Consider the following eight numbers, one of which, labeled x, is unknown: 4, 32, 6, x, 10, 3, 35, 37. Given that the range of these numbers is 64, determine the two possible values of x.

The range of a set of numbers is the difference between the largest and smallest numbers in the set. In this case, we are told that the range of the set of numbers is 64.

To find the two possible values of x, we need to consider two scenarios:

1. x is the largest number in the set. 2. x is the smallest number in the set.

Let's determine x for each scenario.

Scenario 1: x is the largest number in the set. To find x in this case, we take the smallest number in the set and add 64 (the range):

Smallest number: 3 Range: 64 Potential value of x: 3 + 64 = 67

Thus, if x is the largest number, x would be 67.

Scenario 2: x is the smallest number in the set. To find x in this case, we take the largest number in the set and subtract 64:

Largest number: 37 Range: 64 Potential value of x: 37 - 64 = -27

Thus, if x is the smallest number, x would be -27.

Therefore, the two possible values for x, given the range of 64, are 67 and -27.

Extra: Understanding Range: The range is a basic statistical measure that represents the spread of a set of numbers. To find the range, you identify the maximum and minimum numbers in the set and subtract the minimum number from the maximum. Range = Maximum - Minimum.

In the context of the given problem, knowing the range allowed us to explore two extreme scenarios where the unknown number x could either be the minimum or the maximum of the set. By using simple arithmetic and the fact that the range is constant, we could calculate the precise values for x that maintained the given range. This exercise demonstrates a common method of problem-solving in mathematics: making use of known information to determine unknown quantities within a set of constraints.