Question

Find six rational numbers between $\dfrac{{ - 1}}{2}$ and $\dfrac{5}{4}$.

Answer 1

Hint:
Here, we are required to find six rational numbers between the given two numbers. We will take the LCM of the two numbers and make their denominators the same. Then we will find the numbers which lie between then by just looking at the numerator of the two given numbers. By this way, we will find the required six rational numbers.

Complete step by step solution:
For answering this question, we should know what a rational number is.
A rational number is a number which can be written in the form of $\dfrac{p}{q}$ where $p$is the numerator and $q$ is the denominator. Also, the denominator $q \ne 0$.
Hence, in simple terms, rational numbers are those numbers which can be expressed in the form of fractions.
Now, according to the question,
We have to find six rational numbers between $\dfrac{{ - 1}}{2}$ and $\dfrac{5}{4}$.
First of all, we will take the LCM of the denominators of the two rational numbers.
Now, LCM of 2 and 4 is 4 itself. This is because 2 is a factor of 4.
Hence, multiplying the first rational number by 2, we get the two rational numbers as:
$\dfrac{{ - 1}}{2} = \dfrac{{ - 1 \times 2}}{{2 \times 2}} = \dfrac{{ - 2}}{4}$
And $\dfrac{5}{4}$.
Now, the rational numbers which lie between these two numbers are:
$\dfrac{{ - 1}}{4},\dfrac{0}{4},\dfrac{1}{4},\dfrac{2}{4},\dfrac{3}{4},\dfrac{4}{4}$
These can also be written as:
$\dfrac{{ - 1}}{4},0,\dfrac{1}{4},\dfrac{2}{4},\dfrac{3}{4},1$

Hence, these are the six rational numbers which lie between $\dfrac{{ - 1}}{2}$ and $\dfrac{5}{4}$.

Note:
If we were asked to find more such rational numbers between these two numbers, then instead of taking the LCM as 4, we should take a larger number which is divisible by 4 and 2 as the LCM, for instance, it can be 8,12,16,…
In this case, let the LCM be 8.
Then, the two numbers are:
$\dfrac{{ - 1}}{2} = \dfrac{{ - 1 \times 4}}{{2 \times 4}} = \dfrac{{ - 4}}{8}$ and $\dfrac{5}{4} = \dfrac{{5 \times 2}}{{4 \times 2}} = \dfrac{{10}}{8}$
Clearly, between the numerators $ - 4$ and $10$, we can find 13 rational numbers.
Hence, the larger the LCM taken ,the larger will be the number of rational numbers between two given numbers.
Hence, by this way, we can find more and more rational numbers between any two given numbers.