Question

How do you find the quotient of $\dfrac{3}{8}$ divided by $\dfrac{8}{9}$?

Answer 1

Hint: In the above question, we have been given two fractions $\dfrac{3}{8}$ and $\dfrac{8}{9}$ which are to be divided. We know that when one number is divided by the other number, we basically multiply the inverse of the second number with the first number. So in this question, we basically need to multiply the inverse of the fraction $\dfrac{8}{9}$ with the fraction $\dfrac{3}{8}$. The inverse of the fraction $\dfrac{8}{9}$ is equal to $\dfrac{9}{8}$. So we have to multiply $\dfrac{3}{8}$ by $\dfrac{9}{8}$. On multiplying them, we will obtain a new fraction $27/64$ whose quotient can be determined by carrying out the division.

Complete step-by-step solution:
The fractions given in the above question are $\dfrac{3}{8}$ and $\dfrac{8}{9}$. Let us call the first fraction to be $x$ and the second fraction to be $y$. So we can write the equations
\[\begin{align}
  & \Rightarrow x=\dfrac{3}{8}........(i) \\
 & \Rightarrow y=\dfrac{8}{9}........(ii) \\
\end{align}\]
Now, the above question states that $3/8$ is divided by $8/9$. So we divide equation (i) by (ii) to get
$\Rightarrow \dfrac{x}{y}=\dfrac{\dfrac{3}{8}}{\dfrac{8}{9}}$
We know that when one fraction is divided by the other fraction, the inverse of the second fraction is multiplied with the first fraction. So we can write the above equation as
$\begin{align}
  & \Rightarrow \dfrac{x}{y}=\dfrac{3}{8}\times \dfrac{9}{8} \\
 & \Rightarrow \dfrac{x}{y}=\dfrac{27}{64} \\
\end{align}$
Now, for determining the quotient of the given number, we divide $27$ by $64$ as shown below
\[64\overset{0.421875}{\overline{\left){\begin{align}
  & \underline{\begin{align}
  & 27 \\
 & 00 \\
\end{align}} \\
 & 270 \\
 & \underline{256} \\
 & 140 \\
 & \underline{128} \\
 & \underline{\begin{align}
  & 120 \\
 & 64 \\
\end{align}} \\
 & 560 \\
 & \underline{512} \\
 & \underline{\begin{align}
  & 480 \\
 & 448 \\
\end{align}} \\
 & 320 \\
 & \underline{320} \\
 & \underline{0} \\
\end{align}}\right.}}\]
Hence, the quotient of the division of the given pair of fractions is equal to $0.421875$.

Note: If we carefully note the value of the fraction obtained, which is equal to $\dfrac{27}{64}$, then we have both the numerator and the denominator perfect cubes. The numerator, $27$ can be written as ${{\left( 3 \right)}^{3}}$ and the denominator, $64$ can be written as ${{\left( 4 \right)}^{3}}$. So the fraction $\dfrac{27}{64}$ can be written as ${{\left( \dfrac{3}{4} \right)}^{3}}$. So for determining the quotient, we could find out the quotient of the fraction $\dfrac{3}{4}$ and then take its cube to get the final answer.