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# Arrange the following rational number into ascending order $\dfrac{3}{7},\dfrac{4}{5},\dfrac{7}{9},\dfrac{1}{2}$ A. $\dfrac{4}{5},\dfrac{7}{9},\dfrac{3}{7},\dfrac{1}{2}$ B. $\dfrac{3}{7},\dfrac{1}{2},\dfrac{7}{9},\dfrac{4}{5}$ C. $\dfrac{4}{5},\dfrac{7}{9},\dfrac{1}{2},\dfrac{3}{7}$ D. $\dfrac{1}{2},\dfrac{3}{7},\dfrac{7}{9},\dfrac{4}{5}$

Last updated date: 13th Jun 2024
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Hint: We need to arrange these fractions in ascending order and all these fractions have different denominators. To compare the fractions, we need to have the same denominators for all the fractions. As they are different here, we will take the LCM of all the denominators and multiply the denominator and the numerator by the same number such that after multiplication, the denominator becomes equal to the LCM that we calculated. Then we will compare the fractions and arrange them in ascending order. This will give us our answer.

Now, we have the fractions $\dfrac{3}{7},\dfrac{4}{5},\dfrac{7}{9},\dfrac{1}{2}$
We will now take the LCM of the denominators, i.e. , the LCM of 7,5,9 and 2.
Now, we can see that these numbers are co-prime (co-prime numbers are the numbers which do not have any common factor except for 1) and the LCM of co-prime numbers is equal to their product. Thus, the LCM of these numbers will be equal to the product of all these numbers.
Thus, the LCM is given as:
\begin{align} & \Rightarrow LCM=7\times 5\times 9\times 2 \\ & \Rightarrow LCM=630 \\ \end{align}
Thus, we will multiply the denominators and numerators of all the fractions with such a number that the denominator becomes ‘630’.
Now, we will make the denominators of all the fractions the same. So, we will need to multiply the numerator and denominator by 90 in $\dfrac{3}{7}$ , 126 in $\dfrac{4}{5}$, 70 in $\dfrac{7}{9}$ and by 315 in $\dfrac{1}{2}$.
Thus, the fractions will become:
\begin{align} & \dfrac{3\times 90}{7\times 90},\dfrac{4\times 126}{5\times 126},\dfrac{7\times 70}{9\times 70},\dfrac{1\times 315}{2\times 315} \\ & \Rightarrow \dfrac{270}{630},\dfrac{504}{630},\dfrac{490}{630},\dfrac{315}{630} \\ \end{align}
Now that we have made the denominators the same for all the fractions, we can compare them by comparing the numerators. Larger the numerator, larger will be fraction.
On comparing we get:
\begin{align} & \dfrac{270}{630}<\dfrac{315}{630}<\dfrac{490}{630}<\dfrac{504}{630} \\ & \Rightarrow \dfrac{3}{7}<\dfrac{1}{2}<\dfrac{7}{9}<\dfrac{4}{5} \\ \end{align}
Thus, the fractions in ascending order will come out as:
$\dfrac{3}{7},\dfrac{1}{2},\dfrac{7}{9},\dfrac{4}{5}$

So, the correct answer is “Option B”.

Note: We can also solve this question by solving the fractions and converting them to their decimal expansions and comparing those expansions. It will be done as follows:
Decimal expansion of $\dfrac{3}{7}=0.428$
Decimal expansion of $\dfrac{4}{5}=0.8$
Decimal expansion of $\dfrac{7}{9}=0.777$
Decimal expansion of $\dfrac{1}{2}=0.5$
On comparing the decimal expansions, we will get:
$\dfrac{3}{7}<\dfrac{1}{2}<\dfrac{7}{9}<\dfrac{4}{5}$
Thus, the arranged fractions will be:
$\dfrac{3}{7},\dfrac{1}{2},\dfrac{7}{9},\dfrac{4}{5}$
Thus the answer is the same in both cases.