Question

Which one of the following is greater:
$(1)$\[5.0\]
\[\left( 2 \right)\]\[0.5\]
\[\left( 3 \right)\] \[0.005\]
\[\left( 4 \right)\] \[0.05\]

Answer 1

Hint: We have to find which of the following is greater . We solve this by decimal concept and by making the denominator equal for each value by removing the decimal points and simultaneously multiplying the numerator by for each decimal value removed . Then comparing the numerator we find the greater number .

Complete step-by-step answer:
Given : \[4\]decimal values\[5.0\]\[,{\text{ }}0.5{\text{ }},{\text{ }}0.005{\text{ }},{\text{ }}0.05\]
Let us consider that
\[a\]\[ = {\text{ }}5.0\]
\[b\]\[ = {\text{ }}0.5\]
\[c\]\[ = {\text{ }}0.005\]
\[d\]\[ = {\text{ }}0.05\]
Now , removing the decimal of \[a{\text{ }},\]\[b{\text{ }},\] \[c,\]\[d\] and obtaining fractional values
To convert the decimal values to fractional values, we can count the number of values after the decimal point. The number of values upto a non-zero number after the decimal point is equal to the number of zeros to be added in the denominator of the fractional value
Then ,
\[a{\text{ }} = \]\[\;\left( {\dfrac{{50}}{{10}}} \right)\]
\[b{\text{ }} = \] \[\left( {\dfrac{5}{{10}}} \right)\]
\[c{\text{ }} = \] \[\left( {\dfrac{5}{{1000}}} \right)\]
\[d{\text{ }} = \] \[\left( {\dfrac{5}{{100}}} \right)\]
As we have \[1000\] in the denominator of \[c\] we will make value of denominator \[ = {\text{ }}1000\] in \[a{\text{ }},{\text{ }}b\] and \[d\]
Now , taking L.C.M. and making the denominators of \[a{\text{ }},\]\[b{\text{ }},\] \[c{\text{ }},{\text{ }}d\] equal , we get
\[a{\text{ }} = \] \[\left( {\dfrac{{50}}{{10}}{\text{ }}} \right){\text{ }} \times {\text{ }}\left( {\dfrac{{100}}{{100}}{\text{ }}} \right)\]
Then ,
\[a{\text{ }} = \] \[\left( {\dfrac{{5000}}{{1000}}} \right)\]
Similarly for \[b\]and \[d\]
\[b{\text{ }} = \] \[\left( {\dfrac{5}{{10}}} \right){\text{ }} \times {\text{ }}\left( {\dfrac{{100}}{{100}}} \right)\]
Then ,
\[b{\text{ }} = \] \[\left( {\dfrac{{500}}{{1000}}} \right)\]
\[d{\text{ }} = \] \[\left( {\dfrac{5}{{100}}} \right){\text{ }} \times {\text{ }}\left( {\dfrac{{10}}{{10}}} \right)\]
Then ,
\[d{\text{ }} = \] \[\left( {\dfrac{{50}}{{1000}}} \right)\]
Now comparing the values ,
( As the denominator for each value is equal the greatest number can be calculated by the number which has the greatest value in the numerator )
The numerators of \[a{\text{ }},{\text{ }}b{\text{ }},{\text{ }}c\] and \[d\] are \[5000{\text{ }},{\text{ }}500\]\[,{\text{ }}5{\text{ }},{\text{ }}50\] respectively .
( As \[5000{\text{ }} > \]\[500{\text{ }} > \]\[50{\text{ }} > {\text{ }}5\])
\[a{\text{ }} > {\text{ }}b\]\[ > {\text{ }}d\] \[ > {\text{ }}c\]
Hence , \[a\] has the greatest value
Thus , the correct option is \[\left( 1 \right)\]
So, the correct answer is “Option 1”.

Note: Any number can be represented in the form of a decimal number . Also , any decimal number can be represented in the form of fraction by just removing the decimal points , decimal points are always reduced from the left side to the right of the decimal point
For an example \[5.005{\text{ }} = \]\[\left( {\dfrac{{50.05}}{{10}}} \right){\text{ }} = \] \[\left( {\dfrac{{500.5}}{{100}}} \right){\text{ }} = \]\[\left( {\dfrac{{5005}}{{1000}}} \right)\]