While performing calculations on fractions, you are required to find the lowest form of a fraction because it makes the calculations simple and easy. There are three commonly used methods of reducing fractions to the lowest terms. Let’s start finding out what is the lowest form of fraction and how to arrive at it.
Lowest Form of Fraction
The lowest form also called the simplest form of a fraction is the fraction, whose numerator and denominator are relatively prime. It means the numerator and the denominator of the fraction do not have any common factor between them, apart from 1.
For example lowest or simplest form of the fraction $\frac{42}{108}$ is $\frac {2}{3}$. Here the numbers $7$ and $18$ are called relatively prime numbers because there are no common factors between $7$ and $18$ other than $1$.
Note: Two numbers are called relatively prime (or coprime) if there is no number greater than $1$ that divides them both (that is, their greatest common divisor is $1$).
Why Reducing Fractions To Lowest Form Required?
When you solve problems based on fractions, you perform operations like addition and subtraction, or multiplication and division. Performing these operations will be easy and less time-consuming if the fractions are in their lowest or simplest form.
Though we reduce the fractions to simplify, their value remains unchanged. The reduced fraction is equivalent to the original fraction. In fact, the original fraction and the reduced fractions form a pair of equivalent fractions.
For example, $\frac {1}{4}$ is the reduced form of $\frac {3}{12}$, and there is no difference in the value represented by these two fractions. Both represent the same value.
Reducing Fractions To Lowest Terms
Reducing fractions means converting a fraction to a form where the numerator and denominator do not have any factor in common between them other than $1$.
When the numerator and the denominator of a fraction have only $1$ as a factor, then the fraction is in its lowest or simplest form. A fraction in the lowest form cannot be reduced further.
For example $\frac {2}{3}$, $\frac {4}{11}$, $\frac{8}{21}$ are in their lowest or simplest form.
Note: $1$ is a factor of every number.
Methods of Reducing Fractions To Lowest Terms
Reducing a fraction means making a fraction as simple as possible. In order to find the reduced form of fractions, you have to get such values of the numerator and the denominator that are relatively prime (co-prime) to each other.
The following are the methods of reducing fractions to the lowest terms.
- Equivalent Fractions Method
- Prime Factorization Method
- Highest Common Factor Method
Equivalent Fractions Method
Equivalent fractions are the fractions that have the same value irrespective of their numerators and denominators. In this method, we try to find the common factors and cancel out those from the numerator and denominator thus reducing the fraction to its lowest form.
Steps For Equivalent Fractions Method
These are the steps to reduce fractions by the equivalent fractions method.
Step 1: Find any common factor of the numerator and the denominator
Step 2: Divide the numerator and denominator by the common factor
Step 3: Repeat steps 2 and 3 until there are no more common factors other than 1
Examples
Ex 1: $\frac {84}{126}$
In the given fraction both $84$ and $126$ are even numbers, so $2$ is a common factor.
$\frac {84}{126}$ = $\frac {42 \times 2}{63 \times 2}$
Canceling out $2$ from both numerator and denominator, we get $\frac {42}{63}$
Still, the fraction is not in the lowest form, as both the numerator and denominator are divisible by $3$.
$\frac {42}{63}$ = $\frac {14 \times 3}{21 \times 3}$
Canceling out $3$ from both numerator and denominator, we get $\frac {14}{21}$
Now also, the numerator and the denominator have a common factor $7$.
$\frac {14}{21}$ = $\frac {2 \times 7}{3 \times 7}$
Again removing $7$ from the numerator and the denominator, we get $\frac {2}{3}$
The numbers $2$ and $3$ are coprime and hence cannot be reduced further. Therefore, the lowest form of $\frac {84}{126}$ is $\frac {2}{3}$.
Ex 2: $\frac {315}{735}$
$\frac {315}{735} = \frac {63 \times 5}{147 \times 5}$
Canceling $5$, we get $\frac {63}{147}$
Again $\frac {63}{147} = {21 \times 3}{49 \times 3}$
Canceling $3$, the fraction further reduces to $\frac {21}{49}$
$\frac {21}{49} = \frac {3 \times 7}{7 \times 7}$
Finally canceling $7$, the fraction reduces to $\frac {3}{7}$
So, $\frac {3}{7}$ is the lowest form of the fraction $\frac {315}{735}$.
Prime Factorization Method
Prime factorization is a way of expressing any composite number as a product of its prime factors. In this method, the numerator and the denominator are converted to their prime factorization respectively.
Steps For Prime Factorization Method
These are the steps to reduce fractions by the prime factorization method.
Step 1: Find the prime factorization of the numerator and the denominator
Step 2: Cancel out the common prime factors of the numerator and the denominator
Step 3: The remaining numbers in the numerator and the denominator is the reduced fraction
Examples
Ex 1: $\frac {72}{108}$
Prime factorization of $72$ and $108$ in $\frac {72}{108}$ is $\frac {2 \times 2 \times 2 \times 3 \times 3}{2 \times 2 \times 3 \times 3 \times 3}$.
Canceling two $2$s and two $3$s from the numerator and the denominator, the fraction $\frac {72}{108}$ reduces to $\frac {2}{3}$.
Thus the lowest form of $\frac {72}{108}$ is $\frac {2}{3}$.
Ex 2: $\frac {180}{252}$
Prime factorization of $180$ and $252$ in $\frac {180}{252}$ is $\frac {2 \times 2 \times 3 \times 3 \times 5}{2 \times 2 \times 3 \times 3 \times 7}$.
Canceling two $2$s and two $3$s from the numerator and the denominator, the fraction $\frac {180}{252}$ reduces to $\frac {5}{7}$.
Thus the lowest form of $\frac {180}{252}$ is $\frac {5}{7}$.
Highest Common Factor Method
The highest common factor or HCF of two or more numbers is the greatest number that divides exactly the given numbers. Using HCF, you can reduce a fraction to its lowest or simplest form.
Steps For Highest Common Factor Method
These are the steps to reduce fractions by the HCF method.
Step 1: Find the highest common factor (HCF) of the numerator and the denominator
Step 2: Divide the numerator and denominator by the HCF. The fraction so obtained is the reduced fraction
Examples
Ex 1: $\frac {84}{174}$
HCF of $84$ and $174$ is $6$.
Dividing both the numerator and the denominator by $6$.
$\frac {84 \div 6}{174 \div 6}$ = $\frac {14}{29}$
The lowest form of $\frac {84}{174}$ is $\frac {14}{29}$.
Ex 2: $\frac {48}{152}$
HCF of $48$ and $152$ is $8$.
Dividing both the numerator and the denominator by $8$.
$\frac {48 \div 8}{152 \div 8}$ = $\frac {6}{19}$
The lowest form of $\frac {48}{152}$ is $\frac {6}{19}$.
Conclusion
Reducing a fraction to its lowest form generally forms the first step in calculations involving fractions. You can use any of the three methods, viz. equivalent fractions method, prime factorization method, or HCF method to reduce any fraction.
Practice Problems
- Find the lowest form of the fractions using the equivalent fractions method
- $\frac {21}{56}$
- $\frac {48}{64}$
- $\frac {44}{72}$
- Find the lowest form of the fractions using the prime factorization method
- $\frac {120}{360}$
- $\frac {28}{140}$
- $\frac {48}{168}$
- Find the lowest form of the fractions using the HCF method
- $\frac {7}{28}$
- $\frac {9}{72}$
