Question

How do you order the rational numbers from least to greatest: \[-4\dfrac{3}{5},-3\dfrac{2}{5},-4.65,-4.09\]?

Answer 1

Hint: The fraction of the form \[a\dfrac{b}{c}\] is called improper fractions. Before doing any operations like additions, subtraction, multiplication, etc. we have to convert it to a proper fraction. The improper fraction \[\pm a\dfrac{b}{c}\] can be converted to a proper fraction as \[\pm \dfrac{ca+b}{c}\]. For ordering the improper fractions we have to convert them to proper form.

Complete step by step answer:
We are given the rational numbers \[-4\dfrac{3}{5},-3\dfrac{2}{5},-4.65,-4.09\]. Before ordering them let’s convert the given improper fractions to proper fraction form, as follows
The proper fraction form of \[-4\dfrac{3}{5}\] is \[-\dfrac{\left( 4\times 5+3 \right)}{5}=-\dfrac{23}{5}\]. And the proper fraction form of \[-3\dfrac{2}{5}\] is \[-\dfrac{\left( 3\times 5+2 \right)}{5}=-\dfrac{17}{5}\]. So, the given rational numbers are \[-\dfrac{23}{5},-\dfrac{17}{5},-4.65,-4.09\].
For ordering them we have to convert all of them to common form either decimal or fraction. Let’s convert the fraction to decimal form. We already have two numbers in decimal form, we only have to convert the other two in decimal form. The decimal form of a fraction can be expressed by writing the quotient of division of the numerator and denominator.
Hence the decimal form of \[-\dfrac{23}{5}\] is \[-4.60\]. And the decimal form of the fraction \[-\dfrac{17}{5}\] is \[-3.40\].
So, the given numbers in the decimal form are \[-4.60,-3.40,-4.65,-4.09\].
When all numbers are negative, the number having the greatest numerical value is the least and the number having the least numerical value is greatest.
As \[-4.65\] has the greatest numerical value, it is the least. Then comes \[-4.60\], followed by \[-4.09\] and finally \[-3.40\], which is greatest.

Hence the order from least to greatest is \[-4.65,-4\dfrac{3}{5},-4.09,-3\dfrac{2}{5}\].

Note: We can also convert all the compare the given numbers by converting the decimals to fractions. We can do this as follows, as the LCM of the denominators of the other two fractions is 5. We have to multiply and divide the decimals by 5, it will give their fractions form. After this, as the denominators are the same for all the fractions, we can order them by comparing their numerators.