Question

Find the simplest form of the ratio $\dfrac{1}{6}:\dfrac{1}{4}$.

Answer 1

Hint: We need to find the LCM of the denominators and multiply with the individual terms to find the solution as the numerators are all 1. We take the simultaneous factorisation of those two numbers to find the LCM. The ratios become proportional to each other.

Complete step by step solution:
We need to find the simplest form of the ratio $\dfrac{1}{6}:\dfrac{1}{4}$. As the numerators are all 1, we find the LCM of the denominators and multiply with the individual terms to find the solution.
We need to find the LCM of 4 and 6.
We use the simultaneous factorisation to find the greatest common factor of 4 and 6.
We have to divide both of them with possible primes which can divide both of them.
\[\begin{align}
  & 2\left| \!{\underline {\,
  4,6 \,}} \right. \\
 & 1\left| \!{\underline {\,
  2,3 \,}} \right. \\
\end{align}\]
The LCM is $2\times 2\times 3=12$.
Now we multiply 12 with the fractions $\dfrac{1}{6}$ and $\dfrac{1}{4}$ to find its simplest form.
So, the terms become $12\times \dfrac{1}{6}=\dfrac{12}{6}=2$ and $12\times \dfrac{1}{4}=\dfrac{12}{4}=3$ respectively.
The ratio $\dfrac{1}{6}:\dfrac{1}{4}$ changes to $2:3$. Therefore, the simplest form of the ratio $\dfrac{1}{6}:\dfrac{1}{4}$ is $2:3$.

Note: We need to remember that the LCM has to be only one number. It is the least common multiple of all the given numbers. If the given numbers are prime numbers, then the LCM of those numbers will always be the multiple of those numbers. These rules follow for both integers and fractions.