Question

What is the least number among the following numbers?
A. $0.2$
B. $1\div 0.2$
C. $0.\overline{2}$
D. ${{\left( 0.2 \right)}^{2}}$

Answer 1

Hint:In this question, we will evaluate all the terms and write its values and then compare using general comparison rules.

Complete step-by-step answer:
In decimal point numbers, we can write as many zeroes at the end of the number after the decimal point, as we want. Also, to compare different decimal point numbers, it is convenient to have equal numbers of digits after decimal point in all the numbers. To do so, we will reject the number with the highest limit number of decimal digits and write zeros at the end of other numbers to make the number of decimal digits equal.
Now, in the first option, we have 0.2.
In the second option, we have $1\div 0.2$.We can write this as, $\dfrac{1}{0.2}$ .
Multiplying 10 in numerator and denominator, we get,
$\dfrac{1\times 10}{0.2\times 10}=\dfrac{10}{2}=5$
In the third option, we have $0.\overline{2}$.
We know, that bar on a number means that the number is repeating infinite numbers of time, so, $0.\overline{2}=0.22222\ldots $
In fourth option, we have ${{\left( 0.2 \right)}^{2}}$
We can write this as ${{\left( 0.2 \right)}^{2}}=0.2\times 0.2=0.04$
Therefore, the numbers to be compared are: 0.2, 5, $0.\overline{2}22222\ldots $, 0.04.
Writing equal numbers of zeros after all numbers by adding zeros and rounding off $0.\overline{2}$, we have,
0.20, 5.00, 0.22, 0.04.
Comparing digits before decimal point, we have 5 is the largest number.
For remaining numbers, comparing digits after decimal point, we have,
\[0.22\text{ }>\text{ }0.20\text{ }>0.\text{ }4\] .
Therefore, order of the given numbers is : 5 > 0.22 > 0.20> 0.04
$\Rightarrow 1\div 0.2>0.\overline{2}>0.2>{{\left( 0.2 \right)}^{2}}$
Hence, the least number among given option is ${{\left( 0.2 \right)}^{2}}$
Therefore, the correct answer is option (d).

Note: In this type of question, while writing zeroes at the end of the number make sure that zeroes can be written only after the decimal point, we find the given decimal, and then write the zeroes.