Question

By what rational number should we multiply $ \dfrac{20}{-9} $ . So that the product may be $ \dfrac{-5}{9} $ ?

Answer 1

Hint: We first try to find the operation which can be applied to find the required number. We divide $ \dfrac{-5}{9} $ by $ \dfrac{20}{-9} $ . We change from division to multiplication. We find the simplified form of the solution.

Complete step by step solution:
We find the required number by dividing $ \dfrac{-5}{9} $ by $ \dfrac{20}{-9} $ .
Therefore, the number will be $ \dfrac{-5}{9}\div \dfrac{20}{-9} $ .
We need to be careful about the change of operation from division of multiplication. This process also changes the fraction to its inverse form.
Therefore, $ \dfrac{-5}{9}\div \dfrac{20}{-9}=\dfrac{-5}{9}\times \dfrac{-9}{20}=\dfrac{45}{180} $ .
We need to find the simplified form of $ \dfrac{45}{180} $ .
For any fraction $ \dfrac{p}{q} $ , we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $ \dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}} $ .
For our given fraction $ \dfrac{45}{180} $ , the G.C.D of the denominator and the numerator is 45.
 $ \begin{align}
  & 3\left| \!{\underline {\,
  45,180 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15,60 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,20 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,4 \,}} \right. \\
\end{align} $
The GCD is $ 3\times 3\times 5=45 $ .
Now we divide both the denominator and the numerator with 45 and get $ \dfrac{{}^{45}/{}_{45}}{{}^{180}/{}_{45}}=\dfrac{1}{4} $ .
The required number is $ \dfrac{1}{4} $ .
So, the correct answer is “ $ \dfrac{1}{4} $ ”.

Note: The process is similar for both proper and improper fractions. In case of mixed fractions, we need to convert it into an improper fraction and then apply the case. Also, we can only apply the process on the proper fraction part of a mixed fraction.