Answer

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**Hint:**Here in this question, we have to arrange the given numbers from the smallest to the largest. The numbers are in the form of fraction by simplifying the fraction into its simplest form with the help of division or by the table of multiplication. Then we arrange the numbers from the smallest to the largest number.

**Complete step-by-step solution:**

the number is in the form of fraction. The fraction number has two parts numerator and denominator. In the fraction we have 3 kinds

Proper fraction: the value of the numerator is less than the value of the denominator.

Improper fraction: the value of the numerator is greater than the value of the denominator.

Mixed fraction: the combination of a whole number and the fraction term.

While arranging any kind of number we have two kind of arrangements namely,

Ascending order: The numbers are arranged from least to the greatest.

Descending order: The numbers are arranged from greatest to the least.

Now consider the set in the question \[\left\{ {\dfrac{{18}}{9},\dfrac{{32}}{{24}},\dfrac{{25}}{{20}},\dfrac{{33}}{{12}}} \right\}\]

First we simplify each term of the set.

Consider \[\dfrac{{18}}{9}\]

On simplifying we get \[\dfrac{{18}}{9} = 2\]

Consider the second term \[\dfrac{{32}}{{24}}\]

The number 8 is the common factor for the both 32 and 24. Dividing the both numbers by 8 we get \[\dfrac{4}{3}\]. On simplifying we get \[\dfrac{4}{3} = 1.33\]

Consider the third term \[\dfrac{{25}}{{20}}\]

The number 5 is the common factor for the both 25 and 20. Dividing the both numbers by 5 we get \[\dfrac{5}{4}\]. On simplifying we get \[\dfrac{5}{4} = 1.25\]

Consider the fourth term \[\dfrac{{33}}{{12}}\]

The number 3 is the common factor for the both 33 and 12. Dividing the both numbers by 3 we get \[\dfrac{{11}}{4}\]. On simplifying we get \[\dfrac{{11}}{4} = 2.75\]

Therefore the numbers are written as

2, 1.33, 1.25, 2.75.

The ascending order of these numbers is

1.25, 1.33, 2, 2.75.

\[ \Rightarrow \left\{ {\dfrac{{25}}{{20}},\dfrac{{32}}{{24}},\dfrac{{18}}{9},\dfrac{{33}}{{12}}} \right\}\]

This also written as

\[ \Rightarrow \left\{ {\dfrac{{25}}{{20}} < \dfrac{{32}}{{24}} < \dfrac{{18}}{9} < \dfrac{{33}}{{12}}} \right\}\]

**Note:**To simplify the fraction the table of multiplication is needed. The common factor is the number which can divide the both numbers. For the ascending arrangement we use the symbol < and for the descending arrangement we use > symbol. In a proper fraction if the numerator value is less then it will be least and vice versa in the improper fraction.

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