Question

Which of the following is the reciprocal of $\dfrac{7}{9}$ ? \[\]
A. $\dfrac{3}{7}$ \[\]
B. $\dfrac{5}{7}$ \[\]
C.$\dfrac{9}{7}$ \[\]
D. $\dfrac{6}{5}$ \[\]

Answer 1

Hint: We recall the definition of multiplicative inverse otherwise known as reciprocal and show that the reciprocal of rational number $\dfrac{a}{b}$ is a rational with which by multiplying $\dfrac{a}{b}$ we get the multiplicative identity 1 which is given by $\dfrac{b}{a}$. We take $a=7,b=9$ and find $\dfrac{b}{a}$.

Complete step-by-step solution:
We know that the multiplicative identity of a rational number set is 1 because we can multiply any rational number with 1 and get the same number. For example let us multiply 1 with a rational number with numerator $a$ and denominator $b$ where $a$ is any integer and $b$ is a non-zero integer to have
\[\dfrac{a}{b}\times 1=\dfrac{a}{b}\]
The multiplicative inverse otherwise known as reciprocal of a rational number $\dfrac{a}{b}$ is also a rational number with which by multiplying $\dfrac{a}{b}$ we get the multiplicative identity 1. Let $x$ be the multiplicative inverse of $\dfrac{a}{b}$. So we have
\[\begin{align}
  & \dfrac{a}{b}\times x=1 \\
 & \Rightarrow \dfrac{ax}{b}=1 \\
\end{align}\]
We cross multiply in above step to have,
\[ax=b\]
We have to assume $a$ as non-zero number and we divide both side of the equation in the above step to have,
\[\begin{align}
  & \dfrac{ax}{a}=\dfrac{b}{a} \\
 & \Rightarrow x=\dfrac{b}{a} \\
\end{align}\]
So the reciprocal of $\dfrac{a}{b}$ is$\dfrac{b}{a}$ and we can find the reciprocal when both numerator $a$ and denominator $b$ are non-zero numbers. \[\]
We are given the question to find the reciprocal of the rational number $\dfrac{7}{9}$. We see that a given rational number has 7 as numerator and 9 as the denominator and both are non-zero integers. Let us assume $a=7,b=9$.
So the reciprocal of $\dfrac{a}{b}=\dfrac{7}{9}$ is $\dfrac{b}{a}=\dfrac{9}{7}$. Let us verify by multiplying them
\[\dfrac{7}{9}\times \dfrac{9}{7}=\dfrac{7\times 9}{9\times 7}=\dfrac{63}{63}=1\]
So the reciprocal of $\dfrac{7}{9}$ is $\dfrac{9}{7}$ and the correct option is C.
Note: We note that the reciprocal of 0 does not exist. The reciprocal is also denoted as ${{\left( \dfrac{a}{b} \right)}^{-1}}$ and is used in division of rational numbers. We must be careful of the multiplicative inverse and additive inverse. The additive inverse of $\dfrac{a}{b}$ in a rational number set is a rational number which we add to get the additive identity 0 which is given by $\dfrac{-a}{b}$.