let the number be 'x', Then its reciprocal is $\frac{1}{x}$

∴ We have $x+\frac{1}{x}=\frac{17}{4}$

Multiplying every term by 4x, we get 

$= 4x\times x+4x\times \frac{1}{x}= 4x\times \frac{17}{4}$

= 4x²+4=17x

= 4x²-17x+4=0

= 4x²-16x-x+4=0

= 4x(x-4)-1(x-4)=0

= (4x-1)(x-4)=0

$\therefore x=4 or \frac{1}{4}$

Hence, Option C is correct.