Question

Which of the following fractions have the smallest value: $\dfrac{1}{2}$, $\dfrac{3}{4}$, $\dfrac{2}{3}$ and $\dfrac{5}{6}$?

Answer 1

Hint: Fractions are numerical quantities that are a part of a Real number. When a number or an object is split into equal parts, each part becomes a fraction of the whole. A fraction is a term that comes from the Latin language. “Fractus” means “broken” in Latin. Every fraction consists of a numerator (the number above -) and denominator (the number below -). To compare the given fractions, we first make the denominators of all the fractions equal and then compare the numerators of all the fractions.

Complete step by step solution:
The magnitude of fractions may be grouped in either ascending or descending order. Ascending order is the arrangement of numbers from lowest to highest, while descending order is the arrangement of numbers from highest to lowest.
Consider the numbers given : $\dfrac{1}{2}$, $\dfrac{3}{4}$, $\dfrac{2}{3}$ and $\dfrac{5}{6}$.
So, to find the smallest of all the fractional numbers given to us, we compare the numerators after making the denominators
To make the denominator of each fraction equal, we find the LCM of all the denominators of the above numbers.
The Least Common Multiple (LCM) of the denominators $2$, $4$, $3$ and $6$ is $12$.
Now, we multiply the numerator and denominator by the same number in such a way that the denominator of all the fractions becomes equal to the LCM. So, we get,
\[\dfrac{1}{2} \times \dfrac{6}{6} = \dfrac{6}{{12}}\]
\[\dfrac{3}{4} \times \dfrac{3}{3} = \dfrac{9}{{12}}\]
$\dfrac{2}{3} \times \dfrac{4}{4} = \dfrac{8}{{12}}$
$\dfrac{5}{6} \times \dfrac{2}{2} = \dfrac{{10}}{{12}}$
Now, the denominators of all the fractions are equal. The numerators are $6, 9, 8, 10$.
So, we can arrange the fractions in ascending order by comparing the numerators.
Now, let us arrange the above fractions in the ascending order as shown below:
$\dfrac{6}{{12}}$, $\dfrac{8}{{12}}$, $\dfrac{9}{{12}}$, $\dfrac{{10}}{{12}}$
Now, converting back to the corresponding fractions in simplest form, we get, $\dfrac{1}{2}$, $\dfrac{2}{3}$, $\dfrac{3}{4}$ and $\dfrac{5}{6}$
Hence, the smallest fraction is $\dfrac{1}{2}$.

Note:
In mathematics, we use fractions to find the fractional part of a number, calculate decimals and percentages, ratio and proportion, probability, and algebraic equations. We should also know how to compare the unlike fractions by taking LCM of the denominators. One should know the method to calculate the equivalent fractions by taking the LCM of the denominators.