Question

Express the following as a fraction and simplify:
(a) \[2.75\]
(b) \[3.625\]
(c) \[1.25\]
(d) \[6.5\]
(e) \[4.125\]
(f) \[3.5\]

Answer 1

Hint: Here, we need to express the given decimals as a fraction in the simplest form. We will write the decimals 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 answer:
(a)
First, we will write the given decimal as a fraction.
Rewriting \[2.75\] as a fraction, we get
\[ \Rightarrow 2.75 = \dfrac{{275}}{{100}}\]
Now, we will write \[\dfrac{{275}}{{100}}\] 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, 275 and 100 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 2.75 = \dfrac{{\dfrac{{275}}{5}}}{{\dfrac{{100}}{5}}}\\ \Rightarrow 2.75 = \dfrac{{55}}{{20}}\end{array}\]
The numbers 55 and 20 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 2.75 = \dfrac{{\dfrac{{55}}{5}}}{{\dfrac{{20}}{5}}}\\ \Rightarrow 2.75 = \dfrac{{11}}{4}\end{array}\]
Since 11 and 4 do not have any common factor, 11 and 4 are co-prime numbers.
Therefore, we have expressed \[2.75\] as a fraction in simplest form as \[\dfrac{{11}}{4}\].
(b)
Rewriting \[3.625\] as a fraction, we get
\[ \Rightarrow 3.625 = \dfrac{{3625}}{{1000}}\]
Now, we will write \[\dfrac{{3625}}{{1000}}\] in the simplest form.
We can observe that 3625 and 1000 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 3.625 = \dfrac{{\dfrac{{3625}}{5}}}{{\dfrac{{1000}}{5}}}\\ \Rightarrow 3.625 = \dfrac{{725}}{{200}}\end{array}\]
The numbers 725 and 200 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 3.625 = \dfrac{{\dfrac{{725}}{5}}}{{\dfrac{{200}}{5}}}\\ \Rightarrow 3.625 = \dfrac{{145}}{{40}}\end{array}\]
The numbers 145 and 40 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 3.625 = \dfrac{{\dfrac{{145}}{5}}}{{\dfrac{{40}}{5}}}\\ \Rightarrow 3.625 = \dfrac{{29}}{8}\end{array}\]
Since 29 and 8 do not have any common factor, 29 and 8 are co-prime numbers.
Therefore, we have expressed \[3.625\] as a fraction in simplest form as \[\dfrac{{29}}{8}\].
(c)
Rewriting \[1.25\] as a fraction, we get
\[ \Rightarrow 1.25 = \dfrac{{125}}{{100}}\]
Now, we will write \[\dfrac{{125}}{{100}}\] in the simplest form.
We can observe that 125 and 100 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 1.25 = \dfrac{{\dfrac{{125}}{5}}}{{\dfrac{{100}}{5}}}\\ \Rightarrow 1.25 = \dfrac{{25}}{{20}}\end{array}\]
The numbers 25 and 20 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 1.25 = \dfrac{{\dfrac{{25}}{5}}}{{\dfrac{{20}}{5}}}\\ \Rightarrow 1.25 = \dfrac{5}{4}\end{array}\]
Since 5 and 4 do not have any common factor, 5 and 4 are co-prime numbers.
Therefore, we have expressed \[1.25\] as a fraction in simplest form as \[\dfrac{5}{4}\].
(d)
Rewriting \[6.5\] as a fraction, we get
\[ \Rightarrow 6.5 = \dfrac{{65}}{{10}}\]
Now, we will write \[\dfrac{{65}}{{10}}\] in the simplest form.
We can observe that 65 and 10 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 6.5 = \dfrac{{\dfrac{{65}}{5}}}{{\dfrac{{10}}{5}}}\\ \Rightarrow 6.5 = \dfrac{{13}}{2}\end{array}\]
Since 13 and 2 do not have any common factor, 13 and 2 are co-prime numbers.
Therefore, we have expressed \[6.5\] as a fraction in simplest form as \[\dfrac{{13}}{2}\].
(e)
Rewriting \[4.125\] as a fraction, we get
\[ \Rightarrow 4.125 = \dfrac{{4125}}{{1000}}\]
Now, we will write \[\dfrac{{4125}}{{1000}}\] in the simplest form.
We can observe that 4125 and 1000 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 4.125 = \dfrac{{\dfrac{{4125}}{5}}}{{\dfrac{{1000}}{5}}}\\ \Rightarrow 4.125 = \dfrac{{825}}{{200}}\end{array}\]
The numbers 825 and 200 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 4.125 = \dfrac{{\dfrac{{825}}{5}}}{{\dfrac{{200}}{5}}}\\ \Rightarrow 4.125 = \dfrac{{165}}{{40}}\end{array}\]
The numbers 165 and 40 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 4.125 = \dfrac{{\dfrac{{165}}{5}}}{{\dfrac{{40}}{5}}}\\ \Rightarrow 4.125 = \dfrac{{33}}{8}\end{array}\]
Since 33 and 8 do not have any common factor, 33 and 8 are co-prime numbers.
Therefore, we have expressed \[4.125\] as a fraction in simplest form as \[\dfrac{{33}}{8}\].
(f)
Rewriting \[3.5\] as a fraction, we get
\[ \Rightarrow 3.5 = \dfrac{{35}}{{10}}\]
Now, we will write \[\dfrac{{35}}{{10}}\] in the simplest form.
We can observe that 35 and 10 are divisible by 5.
Dividing the numerator and denominator by 5, we get
\[\begin{array}{l} \Rightarrow 3.5 = \dfrac{{\dfrac{{35}}{5}}}{{\dfrac{{10}}{5}}}\\ \Rightarrow 3.5 = \dfrac{7}{2}\end{array}\]
Since 7 and 2 do not have any common factor, 7 and 2 are co-prime numbers.
Therefore, we have expressed \[3.5\] as a fraction in simplest form as \[\dfrac{7}{2}\].

Note: We used the term fractions and co-prime numbers in the 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.
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 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.