It is a rational number because it is a repeating decimal. (a) 124/165

Question

It is a rational number because it is a repeating decimal.

(a) \(\frac{124}{165}\) 

(b) \(\frac{131}{30}\) 

(c) \(\frac{2027}{625}\)

(d) \(\frac{1625}{462}\)

Answer 1

(c) \(\frac{2027}{625}\) 

\(\frac{124}{165}\) = \(\frac{124}{5\times33'}\). we know 5 and 33 are not the factors of 124. It is in its simplest form and it cannot be expressed as the product of (2^m × 5ⁿ) for some non-negative integers m, n. 

So, it cannot be expressed as a terminating decimal.

\(\frac{131}{30}\) = \(\frac{131}{5\times6'}\). we know 5 and 6 are not the factors of 131. It is in its simplest form and it cannot be expressed as the product of (2^m × 5ⁿ) for some non-negative integers m, n. 

So, it cannot be expressed as a terminating decimal.

\(\frac{2027}{625}\) = \(\frac{2027\times2^4}{5^4\times2^4}\) = \(\frac{32432}{10000}\) = 3.2432; as it is of the form (2^m × 5ⁿ), where m, n are non-negative integers. 

So, it is a terminating decimal

\(\frac{1625}{462}\) = \(\frac{1625}{2\times7\times33'}\). we know 2, 7 and 33 are not the factors of 1625. It is in its simplest form and it cannot be expressed as the product of (2^m × 5ⁿ) for some non-negative integers m, n. 

So, it cannot be expressed as a terminating decimal.