Introduction to Mathematical Problem Solving
Mathematical problem solving is a fundamental skill that serves as the foundation for many disciplines, including business, engineering, and science. This article explores a classic problem-solving scenario: a pen box dilemma. We will break down the problem, set up equations, and solve for the number of blue and red pens, ultimately finding the difference between the two.

The Problem Statement

Suppose 4/9 of the number of blue pens in a box is equal to 1/5 of the number of red pens. Given that there are 609 blue and red pens in total, we need to determine how many more red pens than blue pens are in the box.

Defining Variables and Setting Up Equations

Let’s denote the number of red pens as r and the number of blue pens as b. The given conditions can be mathematically expressed as follows:

(frac{4b}{9} frac{r}{5})

From this equation, we can solve for r in terms of b:
(frac{4b}{9} frac{r}{5} rightarrow r frac{20b}{9})

Combining the Equations

Since we also know that the total number of pens is 609, we can write:
(r b 609)

Substituting the expression for r from the first equation into this second equation, we get:
(frac{20b}{9} b 609)

Solving for b

To solve for b, we first combine the terms on the left side. We need a common denominator for the fractions, which is 9:
(frac{20b}{9} frac{9b}{9} 609)

Simplifying the left side:
(frac{29b}{9} 609)

Multiplying both sides by 9 to clear the denominator:
(29b 5481)

Solving for b:
(b frac{5481}{29} 189)

Calculating the Number of Red Pens

Now that we have the number of blue pens, we can find the number of red pens using the first equation:
(r frac{20 times 189}{9} 420)

Final Result and Solution

We now have the number of blue pens (189) and the number of red pens (420). To find how many more red pens than blue pens there are, we can simply subtract the number of blue pens from the number of red pens:
(420 - 189 231)

Conclusion

In conclusion, there are 231 more red pens than blue pens in the box. This problem demonstrates the power of algebraic problem solving and the practical application of equations in real-world scenarios. By understanding mathematical relationships and solving for unknowns, we can derive meaningful insights from seemingly complex problems.