Solving the Marble Ratio Problem: A Comprehensive Guide

When it comes to solving problems related to marble ratios, understanding the relationships between different colored marbles can be crucial. In this article, we will delve into a specific problem involving blue, green, and red marbles. We will break down the steps to find the number of green marbles in a box where the marbles are in a ratio of 5 to 7 to 8, with a total of 240 marbles.

Ratio and Total Marbles

The question states that the ratio of blue, green, and red marbles is 5:7:8. This means that for every 5 blue marbles, there are 7 green marbles and 8 red marbles. The problem also mentions that the total number of marbles is 240.

Step-by-Step Solution

Calculate the Total Parts in the Ratio: Adding the parts of the ratio: 5 7 8 20 parts

This indicates that the total number of parts (or marbles) in the given ratio is 20.

Determine the Value of One Part: The total number of marbles is 240, so to find the value of one part, we divide the total number of marbles by the total parts in the ratio: 240 / 20 12

This means that each part in the ratio is equal to 12 marbles.

Calculate the Number of Green Marbles: Multiplying the value of one part by the proportion of green marbles: 7 parts * 12 marbles per part 84 marbles

Hence, the number of green marbles in the box is 84.

Alternative Methods

There are several alternative methods to solve this problem, which we will briefly describe:

Using Fractions

7/20 × 240 7 × 12 84

Step-by-Step Calculation

Blue marbles: 12 × 5 60 Green marbles: 12 × 7 84 Red marbles: 12 × 8 96

By distributing the total parts of 20 into 12 equal groups, we can identify that the green marbles correspond to 7 groups out of these 20, resulting in 84 green marbles.

Using a Combined Ratio Approach

If we consider the total parts as 20, where 7 parts are green, and distribute 240 marbles accordingly, we get:

7/20 of 240 84 green marbles

Conclusion

The number of green marbles in the box, based on the given ratio and total number of marbles, is 84. This method can be applied to similar problems involving ratios and proportions. Understanding and mastering these techniques will help in efficiently solving real-world problems.