Understanding Ratios and Solving Marble Problems

Often, mathematical problems involve understanding and applying ratios to find solutions. One common type of ratio problem involves marbles or balls. Let's explore a specific problem where the ratio of red, green, and blue marbles in a box is given as 2:6:5. The problem asks us to determine the total number of marbles if there are 20 fewer red marbles than green marbles.

Solution Step-by-Step

To solve this problem, we first need to understand the given ratio and the relationship between the quantities of each type of marble.

Method 1: Adding Ratios and Multiplying by a Constant

The ratio of red:green:blue marbles is 2:6:5. Let's add the ratios to find the total number of parts:

[ 2 6 5 13 ]

If the total number of marbles in the box is 52, we can determine the value of one part by dividing 52 by 13:

[ frac{52}{13} 4 ]

Multiplying each part of the ratio by 4, we get the actual number of each type of marble:

[ 2 times 4 8 quad (red) ] [ 6 times 4 24 quad (green) ] [ 5 times 4 20 quad (blue) ]

Thus, there are 20 green marbles.

Method 2: Setting Up and Solving Equations

Alternatively, we can set up equations to solve the problem. Let the number of red, green, and blue marbles be 2x, 5x, and 6x respectively.

The total number of marbles is given as 52, so:

[ 2x 5x 6x 52 ]

Solving for x:

[ 13x 52 ] [ x frac{52}{13} 4 ]

Then, we can find the number of green marbles:

[ 5x 5 times 4 20 ]

So, there are 20 green marbles.

Verification: Ratio Check

The ratio of marbles is 2:5:6, and the total number of marbles is 52. Checking the ratios:

[ text{Green marbles} frac{5}{13} times 52 20 ]

This confirms that there are 20 green marbles, which is consistent with our previous solutions.

Alternative Solutions

Another way to solve this problem is to use the fact that the given numbers 2, 5, and 6 are in the ratio 2:5:6. Adding these numbers:

[ 2 5 6 13 ]

Since the total number of marbles is 52, each part of the ratio represents 4 marbles (52/13 4). Multiplying the number of green marbles by 4:

[ 5 times 4 20 ]

Thus, there are 20 green marbles.

Conclusion

By using ratios and basic arithmetic, we can determine the number of each type of marble in the box. For this problem, there are 20 green marbles.

For further practice, similar problems can be used to reinforce the concept of ratios and solving equations. Understanding these methods not only helps in solving problems quickly but also builds a strong foundation in mathematics.