Question

A rational number can be expressed as a terminating decimal if the denominator has factors:
A. 2 or 5
B. 2, 3 or 5
C. 3 or 5
D. only 2 and 3

Answer 1

Hint: This number belongs to a set of numbers that mathematicians call rational numbers i.e., Numbers that can be written as the ratio of two integers, where the denominator is not zero. Rational numbers are numbers that can be written as a ratio of two integers and a terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point.

Complete step by step solution:
Any rational number with its denominator is in the form of \[{2^m} \times {5^n}\], where m, n are positive integers and are terminating decimals. The examples are shown below:
\[\dfrac{{15}}{8} = \dfrac{{15}}{{2 \times 2 \times 2}} = \dfrac{{15}}{{{2^3}}} = \dfrac{{15 \times {5^3}}}{{{2^3} \times {5^3}}} = \dfrac{{15 \times 125}}{{8 \times 125}} = \dfrac{{1875}}{{1000}} = 1.875\]

\[\dfrac{{31}}{{20}} = \dfrac{{31}}{{2 \times 2 \times 5}} = \dfrac{{31 \times 5}}{{2 \times 2 \times 5 \times 5}} = \dfrac{{155}}{{100}} = 1.55\]
In both the examples, the result is a terminating decimal because terminating decimal is a decimal which can be expressed in a finite number of figures.
Hence, a rational number can be expressed as a terminating decimal if the denominator has factors 2 or 5.

Hence, the option (A) is the correct answer.

Note: A terminating decimal like 13.2 can be represented as the repeating decimal 13. 2000000.., but when the repeating digit is zero, the number is usually labelled as terminating. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence and conversely, a decimal expansion that terminates or repeats must be a rational number.