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Solution: We find the greatest common factor of 120 and 180:
Question: A web developer is testing response times of two servers: one responds in 0.3 seconds and the other in 0.9 seconds. What is the arithmetic mean of the response times, in seconds?
Solution: The arithmetic mean is calculated as:
Now factor out the common binomial $ (x - 3) $:
Question: An archaeologist is analyzing a sequence of ancient settlement populations modeled by five real numbers in arithmetic progression. If the sum of the numbers is 50 and the product of the two smallest numbers is 48, find the common difference.
Let the five numbers in arithmetic progression be:
We are also told the product of the two smallest numbers is 48:
Since the numbers must be real and in increasing order, and $ a = 10 $, test both:
If $ d = 13 $, the sequence is: $ -16, -11, 10, 23, 36 $, sum is 50, product of first two is $ (-16)(-11) = 176
So the correct common difference is:
