How to Find the Simplest Form of a Fraction Using HCF
Simplifying a fraction means reducing a fraction to its simplest form. A fraction is in its simplest form if its numerator and denominator have no common factors other than 1. An important step in solving fraction problems is reducing them to the simplest form. Though we simplify them, the fraction's value will remain unchanged. This means the simplified fraction and the actual fraction form a pair of equivalent fractions. In this article, you will learn about the simplest form of a fraction and how it can be converted.
Numerator and Denominator
The Simplest Form of a Fraction
A fraction is in its simplest form if its numerator and denominator are co-prime or have no common factors except 1. The simplest form of a fraction is equivalent to the given fraction. For example, the fraction $\dfrac{3}{4}$ is in the simplest form because 3 and 4 have no common factor except 1.
Let's try simplifying the fraction $\dfrac{3}{6}$ step by step.
Simplification of Fraction
In the example, the simplified form is obtained by dividing the numerator and denominator by 3, the greatest number that divides both numbers exactly.
Fun Facts: The simplest form is the number's smallest possible equivalent fraction.
Simplification of Fractions in Step-by-Step Method
There are different ways to simplify fractions, and all these methods are explained below.
Method 1:
To understand how to simplify fractions, adhere to the detailed steps below.
Step 1: List the factors of the numerator and denominator numbers.
Step 2: Determine the common factors of the numerator and denominator.
Step 3: Divide the numerator and denominator by the common factors until they have no common factor except 1. The fraction so obtained is in the simplest form.
Consider the fraction, $\dfrac{8}{24}$ and follow the steps mentioned below to understand how to simplify the fraction $\dfrac{8}{24}$.
Step 1: Write the factors of the numerator and denominator.
The factors 8 and 24 are
Factors of 8: 1, 2, 4, and 8
Factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24
Step 2: Determine the common factors of the numerator and denominator. The common factors of 8 and 24 are 1, 2, 4, and 8.
Step 3: Divide the numerator and denominator by the common factors until they have no common factor except 1. The fraction so obtained is in the simplest form. Let's start dividing by 2, then $\dfrac{8}{24} = \dfrac{\dfrac{8}{2}}{\dfrac{24}{2}} = \dfrac{4}{12}$.
We will divide by 2 until we can't go any further. So, we have
$\dfrac{\dfrac{4}{2}}{\dfrac{12}{2}}$
=$\dfrac{2}{6}$
$\dfrac{\dfrac{2}{2}}{\dfrac{6}{2}}$
= $\dfrac{1}{3}$
Conversion of a Fraction
A fraction can be converted into a decimal, a fraction can be converted into a percentage, improper fractions can be converted to mixed fractions, and vice versa.
$\dfrac{3}{5}, \dfrac{2}{3}, \dfrac{4}{7}, \dfrac{7}{11},$ etc., are the simplest forms of $\dfrac{6}{10}, \dfrac{8}{12}, \dfrac{20}{35},$ and $\dfrac{21}{33}$ respectively.
The simplest form examples are given below:
Percentage: 30%
Fraction: $\dfrac{3}{10}$
Decimal: 0.3
To go from a fraction to a percentage, we can convert to decimal first.
$\dfrac{3}{5} \rightarrow 0.6 \rightarrow 60 \%$
Solved Examples
Q 1. Find the simplest form of the fraction $\dfrac{11}{33}$.
Ans: The highest common factor of 11 and 33 is 11.
So, divide both the numerator and denominator by 11,
i.e $\dfrac{11}{33} = \dfrac{(11 \div 11)}{(33 \div 11)} = \dfrac{1}{3}$.
Therefore, the simplest form of $\dfrac{11}{33}$ is $\dfrac{1}{3}$.
Q 2. Convert $\dfrac{350}{175}$ into simplest form.
Ans: Here, $\dfrac{350}{175}=\dfrac{50}{25}=\dfrac{2}{1}$
Here, equivalent fractions of $\dfrac{350}{175}$ are $\dfrac{50}{25}$ and $\dfrac{2}{1}$. Also, these fractions have a numerator greater than the numerator, as you can see that $350>175,50>25$, and $2>1$. Besides this, the simplest form of the above fraction is $\dfrac{2}{1}$.
Q 2. Convert $\dfrac{200}{550}$ into simplest form.
