Andy, Luke, And Tina Share Some Sweets In The Ratio $1: 6: 14$.Tina Gives $\frac{3}{7}$ Of Her Sweets To Andy.Tina Then Gives $12 \frac{1}{2} \%$ Of The Rest Of Her Sweets To Luke.Tina Says, Now All Three Of Us Have The Same

Introduction

In this article, we will delve into a mathematical puzzle involving the sharing of sweets among three friends, Andy, Luke, and Tina. The problem is presented in a ratio of $1:6:14$, indicating the initial distribution of sweets among the three friends. As the story unfolds, Tina gives away a portion of her sweets to Andy and then to Luke, resulting in an intriguing scenario where all three friends end up with the same number of sweets. We will explore this problem step by step, using mathematical concepts to unravel the mystery.

The Initial Distribution

Let's assume that the total number of sweets is represented by the variable $x$. According to the given ratio, the number of sweets each friend initially receives is as follows:

• Andy: $\frac{1}{1+6+14}x = \frac{1}{21}x$
• Luke: $\frac{6}{1+6+14}x = \frac{6}{21}x$
• Tina: $\frac{14}{1+6+14}x = \frac{14}{21}x$

Tina's First Gift

Tina decides to give $\frac{3}{7}$ of her sweets to Andy. To find the number of sweets Tina gives away, we multiply her initial share by the fraction she gives away:

$\frac{3}{7} \times \frac{14}{21}x = \frac{6}{21}x$

This means that Andy now has $\frac{1}{21}x + \frac{6}{21}x = \frac{7}{21}x$ sweets.

Tina's Second Gift

After giving away $\frac{3}{7}$ of her sweets to Andy, Tina is left with:

$\frac{14}{21}x - \frac{6}{21}x = \frac{8}{21}x$

Tina then decides to give $12 \frac{1}{2}\%$ of the remaining sweets to Luke. To find the number of sweets Tina gives away this time, we first convert the percentage to a decimal:

$12 \frac{1}{2}\% = 12.5\% = 0.125$

Now, we multiply Tina's remaining share by the decimal equivalent of the percentage:

$0.125 \times \frac{8}{21}x = \frac{1}{21}x$

This means that Luke now has $\frac{6}{21}x + \frac{1}{21}x = \frac{7}{21}x$ sweets.

The Final Distribution

After both gifts, the number of sweets each friend has is as follows:

• Andy: $\frac{7}{21}x$
• Luke: $\frac{7}{21}x$
• Tina: $\frac{8}{21}x - \frac{1}{21}x = \frac{7}{21}x$

As we can see, all three friends now have the same number of sweets, which is $\frac{7}{21}x$. This is the solution to the problem.

Conclusion

In this article, we explored a mathematical puzzle involving the sharing of sweets among three friends. We used mathematical concepts to unravel the mystery, step by step. The problem required us to work with ratios, fractions, and percentages, ultimately leading to the solution where all three friends have the same number of sweets. This problem serves as a great example of how mathematical concepts can be applied to real-world scenarios, making math more accessible and interesting.

Mathematical Concepts Used

• Ratios and proportions
• Fractions and decimals
• Percentages and conversions
• Algebraic manipulations

Real-World Applications

This problem can be applied to real-world scenarios where resources are shared among individuals or groups. For example, in a business setting, a company may need to distribute a certain amount of resources among its employees. By using mathematical concepts, the company can ensure that each employee receives a fair share of the resources.

Further Exploration

This problem can be extended to more complex scenarios, such as:

• What if the ratio of sweets is changed?
• What if the percentage of sweets given away is different?
• What if there are more than three friends involved?

Introduction

In our previous article, we explored a mathematical puzzle involving the sharing of sweets among three friends, Andy, Luke, and Tina. The problem required us to work with ratios, fractions, and percentages to unravel the mystery. In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the initial distribution of sweets among the three friends?

A: The initial distribution of sweets is given by the ratio $1:6:14$, which means that Andy receives $\frac{1}{21}x$ sweets, Luke receives $\frac{6}{21}x$ sweets, and Tina receives $\frac{14}{21}x$ sweets, where $x$ is the total number of sweets.

Q: What happens when Tina gives $\frac{3}{7}$ of her sweets to Andy?

A: When Tina gives $\frac{3}{7}$ of her sweets to Andy, she is left with $\frac{14}{21}x - \frac{6}{21}x = \frac{8}{21}x$ sweets. Andy now has $\frac{1}{21}x + \frac{6}{21}x = \frac{7}{21}x$ sweets.

Q: What percentage of her remaining sweets does Tina give to Luke?

A: Tina gives $12 \frac{1}{2}\%$ of her remaining sweets to Luke. This is equivalent to $0.125$ or $12.5\%$.

Q: How many sweets does Luke receive from Tina's second gift?

A: Luke receives $\frac{1}{21}x$ sweets from Tina's second gift.

Q: What is the final distribution of sweets among the three friends?

A: After both gifts, the number of sweets each friend has is as follows:

• Andy: $\frac{7}{21}x$
• Luke: $\frac{7}{21}x$
• Tina: $\frac{7}{21}x$

As we can see, all three friends now have the same number of sweets, which is $\frac{7}{21}x$.

Q: What mathematical concepts are used to solve this problem?

A: The mathematical concepts used to solve this problem include:

• Ratios and proportions
• Fractions and decimals
• Percentages and conversions
• Algebraic manipulations

Q: How can this problem be applied to real-world scenarios?

A: This problem can be applied to real-world scenarios where resources are shared among individuals or groups. For example, in a business setting, a company may need to distribute a certain amount of resources among its employees. By using mathematical concepts, the company can ensure that each employee receives a fair share of the resources.

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

A: Yes, this problem can be extended to more complex scenarios, such as:

• What if the ratio of sweets is changed?
• What if the percentage of sweets given away is different?
• What if there are more than three friends involved?

These extensions can lead to further exploration of mathematical concepts and their applications in real-world scenarios.

Conclusion

In this article, we answered some of the most frequently asked questions related to the sweet sharing conundrum. We hope that this Q&A article has provided a better understanding of the mathematical concepts involved and their applications in real-world scenarios.