Question

Without doing any actual division, find which of the following rational number are terminating decimal representation:

(1) $\dfrac{7}{16}$
(2) $\dfrac{23}{125}$
(3) $\dfrac{9}{14}$
(4) $\dfrac{32}{45}$
(5) $\dfrac{43}{50}$
(6) $\dfrac{17}{40}$
(7) $\dfrac{61}{75}$
(8) $\dfrac{123}{250}$

(a). (1), (3), (5), (6) and (7)
(b). (1), (2), (5), (6) and (8)
(c). (1), (3), (5), (6) and (8)
(d). (1), (2), (5), (6) and (7)

Answer 1

Hint: In Maths, rational numbers are represented in $\dfrac{p}{q}$ form where $q$ is not equal to zero. It is one of the most important maths topics. Any fraction with non-zero denominators is a rational number. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational. Also, learn irrational numbers here.

Complete step by step answer:
We will learn here the properties of rational numbers along with its types, the difference between rational and irrational numbers. Solved examples help to understand the concepts in a better way. Also, learn the various rational numbers examples and learn how to find the rational numbers in a better way. To represent rational numbers on a number line, we need to simplify and write in decimal form first.
A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q is greater than 0. Also, we can say that any fraction fit under the category of rational numbers, where denominator and numerator are integers and the denominator is not equal to zero.
To identify if a number is rational or not, check the below conditions:
It is represented in the form of $\dfrac{p}{q}$ , where $q\ne 0$.
The ratio $\dfrac{p}{q}$ can be further simplified and represented in decimal form.
The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.
For terminating decimal $\dfrac{p}{q},q={{2}^{m}}{{5}^{n}}$ .
A fraction is a terminating decimal if the prime factors of the denominator of the fraction in its lowest form only contain 2s and/or 5s or no prime factors at all.
So let us check,
(1) $\dfrac{7}{16},16={{2}^{4}}{{5}^{0}}$ . So, it is a terminating decimal.
(2) $\dfrac{23}{125},125={{2}^{0}}{{5}^{3}}$ . So, it is a terminating decimal.
(3) $\dfrac{9}{14},14=2(7)$ a fraction is a terminating decimal if the prime factors of the denominator of the fraction in its lowest form only contain 2s and/or 5s or no prime factors at all. So it is not terminating.
(4) $\dfrac{32}{45},45={{3}^{2}}5$ . So, it is not terminating.
(5) $\dfrac{43}{50},50={{2}^{1}}{{5}^{2}}$ . So, it is terminating.
(6) $\dfrac{17}{40}, 40={{2}^{3}}.5$ . So, it is terminating.
(7) $\dfrac{61}{75},75={{3}^{1}}{{5}^{2}}$ and since it contains 3, So it is not a terminating.
(8) $\dfrac{123}{250},250={{2}^{2}}{{.5}^{4}}$. So, it is terminating.
The correct answer is option (b).

Note: Read the question and see what is asked. Your concept regarding terminating decimal should be clear. A proper assumption should be made. Do not make silly mistakes while substituting. Equate it in a proper manner don’t confuse yourself.