Question

Express the following as a fraction and simplify: \[2.45\].
(a) \[\dfrac{{49}}{{20}}\]
(b) \[\dfrac{{20}}{{49}}\]
(c) \[\dfrac{{19}}{{20}}\]
(d) \[\dfrac{{20}}{{19}}\]

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 simplify it by dividing 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 solution:
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 \[2.45\] as a fraction, we get
\[ \Rightarrow 2.45 = \dfrac{{245}}{{100}}\]
Now, we will write \[\dfrac{{245}}{{100}}\] in the simplest form.
A fraction \[\dfrac{a}{b}\] is in the simplest form if \[a\] and \[b\] are co-prime.
We will divide the numerator and denominator 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, 245 and 100 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 2.45 = \dfrac{{\dfrac{{245}}{5}}}{{\dfrac{{100}}{5}}}\\ \Rightarrow 2.45 = \dfrac{{49}}{{20}}\end{array}\]
Since 49 and 20 are not divisible by any same number, we cannot simplify the fraction further.
Thus, 49 and 20 are co-prime numbers.
We have expressed \[2.45\] as a fraction in simplest form as \[\dfrac{{49}}{{20}}\].

\[\therefore \] The correct option is option (a).

Note:
We used the term co-prime numbers in the solution. Two numbers are called co-prime numbers if they do not share a common factor other than 1. For example, the factors of 49 are 1, 7, and 49. The different factors of 20 are 1, 2, 4, 5, 10, 20. We can observe that 49 and 20 are co-prime since they have no common factor other than 1.