7(14x - 2) = 5(14x - 2)

Expanding and simplifying, we get:

98x - 14 = 70x - 10

Solving for x, we get:

98x - 70x = -10 + 14

28x = 4

x = \frac{4}{28}

x = \frac{1}{7}

Conclusion

In this article, we have solved a mathematical problem involving Yaqub and his bag of green and yellow pins. We have applied our understanding of fractions and proportions to determine the number of pins in Yaqub's bag. The solution to the problem is x = 1/7, which means that the total number of pins in the bag is 7(1/7) = 1.

The Final Answer

The final answer to the problem is x = 1/7.

Additional Insights

This problem requires us to apply our understanding of fractions and proportions to arrive at a solution. The problem also requires us to use algebraic techniques, such as cross-multiplication and simplification, to solve for x.

Real-World Applications

This problem has real-world applications in fields such as statistics and data analysis. For example, in a survey, we may want to determine the proportion of respondents who have a certain characteristic. We can use the same techniques used in this problem to arrive at a solution.

Future Directions

This problem can be extended to more complex scenarios, such as multiple pins being picked from the bag or the introduction of new pins with different colors. We can also explore other mathematical concepts, such as probability and statistics, to arrive at a solution.

References

This problem is a classic example of a mathematical puzzle. There are many resources available online that provide solutions to this problem. Some of the references used in this article include:

[1] "Mathematical Puzzles" by Martin Gardner
[2] "The Art of Problem Solving" by Richard Rusczyk
[3] "Mathematics for the Nonmathematician" by Morris Kline

Appendix

This appendix provides additional information and resources related to the problem.

[1] "Fractional Proportions" by Math Open Reference
[2] "Algebraic Techniques" by Khan Academy
[3] "Statistics and Data Analysis" by Coursera
Yaqub's Pin Puzzle: A Mathematical Conundrum - Q&A =====================================================

Introduction

In our previous article, we delved into a fascinating mathematical problem involving Yaqub and his bag of green and yellow pins. We applied our understanding of fractions and proportions to determine the number of pins in Yaqub's bag. In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the initial proportion of green pins in the bag?

A: The initial proportion of green pins in the bag is $$\frac{3}{7}$$.

Q: What happens when Yaqub picks a green pin from the bag?

A: When Yaqub picks a green pin from the bag, the new proportion of green pins in the bag is $$\frac{2}{7}$$.

Q: How do we determine the number of remaining pins in the bag?

A: We determine the number of remaining pins in the bag by subtracting 1 from the total number of pins, which is 7x.

Q: What is the new proportion of yellow pins in the bag?

A: The new proportion of yellow pins in the bag is $$\frac{3}{7}$$.

Q: How do we solve for x?

A: We solve for x by setting up an equation using the proportion of green pins in the bag. The equation is:

\frac{2}{5} \times (7x - 1) = \frac{2}{7} \times (7x - 1)

Q: What is the final answer to the problem?

A: The final answer to the problem is x = 1/7.

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

A: This problem has real-world applications in fields such as statistics and data analysis. For example, in a survey, we may want to determine the proportion of respondents who have a certain characteristic.

Q: Can this problem be extended to more complex scenarios?

A: Yes, this problem can be extended to more complex scenarios, such as multiple pins being picked from the bag or the introduction of new pins with different colors.

Q: What are some additional resources related to this problem?

A: Some additional resources related to this problem include:

[1] "Mathematical Puzzles" by Martin Gardner
[2] "The Art of Problem Solving" by Richard Rusczyk
[3] "Mathematics for the Nonmathematician" by Morris Kline

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

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

Not setting up the equation correctly
Not simplifying the equation correctly
Not solving for x correctly