The ratio of blue tiles to green tiles is given as $(2x+1):(4x+9)$. It is also given that $\frac{1}{4}$ of the tiles are blue. We are asked to find the value of $x$.

1. Problem Description

The ratio of blue tiles to green tiles is given as $(2x+1):(4x+9)$. It is also given that $\frac{1}{4}$ of the tiles are blue. We are asked to find the value of $x$.

2. Solution Steps

Let the number of blue tiles be $2x+1$ and the number of green tiles be $4x+9$. The total number of tiles is the sum of the number of blue and green tiles, which is $(2x+1) + (4x+9) = 6x+10$.

Since $\frac{1}{4}$ of the tiles are blue, we have:

$\frac{\text{Number of blue tiles}}{\text{Total number of tiles}} = \frac{1}{4}$

$\frac{2x+1}{6x+10} = \frac{1}{4}$

Cross-multiplying, we get:

$4(2x+1) = 1(6x+10)$

$8x+4 = 6x+10$

Subtract $6x$ from both sides:

$8x - 6x + 4 = 6x - 6x + 10$

$2x + 4 = 10$

Subtract 4 from both sides:

$2x + 4 - 4 = 10 - 4$

$2x = 6$

Divide both sides by 2:

$\frac{2x}{2} = \frac{6}{2}$

$x=3$

3. Final Answer

$x = 3$