Question

What rational number is halfway between \[\dfrac{1}{6}\] and $\dfrac{1}{2}$ ?

Answer 1

Hint:In this problem express the rational number with a common denominator. That means to find the least common multiple for the given rational numbers. The direct way to find the answer is to find the average of two numbers or mean value. Average is the sum of the two numbers divided by two.

Formula used:
$\text{Average} = \dfrac{\text{sum}}{2}$;
$\text{sum} = a + b$

Complete step by step answer:
The given numbers are $\dfrac{1}{6}$ and $\dfrac{1}{2}$. We have to find the halfway between these given numbers. Therefore we have to make the rational number with a common denominator.So we have to find the least common value for the denominators $6$ and $2$. Here the least common multiple of $6$ and $2$ is,
$2\left| \!{\underline {\,
  {6,2} \,}} \right. $ $ = 3 \times 2 = 6$.
Therefore the least common multiple (LCM) of the given numbers is $ = 6$

Now to make the numbers with the denominator$ = 6$.
In the given numbers $\dfrac{1}{6}$ has the denominator as $6$.but we should make another number with the denominator $6$.Therefore multiply the number $\dfrac{1}{2}$ numerator and denominator by $\dfrac{3}{3}$ .
Because we know that $2 \times 3 = 6$ .
Therefore,
$\dfrac{1}{2} \times \dfrac{3}{3} = \dfrac{3}{6}$.

Here the two numbers having the same denominator are $\dfrac{3}{6}$ and $\dfrac{1}{6}$. We require finding the number halfway between the given numbers.Add the numerators of given numbers and divide by two and divide by the denominator.
$\dfrac{4}{2} \div 6$
$\Rightarrow \dfrac{{1 + 3}}{2} \div 6$
Addition of the numerators one plus three is four.
$\dfrac{4}{2} \div 6$
Divide the number four by two, we have,
$2 \div 6$........($2 \times 2 = 4$)
$\Rightarrow \dfrac{2}{6}$
Divide the number two by six, we have
$\dfrac{1}{3}$

Therefore the halfway between the given numbers is $ = \dfrac{1}{3}$.

Note:Generally, average refers to the ratio of the sum of the given observations to the total number of observations. Since we are given two observations, we used this formula; $Average = \dfrac{{sum}}{2}$where the sum is referred to as the addition of each observation.