Simplify the following ratio: \[\dfrac{1}{4}:\dfrac{1}{6}:\dfrac{1}{8}\]

Hint: Here in this question, we have to solve the given ratio till it becomes simplest, meaning the given ratio having a fraction which we convert into the whole numbers. To solve this, by taking LCM to the denominator of each fraction and multiplying the LCM to the each fraction and simplify using a multiplication and division operation, we get the required solution.

Complete step by step solution:
a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent.
The most common way to write a ratio using a colon i.e., \[a:b\]
Where, $a$ is antecedent and $b$ is the consequent.
Consider the given ratio,
\[ \Rightarrow \,\,\dfrac{1}{4}:\dfrac{1}{6}:\dfrac{1}{8}\]
We have to write the above ratio to the simplest form.
Now, take a LCM to the denominator of each fraction.
\[
  2\left| \!{\underline {\,
  {4,6,8} \,}} \right. \\
  2\left| \!{\underline {\,
  {2,3,4} \,}} \right. \\
  2\left| \!{\underline {\,
  {1,3,2} \,}} \right. \\
  3\left| \!{\underline {\,
  {1,3,1} \,}} \right. \\
  \,\,\,\,1,1,1 \\
 \]
Then, \[LCM = 2 \times 2 \times 2 \times 3\]
\[ \Rightarrow \,LCM = 24\]
Now, multiply the LCM in to each fraction in ratio, then
\[ \Rightarrow \,\,\dfrac{1}{4} \times 24:\dfrac{1}{6} \times 24:\dfrac{1}{8} \times 24\]
On simplification, we get
\[ \Rightarrow \,\,6:4:3\]
Therefore, the simplest form of the ratio \[\dfrac{1}{4}:\dfrac{1}{6}:\dfrac{1}{8}\] is \[6:4:3\].

Note:
When simplifying a ratio of the form \[a:b\] makes it easier to work with, and the simplifying process is fairly straightforward. Find the greatest factor common to both terms of the ratio, and then divide both terms by that factor. If the given ratios are in fraction follow the same procedure. If the given ratios are not in the form of fraction then we divide the each every ratio number by a common number.