Question

Arrange the following fraction in ascending order: \[\dfrac{5}{3},\dfrac{7}{4},\dfrac{6}{5}\] and \[\dfrac{9}{8}\]
A) \[\dfrac{9}{8} < \dfrac{6}{5} < \dfrac{5}{3} < \dfrac{7}{4}\]
B) \[\dfrac{9}{8} < \dfrac{6}{5} < \dfrac{7}{4} < \dfrac{5}{3}\]
C) \[\dfrac{7}{4} < \dfrac{5}{3} < \dfrac{9}{8} < \dfrac{6}{5}\]
D) \[\dfrac{6}{5} < \dfrac{9}{8} < \dfrac{5}{3} < \dfrac{7}{4}\]

Answer 1

Hint: Ascending order means in increasing order. Smallest to largest fraction will be the order.
Make the same LCM of all the fractions and then compare them. The larger the numerator, the larger the fraction with same LCMs.

Complete step-by-step answer:
We are given fractions \[\dfrac{5}{3},\dfrac{7}{4},\dfrac{6}{5}\] and \[\dfrac{9}{8}\].
We have to arrange in the ascending order.
Since the denominator of each fraction is different, we cannot compare them therefore, first make the denominators same for each fraction.
In order to make the same denominator take the LCM of all the denominators.
Evaluate the LCM of $3,4,5,8$
The LCM is the least common multiple.
So, we have to find the least common multiple of $3,4,5,8$
The least common multiple will be $3 \times 5 \times 8 = 120$
Therefore, the LCM of $3,4,5,8$ is $120$.
Now we make each fraction having the denominator $120$so that we can compare them easily.
Multiply and divide by $40$in $\dfrac{5}{3}$.
\[\dfrac{{5 \times 40}}{{3 \times 40}} = \dfrac{{200}}{{120}}\]
Multiply and divide by $30$in $\dfrac{7}{4}$.
\[\dfrac{{7 \times 30}}{{4 \times 30}} = \dfrac{{210}}{{120}}\]
Multiply and divide by $24$in $\dfrac{6}{5}$.
\[\dfrac{{6 \times 24}}{{5 \times 24}} = \dfrac{{144}}{{120}}\]
Multiply and divide by $15$in $\dfrac{9}{8}$.
\[\dfrac{{9 \times 15}}{{8 \times 15}} = \dfrac{{135}}{{120}}\]
Now, the fractions are in the form $\dfrac{{200}}{{120}},\dfrac{{210}}{{120}},\dfrac{{144}}{{120}},\dfrac{{135}}{{120}}$
We know that the larger the numerator, larger the fraction with same LCMs.
Therefore, the ascending order of the following fractions will be,
$\dfrac{{135}}{{120}} < \dfrac{{144}}{{120}} < \dfrac{{200}}{{120}} < \dfrac{{210}}{{120}}$
Write the original fractions of all the fractions.
\[\dfrac{9}{8} < \dfrac{6}{5} < \dfrac{5}{3} < \dfrac{7}{4}\]
Therefore, option (A) is correct.

Note: We can solve this question by another method which is shown below,
Convert all the fractions in the decimal form.
\[
  \dfrac{5}{3} \approx 1.67 \\
  \dfrac{7}{4} = 1.75 \\
  \dfrac{9}{8} = 1.125 \\
  \dfrac{6}{5} = 1.2 \\
 \]
 The larger the value of the decimal, the larger will be the fraction.
Write the above fractions in the ascending order according to their decimal values.
\[\dfrac{9}{8} < \dfrac{6}{5} < \dfrac{5}{3} < \dfrac{7}{4}\]