Question

How do you write \[\dfrac{40}{32}\] in simplest form?

Answer 1

Hint: This question is from the topic of algebra. To make the solution of this question easier, we are going to use prime factorization for solving this question. So, we will perform the prime factorisation of 40 and 32 separately, then we will cancel off the same factors and reduce it to the simplest form.

Complete step-by-step answer:
Let us solve this question.
In this question, we have asked to make the term \[\dfrac{40}{32}\] in simplest form. Or, we can say, we have to simplify the term\[\dfrac{40}{32}\].
So, let us simplify the term \[\dfrac{40}{32}\].
For simplifying the term \[\dfrac{40}{32}\] or we can say for making the term \[\dfrac{40}{32}\] in simplest term, we will find the prime factorization of 40 and 32.
Let us first find the prime factorization of 40.
\[\begin{align}
  & 2\left| \!{\underline {\,
  40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Hence, the prime factorization of 40 will be
\[40=2\times 2\times 2\times 5\times 1\]
Now, finding for the prime factorization of 32, we get
\[\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Hence, prime factorization of 32 will be:
\[32=2\times 2\times 2\times 2\times 2\times 1\]
So, the term \[\dfrac{40}{32}\] which we have to simplify can also be written as
\[\dfrac{40}{32}=\dfrac{2\times 2\times 2\times 5\times 1}{2\times 2\times 2\times 2\times 2\times 1}\]
We can see in the above that three 2’s can be cancelled out. After cancelling out the three 2’s, we can the above term as
\[\Rightarrow \dfrac{40}{32}=\dfrac{2\times 2\times 2\times 5\times 1}{2\times 2\times 2\times 2\times 2\times 1}=\dfrac{5\times 1}{2\times 2\times 1}\]
The above term can also be written as
\[\Rightarrow \dfrac{40}{32}=\dfrac{5}{4}\]
Now, we have simplified the term \[\dfrac{40}{32}\], and we have got \[\dfrac{5}{4}\].
So, we can say that the simplest form of \[\dfrac{40}{32}\] is \[\dfrac{5}{4}\].

Note: For solving this type of question, we should know how to find the prime factorization.
For solving this question easily, we should just know the table of 8.
From the table of 8, we know that when 8 is multiplied with 5, we get the multiplication of 40.
And, when 8 is multiplied with 4, we get the multiplication of 32.
So, we can write
\[\dfrac{40}{32}=\dfrac{8\times 5}{8\times 4}\]
From the above, we can cancel out 8 on the right side of the equation. So, we can write
\[\Rightarrow \dfrac{40}{32}=\dfrac{5}{4}\]
So, the simplified value of \[\dfrac{40}{32}\] is \[\dfrac{5}{4}\].