Jordan's bathroom floor has been decorated with purple and white tiles. The ratio of purple tiles to white tiles is \( (2 x+1):(4 x+13) \) If \( \frac{1}{4} \) of the tiles are purple, work out the value of \( x \).

To find the value of \( x \), we start with the given ratio of purple tiles to white tiles, which is \( (2x + 1):(4x + 13) \). We can express this in terms of a variable \( k \): - The number of purple tiles = \( 2x + 1 \) = \( 2k \) - The number of white tiles = \( 4x + 13 \) = \( 4k \) According to the problem, \( \frac{1}{4} \) of the total tiles are purple. Therefore, we can write: \[ \frac{2k}{2k + 4k} = \frac{1}{4} \] This simplifies to: \[ \frac{2k}{6k} = \frac{1}{4} \] Cross-multiplying gives: \[ 8k = 6k \] This simplifies to \( 2k = 0 \), which can be ignored because \( k \) can't be zero. Instead, we compute the sums: The total number of tiles is: \[ (2x + 1) + (4x + 13) = 6x + 14 \] Thus, since \( \frac{1}{4} \) of the total is purple: \[ 2x + 1 = \frac{1}{4}(6x + 14) \] Multiplying through by 4 to eliminate the fraction: \[ 4(2x + 1) = 6x + 14 \] This expands to: \[ 8x + 4 = 6x + 14 \] Rearranging leads to: \[ 8x - 6x = 14 - 4 \] \[ 2x = 10 \] Dividing by 2 yields: \[ x = 5 \] So, the value of \( x \) is \( \boxed{5} \).