Question

How do you write the answer \[\dfrac{5}{12}\times \dfrac{24}{25}\] in simplest form?

Answer 1

Hint: We complete the multiplication where \[\dfrac{a}{b}\times \dfrac{c}{d}=\dfrac{ac}{bd}\]. Then we try to describe the relation between the denominator and the numerator to find the simplified form. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1.

Complete step-by-step solution:
We complete the multiplication of \[\dfrac{5}{12}\times \dfrac{24}{25}\] following the method of \[\dfrac{a}{b}\times \dfrac{c}{d}=\dfrac{ac}{bd}\].
So, \[\dfrac{5}{12}\times \dfrac{24}{25}=\dfrac{5\times 24}{12\times 25}=\dfrac{120}{300}\].
We need to find the simplified form of the proper fraction $\dfrac{120}{300}$.
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
This means we can’t eliminate any more common root from them other than 1.
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 $\dfrac{\dfrac{p}{d}}{\dfrac{q}{d}}$.
For our given fraction $\dfrac{120}{300}$, the G.C.D of the denominator and the numerator is 60.
$\begin{align}
  & 2\left| \!{\underline {\,
  120,300 \,}} \right. \\
 & 2\left| \!{\underline {\,
  60,150 \,}} \right. \\
 & 3\left| \!{\underline {\,
  30,75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  10,25 \,}} \right. \\
 & 1\left| \!{\underline {\,
  2,5 \,}} \right. \\
\end{align}$
Now we divide both the denominator and the numerator with 60 and get $\dfrac{\dfrac{120}{60}}{\dfrac{300}{60}}=\dfrac {2}{5}$.
Therefore, the simplified form of $\dfrac{120}{300}$ is $\dfrac{2}{5}$.

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.
We can also prime factorise the numbers as \[\dfrac{5}{12}\times \dfrac{24}{25}=\dfrac{5\times 2\times 12}{12\times 5\times 5}=\dfrac{2}{5}\].