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As there are two numbers, the range is the difference between them

As there are 2 numbers the mean is exactly in the middle of them.

Thus half the range is less than the mean and half the range is greater than the mean.

Thus the two numbers are 10 ± (4/2) = 10 ± 2 = 8 and 12.

Algebraically, this can be seen:

Let the two numbers be x and y with y > x

Then:

  1. their range is y - x = 4; and
  2. their mean is (x + y) / 2 = 10
  3. {2} can be rearrange to give x + y = 10 × 2 → y + x = 20
  4. adding {3} to {1} yields: (y + y) + (-x + x) = 20 + 4 → 2y = 24 → y = 12
  5. subtracting {1} from {3} yields (y - y) + (x - -x) = 20 - 4 → 2x = 16 → x = 8
→ the two numbers are 8 and 12.
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Q: Which 2 numbers have a mean of 10 and a range of 4?
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