Solving the Pen Ratio Problem: A Comprehensive Guide

In this article, we'll solve the problem of finding the ratio of blue to black pens in a box containing 100 pens in total, of which 48 are red and 20 are blue. Through a step-by-step approach, we'll break down the process of finding the number of black pens and the ratio of blue to black pens.

Introduction to the Problem

We have a box containing 100 pens. The distribution of these pens is as follows:

48 red pens 20 blue pens The remaining pens are black

The problem at hand is to find the ratio of the number of blue pens to the number of black pens.

Step-by-Step Solution

1. Determine the Number of Black Pens

First, we need to find out how many black pens are in the box. We start with the total number of pens and subtract the number of red and blue pens:

Total pens 100, Red pens 48, Blue pens 20.

Number of black pens Total pens - Red pens - Blue pens

Number of black pens 100 - 48 - 20 32

So, there are 32 black pens in the box.

2. Find the Ratio of Blue to Black Pens

Now that we know the number of blue and black pens, we can calculate the ratio of blue to black pens. Initially, we have:

Blue pens : Black pens 20 : 32

To simplify this ratio, we divide both the numerator and the denominator by their greatest common divisor, which is 4:

20 div 4 : 32 div 4 5 : 8

Thus, the simplified ratio of blue to black pens is 5:8.

Conclusion

In summary, the ratio of blue to black pens in the box is 5:8. This solution is arrived at by first determining the number of black pens (32) and then simplifying the ratio (20:32) to (5:8).

This type of problem is common in mathematics and can help in understanding the distribution of items in a set. It is particularly useful in areas such as data analysis, inventory management, and statistical modeling.

Key Takeaways

The total number of pens is 100. There are 48 red pens, 20 blue pens, and 32 black pens. The ratio of blue to black pens simplifies to 5:8.

Further Reading

For more information on ratios and other mathematical problem-solving techniques, please refer to standard mathematics textbooks or online resources such as Khan Academy, Math is Fun, and many others.