Solving a System of Equations: Determining the Number of Red Balls in a Box

Today, let's explore a mathematical problem that involves solving a system of equations. Imagine you are given a box with various colored balls: red, green, and yellow. The tasks at hand are to determine the number of red balls, while ensuring the solution is comprehensive and easy to follow.

Problem Overview

The problem states that 3/5 of the balls in a box are red. The remaining balls are green and yellow with a specific ratio of 5:3. Additionally, there are 10 fewer yellow balls than green balls. How can we calculate the number of red balls in the box?

Understanding the Problem

Let's denote the total number of balls in the box as N. According to the problem, 3/5 of the balls are red. This means:

R 3/5 * N

Where:

R is the number of red balls. N is the total number of balls in the box.

The remaining ball count represents green and yellow balls:

N - R 2/5 * N

Let G be the number of green balls and Y be the number of yellow balls. We are given the following relationships:

G/Y 5/3 Y G - 10

Solving the Problem

First, we express G in terms of Y using the ratio:

G (5/3)Y

Next, we use the information that there are 10 fewer yellow balls than green balls:

Y G - 10

Substituting the first equation into the second:

Y (5/3)Y - 10

Multiplying both sides by 3 to eliminate the fraction:

3Y 5Y - 30

Rearranging to solve for Y:

30 5Y - 3Y

30 2Y

Y 15

Now we can find G:

G Y 10 15 10 25

Next, we calculate the total number of non-red balls (green yellow):

G Y 25 15 40

Since 2/5N 40, we can solve for N:

N 40 * (5/2) 100

Finally, we determine the number of red balls:

R 3/5 * N 3/5 * 100 60

Conclusion

The number of red balls in the box is 60. This method showcases the power of solving a system of equations and provides a detailed thought process for each step in the problem-solving journey.