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Arrange in ascending order.
$\dfrac{{14}}{{25}},\dfrac{{18}}{{27}},\dfrac{{16}}{{21}},\dfrac{{12}}{{24}},\dfrac{{17}}{{29}}$

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Last updated date: 26th Feb 2024
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IVSAT 2024
Answer
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Hint: Here, we are required to arrange the given numbers in ascending order. We will first of all simplify these fractions and then, find the LCM of their denominators. Then we will make the denominators the same and compare the numerators to arrange the fractions in ascending order. This will give us the required arrangement of the given fractions in ascending order.

Complete step-by-step answer:
In order to arrange the given numbers in ascending order, first of all, we will find the LCM of the denominators to make the denominators of the numbers the same.
The simplified forms of the given fractions are:
$\dfrac{{14}}{{25}},\dfrac{{18}}{{27}},\dfrac{{16}}{{21}},\dfrac{{12}}{{24}},\dfrac{{17}}{{29}} = \dfrac{{14}}{{25}},\dfrac{2}{3},\dfrac{{16}}{{21}},\dfrac{1}{2},\dfrac{{17}}{{29}}$
Hence, for the LCM of 25, 3, 21, 2 and 29, we will do the prime factorisation of each number.
$25 = 5 \times 5 = {5^2}$
$3 = 1 \times 3$
$21 = 3 \times 7$
$2 = 1 \times 2$
$29 = 1 \times 29$
Hence, we will take all the factors present in these numbers and the highest power of the common factors respectively.
Hence, L.C.M. of these numbers $ = 2 \times 3 \times {5^2} \times 7 \times 29 = 6 \times 25 \times 203 = 30,450$
Now, we will multiply the numerators and the denominators of the numbers in such a way so that we are able to make their denominator the same.
Hence, we get,
$\dfrac{{14 \times 1,218}}{{25 \times 1,218}} = \dfrac{{17,052}}{{30,450}}$
$\dfrac{{2 \times 10,150}}{{3 \times 10,150}} = \dfrac{{20,300}}{{30,450}}$
$\dfrac{{16 \times 1,450}}{{21 \times 1,450}} = \dfrac{{23,200}}{{30,450}}$
$\dfrac{{1 \times 15,225}}{{2 \times 15,225}} = \dfrac{{15,225}}{{30,450}}$
$\dfrac{{17 \times 1,050}}{{29 \times 1,050}} = \dfrac{{17,850}}{{30,450}}$
Hence, these numbers with the same denominator are:
$\dfrac{{17,052}}{{30,450}},\dfrac{{20,300}}{{30,450}},\dfrac{{23,200}}{{30,450}},\dfrac{{15,225}}{{30,450}},\dfrac{{17,850}}{{30,450}}$
Now, since the denominators are the same. We will compare the numerators and write them from the smallest number to the largest number.
Hence, we get,
$\dfrac{{15,225}}{{30,450}} < \dfrac{{17,052}}{{30,450}} < \dfrac{{17,850}}{{30,450}} < \dfrac{{20,300}}{{30,450}} < \dfrac{{23,200}}{{30,450}}$
Therefore, writing the original given form of these fractions, we get,
$\dfrac{{12}}{{24}} < \dfrac{{14}}{{25}} < \dfrac{{17}}{{29}} < \dfrac{{18}}{{27}} < \dfrac{{16}}{{21}}$
Hence, the given numbers can be written in ascending order as:
$\dfrac{{12}}{{24}} < \dfrac{{14}}{{25}} < \dfrac{{17}}{{29}} < \dfrac{{18}}{{27}} < \dfrac{{16}}{{21}}$
Thus, this is the required answer.

Note:
In this question, in order to write the given numbers in ascending order, we were required to find the LCM of the denominators. Hence, we should know that prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Now, in order to express the denominators given fractions as a product of their prime factors, we are required to do the prime factorization of the given denominators. Now, factorization is a method of writing an original number as the product of its various factors. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.