6) In a bag, there are only red counters, blue counters, green counters and yellow councers. The total number of counters in the bag is 80 In the bag the number of red counters is \( x+7 \) the number of blue coumters is \( x-11 \) the number of green counters is \( 3 x \) Jude takes at random a counter from the bag. The probability that he takes a red counter is \( \frac{1}{4} \) Work out the probability that Jude takes a yellow counter.

To find out the probability that Jude takes a yellow counter, we first need to determine the values of \( x \) and then the number of yellow counters in the bag. Given that the total number of counters is 80, we can set up the equation: \[ (x + 7) + (x - 11) + (3x) + y = 80 \] Here, \( y \) represents the number of yellow counters. This simplifies to: \[ 5x - 4 + y = 80 \Rightarrow 5x + y = 84 \] Next, we know the probability of selecting a red counter is \( \frac{1}{4} \), which can be represented as: \[ \frac{x + 7}{80} = \frac{1}{4} \] Cross-multiplying gives us: \[ 4(x + 7) = 80 \Rightarrow 4x + 28 = 80 \Rightarrow 4x = 52 \Rightarrow x = 13 \] Now substituting \( x \) back into the equations for the number of counters: - Red counters: \( x + 7 = 13 + 7 = 20 \) - Blue counters: \( x - 11 = 13 - 11 = 2 \) - Green counters: \( 3x = 3 \times 13 = 39 \) Now we sum the counters we have: \[ 20 + 2 + 39 = 61 \] To find the number of yellow counters: \[ y = 80 - 61 = 19 \] Finally, the probability that Jude takes a yellow counter is: \[ \frac{y}{80} = \frac{19}{80} \] So, the final answer is: \[ \frac{19}{80} \]