William has the following data: six, eight, 𝑘, eight, eight, nine. If the range is seven, which number could 𝑘 be?
We’re given five options with the values five, six, nine, two, or eight. Let’s start by reminding ourselves that the range of a set of data is the largest value subtract the smallest value. So if we look at our data set, ignoring 𝑘 for the time being, then the range can be found by our largest value, nine, subtract our smallest value, six, which is a range of three. This indicates that our unknown value 𝑘 must be larger or smaller than the existing values in the set.
So let’s take our value 𝑘 to be larger than the values in our set. Since we’re told that the range is seven, then the difference between our largest value 𝑘 and the smallest value of six must be seven. So six plus seven must equal 𝑘. So an answer for 𝑘 could be 13. But this answer isn’t given in our answer options. So let’s see if we can find the other potential value for 𝑘.
Here we would choose a value for 𝑘 that’s smaller than the other values in our data set. Our range would still be seven. So the difference between 𝑘 and nine would be seven. In other words, 𝑘 plus seven equals nine. We can easily solve this to be 𝑘 equals two. And since two is given in option D, then we know that this is our final answer.
If we take a quick look at the other potential answers, we can see that the options of six, nine, and eight wouldn’t change the values in our set, meaning that the range would still be three. If we chose a value of 𝑘 equals five, then the range in this case would be nine take away five. Giving us four, which doesn’t fit with a range of seven and confirms our answer that 𝑘 must be two.