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$\dfrac{1}{3},\dfrac{-2}{9},\dfrac{-4}{3}$

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Hint: Firstly make the denominators of all the rational numbers equal by taking their L.C.M. and the use the concept i.e.â€™ If the rational numbers have all the denominators equal the number with a bigger numerator will be biggerâ€™ to get the ascending order.

Complete step-by-step answer:

Firstly we will rewrite the rational numbers given in the problems,

$\dfrac{1}{3},\dfrac{-2}{9},\dfrac{-4}{3}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (A)

To arrange them in ascending order we have to equate the denominators of all the rational numbers given above, and for that we should calculate the L.C.M. of the denominators as follows,

The denominators of the above numbers are 3, 9, and 3 respectively and to calculate the L.C.M. we will write their factors as follows,

Factors of 3: $3\times 1$

Factors of 9: $3\times 3\times 1$

Factors of 3: $3\times 1$

Therefore, L.C.M. $=3\times 3\times 1=9$

As L.C.M. of the denominators is 9 therefore we have to make denominator of each rational number equal to 9 as follows,

Multiply and divide by 3 to the first rational number,

$\therefore \dfrac{1}{3}=\dfrac{1\times 3}{3\times 3}=\dfrac{3}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

As the denominator of the second rational number is already 9 therefore there is no need of adjustments and therefore we can write this number as it is.

$\therefore \dfrac{-2}{9}=\dfrac{-2}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

Multiply and divide by 3 to the third rational number to equate its denominator to 9,

$\therefore \dfrac{-4}{3}=\dfrac{-4\times 3}{3\times 3}=\dfrac{-12}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

From equation (1) equation (2) and equation (3) we can write the rational number with denominator equal to 9 as,

$\dfrac{3}{9},\dfrac{-2}{9},\dfrac{-12}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (B)

To proceed further in the solution we should know the concept given below,

Concept: If the rational numbers have all the denominators equal the number with a bigger numerator will be bigger.

To decide the ascending order of the rational numbers we should plot them on a number line to make it very simple.

From the above number line we can easily observe that,

3 > -2 > -12

And therefore by using the concept we can say that,

$\dfrac{3}{9}>\dfrac{-2}{9}>\dfrac{-12}{9}$

Therefore by comparing above inequation with (A) and (B) we can write,

$\dfrac{1}{3}>\dfrac{-2}{9}>\dfrac{-4}{3}$

Therefore the Ascending order of the given rational numbers is$\dfrac{-4}{3},\dfrac{-2}{9},\dfrac{1}{3}$.

Note: Do not order the rational numbers in ascending order without equating their denominators because without them your answer will become wrong. Also remember that the greater number having negative sign is always lesser i.e. $\dfrac{-2}{9}>\dfrac{-4}{3}$.

Complete step-by-step answer:

Firstly we will rewrite the rational numbers given in the problems,

$\dfrac{1}{3},\dfrac{-2}{9},\dfrac{-4}{3}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (A)

To arrange them in ascending order we have to equate the denominators of all the rational numbers given above, and for that we should calculate the L.C.M. of the denominators as follows,

The denominators of the above numbers are 3, 9, and 3 respectively and to calculate the L.C.M. we will write their factors as follows,

Factors of 3: $3\times 1$

Factors of 9: $3\times 3\times 1$

Factors of 3: $3\times 1$

Therefore, L.C.M. $=3\times 3\times 1=9$

As L.C.M. of the denominators is 9 therefore we have to make denominator of each rational number equal to 9 as follows,

Multiply and divide by 3 to the first rational number,

$\therefore \dfrac{1}{3}=\dfrac{1\times 3}{3\times 3}=\dfrac{3}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

As the denominator of the second rational number is already 9 therefore there is no need of adjustments and therefore we can write this number as it is.

$\therefore \dfrac{-2}{9}=\dfrac{-2}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

Multiply and divide by 3 to the third rational number to equate its denominator to 9,

$\therefore \dfrac{-4}{3}=\dfrac{-4\times 3}{3\times 3}=\dfrac{-12}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

From equation (1) equation (2) and equation (3) we can write the rational number with denominator equal to 9 as,

$\dfrac{3}{9},\dfrac{-2}{9},\dfrac{-12}{9}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (B)

To proceed further in the solution we should know the concept given below,

Concept: If the rational numbers have all the denominators equal the number with a bigger numerator will be bigger.

To decide the ascending order of the rational numbers we should plot them on a number line to make it very simple.

From the above number line we can easily observe that,

3 > -2 > -12

And therefore by using the concept we can say that,

$\dfrac{3}{9}>\dfrac{-2}{9}>\dfrac{-12}{9}$

Therefore by comparing above inequation with (A) and (B) we can write,

$\dfrac{1}{3}>\dfrac{-2}{9}>\dfrac{-4}{3}$

Therefore the Ascending order of the given rational numbers is$\dfrac{-4}{3},\dfrac{-2}{9},\dfrac{1}{3}$.

Note: Do not order the rational numbers in ascending order without equating their denominators because without them your answer will become wrong. Also remember that the greater number having negative sign is always lesser i.e. $\dfrac{-2}{9}>\dfrac{-4}{3}$.

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