The base (or radix) of the number system such that the following equation…

Answer: 5

Work:

• Interpret the numbers in base b:

• 312 in base b equals 3b^2 + 1b + 2.

• 20 in base b equals 2b.

• 13.1 in base b equals 1b + 3 + 1/b.

• Set up the equation: (3b^2 + b + 2) / (2b) = b + 3 + 1/b.

• Multiply both sides by 2b to clear the denominator: 3b^2 + b + 2 = 2b^2 + 6b + 2.

• Bring all terms to one side: b^2 - 5b = 0, so b(b - 5) = 0.

• Possible solutions are b = 0 or b = 5. Base 0 is not valid, and the base must be an integer greater than the largest digit appearing (largest digit here is 3).

• Therefore the base is 5.