Question

Find five rational numbers between $ 1 $ and $ 2 $ .
(A) $ \dfrac{1}{{10}},\dfrac{2}{{10}},\dfrac{3}{{10}},\dfrac{4}{{10}},\dfrac{5}{{10}} $
(B) $ \dfrac{1}{5},\dfrac{2}{5},\dfrac{3}{5},\dfrac{4}{5},\dfrac{5}{5} $
(C) $ \dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},\dfrac{1}{6} $
(D) $ \dfrac{8}{7},\dfrac{9}{7},\dfrac{{10}}{7},\dfrac{{11}}{7},\dfrac{{12}}{7} $

Answer 1

Hint: If the numerator is less than the denominator then the value is always less than $ 1 $ . We will use this information to neglect wrong options. To find required five rational numbers between $ 1 $ and $ 2 $ , we will write these both numbers in the form of $ \dfrac{p}{q} $ .

Complete step-by-step answer:
We know that if the numerator is less than the denominator then the value is always less than $ 1 $ . If we observe the given options then we can say that in options (A), (B) and (C) the values are less than $ 1 $ and we have to find the rational numbers which are greater than $ 1 $ (that is, between $ 1 $ and $ 2 $ ). Hence, we can say that options (A), (B) and (C) are wrong.
Now to find required five rational numbers between $ 1 $ and $ 2 $ , let us write the numbers $ 1 $ and $ 2 $ in the form of $ \dfrac{p}{q} $ such that denominator $ q $ will be $ 7 $ . So, we can write the number $ 1 $ as $ \dfrac{7}{7} $ and the number $ 2 $ as $ \dfrac{{14}}{7} $ . So, now we have to write five rational numbers between $ \dfrac{7}{7} $ and $ \dfrac{{14}}{7} $ . We know that the whole numbers $ 8,9,10,11,12 $ are lying between $ 7 $ and $ 14 $ . Hence, the rational numbers $ \dfrac{8}{7},\dfrac{9}{7},\dfrac{{10}}{7},\dfrac{{11}}{7},\dfrac{{12}}{7} $ are lying between $ \dfrac{7}{7} $ and $ \dfrac{{14}}{7} $ . That is, five rational numbers are $ \dfrac{8}{7},\dfrac{9}{7},\dfrac{{10}}{7},\dfrac{{11}}{7},\dfrac{{12}}{7} $ between $ 1 $ and $ 2 $ .
So, the correct answer is “Option D”.

Note: One can say that the five rational numbers are $ 1.1,1.2,1.3,1.4,1.5 $ between $ 1 $ and $ 2 $ . That is, the five rational numbers are $ \dfrac{{11}}{{10}},\dfrac{{12}}{{10}},\dfrac{{13}}{{10}},\dfrac{{14}}{{10}},\dfrac{{15}}{{10}} $ between $ 1 $ and $ 2 $ . To find five rational numbers between $ 1 $ and $ 2 $ , one can write the number $ 1 $ as $ \dfrac{6}{6} $ and the number $ 2 $ as $ \dfrac{{12}}{6} $ . Also remember that a rational number between two numbers $ x $ and $ y $ is obtained by $ \dfrac{{x + y}}{2} $ .