Arrange the given rational number in ascending order.

$ \frac{4}{-9}, \frac{-5}{12} $ and $ \frac{2}{-3} $

Given: \( \frac{4}{-9}, \frac{-5}{12} \) and \( \frac{2}{-3} \)

To arrange: The given rational number in ascending order.

Solution:

To arrange the given rational numbers in ascending order we first need to convert these numbers having same denominator.

So, finding LCM of 9, 12 and 3:

Writing down the numbers as a product of their prime factors:

Prime factorization of 9:

• 3 $\times$ 3 = 3²

Prime factorization of 12:

• 2 $\times$ 2 $\times$ 3 = 2² $\times$ 3¹

Prime factorization of 3:

• 3 = 3¹

Finding highest power of each prime number:

3² , 2²

Multiplying these values together:

3² $\times$ 2² = 36

Thus,

LCM(9, 12, 3) = 36

Now,

$ \begin{array}{l}
\frac{4}{-9} \ =\ \frac{4}{-9} \ \times \ \frac{-4}{-4} \ =\ \frac{-16}{36}\\
\\
\\
\frac{-5}{12} \ =\ \frac{-5}{12} \ \times \ \frac{3}{3} \ =\ \frac{-15}{36}\\
\\
\\
\frac{2}{-3} \ =\ \frac{2}{-3} \ \times \ \frac{-12}{-12} \ =\ \frac{-24}{36}
\end{array}$

As all the numbers have same denominator then we just have to compare numerator to arrange them in ascending order. So,

$\frac{-24}{36} < \frac{-16}{36} < \frac{-15}{36}$

Hence, the required ascending order is:

$\frac{2}{-3} < \frac{4}{-9} < \frac{-5}{12}$

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Updated on: 10-Oct-2022

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