Question

How do you order the fractions from least to greatest: $\dfrac{{11}}{{12}}$, $\dfrac{7}{8}$, $\dfrac{{15}}{{16}}$?

Answer 1

Hint: Here we need to find the order of the given fractions. For that, we will first make the denominator of all the fractions the same. To make the denominator of the fractions the same, we will find the LCM of the numbers present in the denominator of the fractions. Then we will multiply the numerator and denominator of the fraction with the number such that all the denominators will become equal. Then we will compare the numerator of the fractions and set the order of the fractions accordingly.

Complete step by step solution:
 Here we need to find the order of the given fractions. The given fractions are:- $\dfrac{{11}}{{12}}$, $\dfrac{7}{8}$ and $\dfrac{{15}}{{16}}$.
For that, we will first make the denominator of all the fractions the same. To make the denominator of the fractions the same, we will find the LCM of the numbers present in the denominator of the fractions.
Therefore, we will find the LCM of 12, 8, and 16.
To find the LCM, we will find the factors of these numbers.
$ 12 = 2 \times 2 \times 3 \\
  8 = 2 \times 2 \times 2 \\
 16 = 2 \times 2 \times 2 \times 2 \\ $
We can see that the LCM of 12, 8 and 16 is equal to 48.
We have to make the denominator of every fraction equal to 48.
So we will multiply the numerator and denominator of the first fraction $\dfrac{{11}}{{12}}$ by 4.
 $ \dfrac{{11 \times 4}}{{12 \times 4}} = \dfrac{{44}}{{48}}$
Now, we will multiply the numerator and denominator of the second fraction $\dfrac{7}{8}$ by 6.
$ \Rightarrow \dfrac{{7 \times 6}}{{8 \times 6}} = \dfrac{{42}}{{48}}$
Now, we will multiply the numerator and denominator of the third fraction, $\dfrac{{15}}{{16}}$ by 3.
$ \Rightarrow \dfrac{{15 \times 3}}{{16 \times 3}} = \dfrac{{45}}{{48}}$
Therefore, the fractions become $\dfrac{{44}}{{48}}$, $\dfrac{{42}}{{48}}$ and $\dfrac{{45}}{{48}}$.
We know that when the fractions have the same denominator, then we compare the fractions according to their numerator.
As we can see that $42 < 44 < 45$
So the order of the fractions become $\dfrac{{42}}{{48}} < \dfrac{{44}}{{48}} < \dfrac{{45}}{{48}}$

Hence, the required order of the given fractions is equal to $\dfrac{7}{8} < \dfrac{{11}}{{12}} < \dfrac{{15}}{{16}}$.

Note:
Here we have determined the correct order of the fractions given in the question. We need to keep in mind that the fractions which have different denominators, cannot compare them directly. We have to find the LCM and then we have to make the denominators the same and then we compare the fractions according to their numerator. We can also convert the fraction into a decimal by dividing the numerator of the fraction by its denominator and then compare the numbers. Here we can make a mistake if we represent the numbers from greater to least.