Ans: Here, $\dfrac{200}{550}=\dfrac{20}{55}=\dfrac{4}{11}$
Here, equivalent fractions of $\dfrac{200}{550}$ are $\dfrac{20}{55}$ and $\dfrac{4}{11}$. Besides this, the simplest form of the above fraction is $\dfrac{4}{11}$.
Practice Problems
Q 1. Change $\dfrac{45}{60}$ to its simplest form.
Ans: $\dfrac{3}{4}$
Q 2. Change $\dfrac{15}{75}$ to its simplest form.
Ans: $\dfrac{1}{5}$
Summary
In this article, we have learned about the simplest fraction. It is just like simplifying a fraction. We have seen how we can convert the fraction into its simplest form by solving it. So to reduce the fractions, we just have to divide the numerator and denominator by their highest common factor. At the end of the article, we have added some solved examples and practice problem to have clear concepts regarding the simplest form.
FAQs on Simplest Form of a Fraction Explained Clearly
1. What is the simplest form of a fraction?
The simplest form of a fraction is when the numerator and denominator have no common factor other than 1. In this form, the fraction cannot be reduced further.
- A fraction is in simplest form if the HCF (GCD) of numerator and denominator is 1.
- Example: 3/4 is in simplest form because 3 and 4 have no common factors except 1.
- Example: 6/8 is not in simplest form because both are divisible by 2.
2. How do you reduce a fraction to its simplest form?
To reduce a fraction to its simplest form, divide both the numerator and denominator by their greatest common divisor (GCD).
- Step 1: Find the GCD of numerator and denominator.
- Step 2: Divide both by the GCD.
- Example: 12/18 → GCD is 6 → 12 ÷ 6 = 2 and 18 ÷ 6 = 3 → simplest form is 2/3.
3. What is the formula to find the simplest form of a fraction?
The formula to find the simplest form of a fraction is: (Numerator ÷ GCD) / (Denominator ÷ GCD).
- First calculate the GCD (Greatest Common Divisor).
- Divide both parts of the fraction by this value.
- Example: 20/30 → GCD = 10 → (20 ÷ 10)/(30 ÷ 10) = 2/3.
4. How do you know if a fraction is already in simplest form?
A fraction is already in simplest form if the numerator and denominator share no common factor other than 1. This means their GCD is 1.
- Check common factors of both numbers.
- If only 1 is common, the fraction cannot be reduced.
- Example: 5/9 is in simplest form because GCD(5,9) = 1.
5. Can you give an example of simplifying a fraction step by step?
Yes, simplifying a fraction step by step involves dividing by the greatest common factor until no common factors remain.
- Example: Simplify 24/36.
- Step 1: Find GCD of 24 and 36 = 12.
- Step 2: Divide both by 12 → 24 ÷ 12 = 2, 36 ÷ 12 = 3.
- Final answer: 2/3.
6. What is the difference between simplest form and lowest terms?
There is no difference between simplest form and lowest terms; both mean the fraction cannot be reduced further. They are interchangeable terms used in mathematics.
- Both require numerator and denominator to have GCD = 1.
- Example: 7/8 is in simplest form or lowest terms.
7. Why do we need to write fractions in simplest form?
We write fractions in simplest form to make them easier to understand, compare, and calculate. Simplified fractions are clearer and standard in mathematics.
- Helps in comparing fractions easily.
- Makes calculations like addition and subtraction simpler.
- Reduces chances of calculation errors.
8. How do you simplify improper fractions?
To simplify an improper fraction, divide numerator and denominator by their GCD, just like any other fraction. You may also convert it into a mixed number.
- Example: 15/9 → GCD = 3 → 15 ÷ 3 = 5, 9 ÷ 3 = 3 → simplest form is 5/3.
- As a mixed number: 5/3 = 1 2/3.
9. What are common mistakes when simplifying fractions?
A common mistake when simplifying fractions is not dividing both numerator and denominator by the same number. Both must be divided by their common factor.
- Incorrect: Dividing only the numerator.
- Incorrect: Cancelling digits instead of factors (e.g., 16/64 → not 1/4 by cancelling 6).
- Correct method: Use the GCD to reduce properly.
10. Can all fractions be written in simplest form?
Yes, every fraction can be written in a simplest form because any two integers have a greatest common divisor. If the GCD is 1, the fraction is already simplified.
- Example: 10/25 → GCD = 5 → simplest form is 2/5.
- Example: 3/7 → GCD = 1 → already in simplest form.
