Solution:

To find out the base of the number system, we need to convert the given numbers into the decimal system.

Given: 312/20 = 13.1

Step 1: Convert the integer part of the quotient into the decimal system.

The integer part of the quotient is 13, which is already in the decimal system.

Step 2: Convert the fractional part of the quotient into the decimal system.

The fractional part of the quotient is 0.1.

0.1 = 1/10

Step 3: Convert the dividend (312) and divisor (20) into the decimal system.

312 = 3 * 100 + 1 * 10 + 2 * 1 = 302

20 = 2 * 10 + 0 * 1 = 20

Step 4: Write the equation in the decimal system.

302/20 = 13.1

Step 5: Find the base of the number system.

Let the base of the number system be 'b'.

We can write 302 in terms of base 'b' as:

302 = 3 * b^2 + 0 * b + 2 * 1 = 3b^2 + 2

We can write 20 in terms of base 'b' as:

20 = 2 * b + 0 * 1 = 2b

Substituting these values in the equation, we get:

(3b^2 + 2)/(2b) = 13.1

Multiplying both sides by 2b, we get:

3b^2 + 2 = 26.2b

3b^2 - 26.2b + 2 = 0

Using the quadratic formula, we get:

b = (26.2 ± sqrt(26.2^2 - 4 * 3 * 2))/(2 * 3)

b = (26.2 ± sqrt(664.84 - 24))/(6)

b = (26.2 ± sqrt(640.84))/(6)

b = (26.2 ± 25.3)/(6)

b = 5.25 or b = 0.17

Since the base of the number system cannot be less than 2, the only valid solution is b = 5.

Therefore, the base of the number system such that the equation 312/20 = 13.1 holds is 5.