Unveiling the Mystery: Understanding Fractions, Percentages and Ratios with Marbles

When dealing with basic arithmetic problems, fractions, percentages, and ratios play a crucial role in resolving the queries effectively. Let's explore a common problem involving marbles and understand how to calculate fractions, percentages, and ratios step by step.

Problem Statement

Imagine a bag of 35 marbles, out of which 28 are blue and the rest are red. We are tasked with finding two things:

What fraction of the marbles are blue? What percentage of the marbles are blue?

Solution to the Problem

Finding the Fraction of Blue Marbles

To determine the fraction of blue marbles:

Identify the total number of marbles. In this case, there are 35 marbles in total. Identify the number of blue marbles. Here, there are 28 blue marbles. Calculate the fraction by dividing the number of blue marbles by the total number of marbles:

28 ÷ 35

Upon dividing, we find that the fraction is 28/35. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which in this case is 7:

28 ÷ 7 4

35 ÷ 7 5

So, the simplified fraction of blue marbles is:

4/5

Finding the Percentage of Blue Marbles

To convert the fraction to a percentage, follow these steps:

Take the simplified fraction, 4/5. Multiply the fraction by 100 to convert it to a percentage:

(4/5) × 100

The multiplication yields:

(4 × 100) ÷ 5 400 ÷ 5 80%

Hence, 80% of the marbles are blue.

Alternative Methods

Let's explore another way to solve this problem using a step-by-step approach:

Divide the total number of marbles by 7 to find the ratio:

35 ÷ 7 5

Now, multiply the ratio by the number of blue marbles to find the number of blue marbles:

4 × 5 20

Alternatively, the number of red marbles can be found using the remainder:

1 × 5 5

So, the total number of blue marbles is 20, and the number of red marbles is 5. This confirms our earlier calculation that 80% of the marbles are blue.

Other Considerations

It's worth noting that another way to approach the problem, using mental calculations, is:

Step Description 1 Calculate the number of blue marbles in a subset of 45 marbles: 2 Result: 18 blue marbles 3 Divide the number of blue marbles by the total number of marbles to find the percentage. 4 Result: 40% of the lot are blue marbles

This method, while simpler, may not always provide the exact fraction or percentage but can serve as a quick estimation.

Conclusion

Through these methods, we can effectively determine the fraction, percentage, and ratio of blue marbles in a set of 35 marbles, with 28 of them being blue. This problem showcases the importance of understanding and applying basic arithmetic concepts in real-world scenarios.

Understanding how to calculate fractions, percentages, and ratios is not only fundamental for academic purposes but also highly valuable in everyday life and various professional contexts. Whether you're a student, a professional, or simply someone looking to sharpen your math skills, mastery of these concepts is a must.