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34 votes
34 votes
The base (or radix) of the number system such that the following equation holds is____________.

$\frac{312}{20} = 13.1$

3 Answers

Best answer
51 votes
51 votes

Let ‘x’ be the base or radix of the number system .

The equation is : $\dfrac{3.x^{2}+1.x^{1} +2.x^{0} }{2.x^{1}+0.x^0} =1.x^{1} +3.x^{0}+1.x^{-1}$

$\qquad \implies \dfrac{3.x^{2}+x +2}{2.x}=x+3 +1/x$

$\qquad \implies \dfrac{3.x^{2}+x +2}{2.x}=\dfrac{x^{2}+3x +1}{x}$

$\qquad \implies 3.x^{2}+x +2=2.x^{2}+6x +2$

$\qquad \implies x^{2}+-5x = 0$

$\qquad \implies x(x-5) = 0$

$\qquad \implies x = 0 \text{ or } x = 5$

As base or radix of a number system cannot be zero, here x = 5.

edited by
4 votes
4 votes

Let x (≠ 0) be the base of the given equation.

We have,

LHS = ( 3x2 + x + 2 ) / ( 2x) = ( 3x / 2 ) + ( 1 / 2 ) + ( 1 / x )

RHS = x + 3 + ( 1 / x )

Now, for the equation to hold true, LHS = RHS

( 3x / 2 ) + ( 1 / 2 ) + ( 1 / x ) = x + 3 + ( 1 / x )

⇒ 3x + 1 = 2x + 6

⇒ x = 5

So, the base is 5.

–1 votes
–1 votes
312/20 = 13.1 can be rewritten as:

312 = 13.1 * 20

312 = 131 * 2

now comparing each digit:

31[2] = 13[1] * [2]

3[1]2 = 1[3]1 * [2]   => 11(base x) = 3*2 (base 10)  => 11(base x) = 6 (base 10)

So base x is 5.
Answer:

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