Question

The rational form of -25.6875 is
$
  (a){\text{ }}\dfrac{{ - 411}}{{16}} \\
  (b){\text{ }}\dfrac{{ - 421}}{{16}} \\
  (c){\text{ }}\dfrac{{ - 431}}{{16}} \\
  (d){\text{ }}\dfrac{{ - 441}}{{16}} \\
 $

Answer 1

Hint: In this question convert -25.6875 into the rational form of $\dfrac{p}{q},q \ne 0$, by removing the decimal point and dividing with 10 raise to the power number of places it is being moved. Then simplify to get the desired rational form.

Complete step-by-step solution -
Given number
-25.6875
Now as we know that rational number is in the form of $\dfrac{p}{q},q \ne 0$ when p and q have not any common factors except 1.
So the given number is written as
$ \Rightarrow - 25.6875 = \dfrac{{ - 256875}}{{10000}}$...................... (1)
Now factorize the numerator and denominator and cancel out the common terms so we have,
 So factors of 256875 are
$ \Rightarrow 256875 = 5 \times 5 \times 5 \times 5 \times 3 \times 137$
And factors of 10000 are
$ \Rightarrow 10000 = 5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2$
So from equation (1) we have,
$ \Rightarrow \dfrac{{ - 256875}}{{10000}} = - \dfrac{{5 \times 5 \times 5 \times 5 \times 3 \times 137}}{{5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2}}$
Now cancel out the common factors we have,
$ \Rightarrow \dfrac{{ - 256875}}{{10000}} = \dfrac{{ - 3 \times 137}}{{2 \times 2 \times 2 \times 2}} = - \dfrac{{411}}{{16}}$
So this is the required rational form of -25.6875.
Hence option (A) is correct.

Note: There is a misconception that negative numbers can’t be expressed in the form of $\dfrac{p}{q}$ that is rational numbers. Any integer can be expressed as a rational number, just the denominator part should not be zero as it makes the term not defined. Even zero is a rational number because 0 can be represented as $\dfrac{0}{1}$, clearly the denominator in this case is non-zero.