The following five numbers have a median of 6 and a range of 9 - Junior Cycle Mathematics - Question 1 - 2016

Given the six numbers with a median of 15, mean of 18, and range of 30:

1. Calculate the Median: The median of six numbers (even number of terms) is the average of the third and fourth terms. Thus: $\frac{b + 14}{2} = 15$ Therefore, $b + 14 = 30$ hence, $b = 16$.

2. Calculate the Mean: The mean is given by: $\frac{a + 8 + 14 + 16 + 26 + c}{6} = 18$ Multiplying through gives: $a + 8 + 14 + 16 + 26 + c = 108$ Thus, $a + c + 64 = 108$ Simplifying gives: $a + c = 44$

3. Calculate the Range: The range is calculated as: $c - a = 30$ We can now set up two equations:

   1. $a + c = 44$
   2. $c - a = 30$ By solving these equations:
   • Adding both equations, we get: $2c = 74$ thus, $c = 37$
   • Substituting c into the first equation: $a + 37 = 44$ therefore, $a = 7$.

In conclusion, the values are: $a = 7, \ b = 16, \ c = 37$.