Question

How do you order the following from least to greatest $0.19,0.09,0.9,1,0$?

Answer 1

Hint: We first try to convert the decimals and integers to their fraction forms. We try to keep their denominator the same to compare them. Then from the value of their numerators we find the relation between the original numbers.

Complete step-by-step solution:
The given five numbers which consist both decimal and integers have to be arranged in the form of least to greatest.
We first try to convert the decimals into their fraction forms.
All the decimals will be converted using the multiplication form.
$0.19$ has two digits after decimals which gives that we are going to multiply ${{10}^{-2}}=\dfrac{1}{100}$ with the number 19. Therefore, $0.19=19\times {{10}^{-2}}=\dfrac{19}{100}$.
$0.09$ has two digits after decimals which gives that we are going to multiply ${{10}^{-2}}=\dfrac{1}{100}$ with the number 9. Therefore, $0.09=9\times {{10}^{-2}}=\dfrac{9}{100}$.
$0.9$ has one digit after decimals which gives that we are going to multiply ${{10}^{-1}}=\dfrac{1}{10}$ with the number 9. Therefore, $0.9=9\times {{10}^{-1}}=\dfrac{9}{10}$.
The integers can be represented as fractions by taking their denominators as 1 to give $\dfrac{1}{1},\dfrac{0}{1}$
Now we try to form the numbers as $\dfrac{19}{100},\dfrac{9}{100},\dfrac{9}{10},\dfrac{1}{1},\dfrac{0}{1}$. We can compare them only when their denominators are the same. Therefore, we are going to make all the denominators of the fractions and integers as 100.
For $\dfrac{9}{10}$, we multiply 10 in both numerator and denominator to get $\dfrac{9\times 10}{10\times 10}=\dfrac{90}{100}$.
For both $\dfrac{1}{1},\dfrac{0}{1}$, we multiply 100 in both numerator and denominator to get $\dfrac{1\times 100}{1\times 100}=\dfrac{100}{100};\dfrac{0\times 100}{1\times 100}=\dfrac{0}{100}$ respectively.
Now we can compare the fractions according to their numerator value.
The order for least to greatest will be $\dfrac{0}{100} < \dfrac{9}{100} < \dfrac{19}{100} < \dfrac{90}{100} < \dfrac{100}{100}$ which gives for actual valued numbers is $0 < 0.09 < 0.19 < 0.9 < 1$.

Note: We can also directly compare them in decimals. The comparison only for decimals would work as the digit values from the start. Among $0.19,0.09,0.9$, the first digit after decimal with greater value is $0.9$. Then we have $0.19$ and lastly $0.09$. The order will be $0 < 0.09 < 0.19 < 0.9 < 1$.