Question

Express the following as a fraction in the simplest form: \[0.365\].

Answer 1

Hint: Here, we need to express the given decimal as a fraction in the simplest form. We will write the decimal as a fraction, and then divide the numerator and denominator by the same number until they become co-prime. A fraction \[\dfrac{a}{b}\] is in the simplest form if \[a\] and \[b\] are co-prime.

Complete step-by-step answer:
A fraction is a number which represents a part of a group. It is written as \[\dfrac{a}{b}\], where \[a\] is called the numerator and \[b\] is called the denominator. The group is divided into \[b\] equal parts. The fraction \[\dfrac{a}{b}\] shows that \[a\] out of \[b\] equal parts of the group.
First, we will write the given decimal as a fraction.
Rewriting \[0.365\] as a fraction, we get
\[ \Rightarrow 0.365 = \dfrac{{365}}{{1000}}\]
Now, we will write \[\dfrac{{365}}{{1000}}\] in the simplest form.
A fraction \[\dfrac{a}{b}\] is in the simplest form if \[a\] and \[b\] are co-prime.
We will divide 365 and 1000 by the same number till they become co-prime.
We know that a number with 0 or 5 in the unit’s place is divisible by 5.
Therefore, 365 and 1000 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 0.365 = \dfrac{{\dfrac{{365}}{5}}}{{\dfrac{{1000}}{5}}}\\ \Rightarrow 0.365 = \dfrac{{73}}{{200}}\end{array}\]
Now, 73 is a prime number.
This means that 73 is divisible by only 1 and itself.
Since 200 is not divisible by 73, we cannot simplify the fraction further.
73 and 200 are co-prime.
Therefore, we have expressed \[0.365\] as a fraction in simplest form as \[\dfrac{{73}}{{200}}\].

Note: We used the terms prime and coprime in the solution. A number is a prime number which is divisible by 1 and itself only. For example, 73 is not divisible by any number other than 1 and 73. Two numbers are called co-prime numbers if they do not share any common factor other than 1. The factors of 73 are 1 and 73. The different factors of 200 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200. We can observe that 73 and 200 are co-prime since they have no common factor other than 1.