If (1012)₃ = (112)_R the value of the radix ‘R’ is

Concept:

To convert any number which is in a different base to a decimal number system we use binary-weighted representation.

Example: Let the number be ( abc⋯ ⋯ yz)r

Now to convert the above number into the decimal number system, we write:

a × rn-1 + b × rn-2 + ⋯ ⋯ + y × r1 + z × r0

Application:

(112)R = (1012)3

If the two numbers are equivalent, they will remain equal in other number systems as well.

Converting both the sides to decimal and equating, we get:

1×R2 + 1×R1 + 2×R0 = 1×33 + 0×32 + 1×31 + 2×3⁰

R2 + R + 2 = 27 + 3 + 2 

R2 + R - 30 = 0

R2 + 6R - 5R - 30 = 0

R(R + 6) - 5(R + 6) = 0

(R + 6)(R - 5) = 0

The valid value of R = 5