Find, in terms of 𝑥, the arithmetic mean of the two numbers eight 𝑥 plus eight and six 𝑥 plus 22.
We recall that in order to calculate the arithmetic mean of a data set, we divide the sum of the values by the number of values. When dealing with two values 𝑥 and 𝑦, their mean is equal to 𝑥 plus 𝑦 divided by two. In this case, our two numbers are eight 𝑥 plus eight and six 𝑥 plus 22. We need to find the sum of these two values and then divide by two. We can simplify the numerator by grouping or collecting like terms. Eight 𝑥 plus six 𝑥 is equal to 14𝑥. Eight plus 22 is equal to 30. Therefore, the numerator simplifies to 14𝑥 plus 30.
In order to divide this by two, we can halve each term on the numerator individually. 14𝑥 divided by two is equal to seven 𝑥. 30 divided by two is 15. The arithmetic mean of two numbers eight 𝑥 plus eight and six 𝑥 plus 22 is seven 𝑥 plus 15. We might notice here that seven 𝑥 is halfway between six 𝑥 and eight 𝑥. 15 is halfway between eight and 22. It is seven away from both numbers. This means that we could’ve worked out the arithmetic mean by inspection.