Question

Arrange the following rational numbers in descending order
\[\dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{{12}},\dfrac{7}{{18}}\]

Answer 1

Hint:In the above problem, we are given 4 rational numbers. They are \[\dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{{12}},\dfrac{7}{{18}}\] . But they are not in a well ordered manner, i.e. not in ascending or descending order. We need to arrange them in a well mannered descending order. Descending order means an order in which the contained elements are arranged from higher to lower value. The highest valued element or number is placed in the first position and then the value decreases further with the lowest element or number being the last element. Now, we can convert each rational number by multiplying and dividing by a suitable whole number such that all four rational numbers have the same denominator. In that situation we can easily arrange them in ascending or descending order.

Complete step by step answer:
Given four rational numbers are \[\dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{{12}},\dfrac{7}{{18}}\]. We have to arrange them in descending order. Now, we can see that their denominators are \[3,9,12,18\]. The nearest number that is divisible by all four denominators is \[36\], i.e. \[36\] is the LCM of \[3,9,12,18\]. Now we have to convert all the rational numbers so that their denominator is \[36\]. Therefore we can write them as,
\[ \Rightarrow \dfrac{2}{3} = \dfrac{2}{3} \times \dfrac{{12}}{{12}}\]

So that,
\[ \Rightarrow \dfrac{2}{3} = \dfrac{{24}}{{36}}\]
Similarly, we can write
\[ \Rightarrow \dfrac{4}{9} = \dfrac{4}{9} \times \dfrac{4}{4}\]
So that,
\[ \Rightarrow \dfrac{4}{9} = \dfrac{{16}}{{36}}\]
Also, we can write as
\[ \Rightarrow \dfrac{5}{{12}} = \dfrac{5}{{12}} \times \dfrac{3}{3}\]
So we get,
\[ \Rightarrow \dfrac{5}{{12}} = \dfrac{{15}}{{36}}\]

Again, we can write
\[ \Rightarrow \dfrac{7}{{18}} = \dfrac{7}{{18}} \times \dfrac{2}{2}\]
So we get,
\[ \Rightarrow \dfrac{7}{{18}} = \dfrac{{14}}{{36}}\]
Now the new rational numbers are \[\dfrac{{24}}{{36}},\dfrac{{16}}{{36}},\dfrac{{15}}{{36}},\dfrac{{14}}{{36}}\]
Now we can easily arrange them in a descending order as
\[\dfrac{{24}}{{36}} > \dfrac{{16}}{{36}} > \dfrac{{15}}{{36}} > \dfrac{{14}}{{36}}\] .
Now we know the descending order, so now we can replace the new rational numbers with the original rational numbers. Replacing the order with the original rational numbers, we can write
\[\dfrac{2}{3} > \dfrac{4}{9} > \dfrac{5}{{12}} > \dfrac{7}{{18}}\] .

Therefore, the required descending order of the rational numbers \[\dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{{12}},\dfrac{7}{{18}}\] is \[\dfrac{2}{3} > \dfrac{4}{9} > \dfrac{5}{{12}} > \dfrac{7}{{18}}\].

Note:There is an alternate method of finding the greater rational number between two different rational numbers.
When \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] are two different rational numbers, then
If \[ad > bc\] then \[\dfrac{a}{b} > \dfrac{c}{d}\].
And,
If \[ad < bc\] then \[\dfrac{a}{b} < \dfrac{c}{d}\].
And ,
If \[ad = bc\] then \[\dfrac{a}{b} = \dfrac{c}{d}\].