Question

Give four rational numbers equivalent to $\dfrac{-2}{7}$

Answer 1

Hint: To find the equivalent rational number, we will multiply denominator and numerator with the same number.

Complete step-by-step answer:
A rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. In the question given, we have to find those rational numbers which are equivalent to $\left( \dfrac{-2}{7} \right)$ .Equivalent rational numbers are rational numbers that have the same value but which are represented in a different form. We can obtain equivalent numbers of any given rational number by multiplying both the numerator and denominator with the same constant number. Thus, we can say that the rational number equivalent to the $\left( \dfrac{-2}{7} \right)$ will be of the form:
$\dfrac{-2}{7}\times \dfrac{k}{k}$
Let us take different values of k and try to obtain different equivalent rational numbers:
Case 1: When the value of k is 7. The equivalent rational number obtained in this case will be
$\dfrac{-2\times 7}{7\times 7}=\dfrac{-14}{49}$
When this is simplified, it value will be equal to $\left( \dfrac{-2}{7} \right)$
Case 2: When the value of k is -5. The equivalent rational number obtained in this case will be
$\dfrac{-2\times \left( -5 \right)}{7\times \left( -5 \right)}=\dfrac{10}{-35}=\dfrac{-10}{35}$
When this is simplified, its value will be equal to $\left( \dfrac{-2}{7} \right)$
Case 3: When the value of k is 3. The equivalent rational number obtained in this case will be
$\dfrac{-2\times 3}{7\times 3}=\dfrac{-6}{21}$
When this is simplified, it value will be equal to $\left( \dfrac{-6}{21} \right)$
Case 4: When the value of k is -11. The equivalent rational number obtained in this case will be
$\dfrac{-2\times \left( -11 \right)}{7\times \left( -11 \right)}=\dfrac{22}{-77}=\dfrac{-22}{77}$
Thus, the four equivalent rational numbers are: $\dfrac{-14}{49},\dfrac{-10}{35},\dfrac{-6}{21}and\dfrac{-22}{77}$

Note: We cannot take the value of k = 0. When we multiply k in numerator and denominator, both become zero. In this case we will not get any equivalent rational number, we will get an indeterminate form.