Question

List five rational numbers between $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$

Answer 1

Hint: First, we need to know about the concept of rational numbers. To find the rational numbers, first we are going to convert the numbers to the decimal form and we are going to find the rational numbers between them. After changing it, we are going to multiply and divide with \[3\] and \[5\]. Now, we will find the rational number between these two converted numbers. By increasing the denominator value, we can easily find \[5\] rational numbers between the given numbers.

Complete step by step answer:
Since we need to find the five rational numbers between $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$. But note that there are infinitely many rational numbers between any two numbers.
To find the value of the given terms we convert the problem into decimal form
Where $\dfrac{{ - 4}}{5}$can be written as decimal form of $ - 0.8$and $\dfrac{{ - 2}}{3}$ can be written as decimal form of $ - 0.667$
And hence the five required rational numbers must be between the numbers $ - 0.8$ to $ - 0.667$ in decimal.
Now finding the LCM of the given, which is the least common multiple of the two numbers $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$
Applying the cross multiplication, we get $5 \times 3 = 15$ as a denominator.
Hence multiplying the numbers $3$ and $5$ respectively to the numbers $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$. Thus, we get $\dfrac{{ - 4}}{5} \times \dfrac{3}{3} = \dfrac{{ - 12}}{{15}}$ and $\dfrac{{ - 2}}{3} \times \dfrac{5}{5} = \dfrac{{ - 10}}{{15}}$
Hence now it was easy to find the rational numbers between the converted numbers $\dfrac{{ - 12}}{{15}},\dfrac{{ - 10}}{{15}}$
Thus, we get the rational numbers as $\dfrac{{ - 11}}{{15}}$ (for the denominator $15$)
Now increasing the denominator so that we get many rational numbers between $\dfrac{{ - 12}}{{15}},\dfrac{{ - 10}}{{15}}$
Hence, we get $\dfrac{{ - 11}}{{16}},\dfrac{{ - 12}}{{16}}$ also we have $\dfrac{{ - 12}}{{17}},\dfrac{{ - 13}}{{17}}$ are decimal numbers between $ - 0.8$ to $ - 0.667$
Therefore, the five rational numbers between the numbers $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$ are $\dfrac{{ - 11}}{{15}},\dfrac{{ - 11}}{{16}},\dfrac{{ - 12}}{{16}},\dfrac{{ - 12}}{{17}},\dfrac{{ - 13}}{{17}}$

Note:
When multiplied and divided here by the given number between $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$, we use the LCM and find the denominator and then we can increase the denominator to get more rational numbers between the given set of questions.
 Note that every whole number has number one as its denominator like $2 = \dfrac{2}{1}$ and hence which is also the rational number, we can say that every natural number are the rational number.
Also, there are infinitely rational numbers between any two numbers. Hence the given rational numbers between $\dfrac{{ - 4}}{5}$ and $\dfrac{{ - 2}}{3}$ are also has infinitely many rational numbers, the six rational numbers are shown above.