Question

How do you find the quotient of $\dfrac{2}{9}$ divided by $\dfrac{5}{4}$?

Answer 1

Hint: We have to divide a fraction by another fraction. In such cases we multiply the dividend by the reciprocal of the divisor and then evaluate the resulting fraction in decimal terms by dividing numerator by the denominator. There will be no remainder left as we will try to find the quotient in decimal numbers.

Complete step-by-step solution:
We have to divide the fraction $\dfrac{2}{9}$ by $\dfrac{5}{4}$ and find the quotient.
i.e. we have to evaluate \[\dfrac{2}{9} \div \dfrac{5}{4}\]
When we have to divide a fraction by another fraction we can simply multiply the dividend by the reciprocal of the divisor. This can be explained as follows,
\[
  \dfrac{2}{9} \div \dfrac{5}{4} \\
   = \dfrac{{\left( {\dfrac{2}{9}} \right)}}{{\left( {\dfrac{5}{4}} \right)}} \\
   = \dfrac{2}{9} \times \dfrac{4}{5} \\
   = \dfrac{{2 \times 4}}{{9 \times 5}} = \dfrac{8}{{45}} \\
\]
Thus, we get \[\dfrac{2}{9} \div \dfrac{5}{4} = \dfrac{8}{{45}}\]
Now we evaluate $\dfrac{8}{{45}}$ in decimal terms to find the quotient.
We will find the quotient rounded up to 3 decimal places. Since the denominator is more than the numerator, we multiply and divide the dividend by ${10^3}$. We can write,
$\dfrac{8}{{45}} = \dfrac{{8 \times {{10}^3}}}{{45 \times {{10}^3}}} = \dfrac{{8000}}{{45}} \times \dfrac{1}{{1000}}$
We can write $8000$ as,
$8000 = (177 \times 45) + 35$
We can observe that $8000$ divided by $45$ gives $177$ as quotient and $35$ as remainder. We can ignore the remainder term from this.
We can write,
$
  \dfrac{{8000}}{{45}} \times \dfrac{1}{{1000}} \\
   = \dfrac{{(177 \times 45) + 10}}{{45}} \times \dfrac{1}{{1000}} \\
   \approx \left( {\dfrac{{177 \times 45}}{{45}}} \right) \times \dfrac{1}{{1000}} \\
   = 177 \times \dfrac{1}{{1000}} \\
   = \dfrac{{177}}{{1000}} \\
   = 0.177 \\
$
Thus, we get the quotient as $0.177$
We can write,
$\dfrac{2}{9} = 0.177 \times \dfrac{5}{4}$

Hence, the quotient of $\dfrac{2}{9}$ divided by $\dfrac{5}{4}$ is $0.177$ when rounded to $3$ decimal places.

Note: To divide one fraction by another we multiplied the dividend by the reciprocal of the divisor. We may not get an exact answer in decimals in some cases in which case we can round up the result to $2$ to $3$ decimal places. In case we find quotients in decimal terms there is no remainder left.