What Is the Simplest Form?
In mathematics, the simplest form refers to the most reduced or simplified representation of a fraction. It is when the numerator and denominator have no common factors other than 1.
A fraction written in the simplest form cannot be further reduced. On reducing the fraction into simplest form, the value of the fraction remains the same.
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Example: The fraction $frac{8}{12}$ is not in simplest form because both 8 and 12 can be divided by 4. By dividing both the numerator and denominator by 4, we get the simplified fraction $frac{2}{3}$, which is in its simplest form.
Simplifying fractions to their simplest form makes them easier to work with and compare. It provides a clear and concise representation of the relationship between the numerator and denominator.
We can further say that the simplified fraction and the actual fraction are equivalent fractions.
Example:
Simplest form of $frac{4}{8}$ is $frac{1}{2}$.
Simplest form of $frac{2}{4}$ is $frac{1}{2}$.
Simplest form of $frac{3}{6}$ is $frac{1}{2}$.
Simplest form of $frac{5}{10}$ is $frac{1}{2}$.
All these fractions are equivalent fractions and $frac{1}{2}$ is the smallest equivalent fraction among them!
Fractions in the Simplest Form: Definition
A fraction is said to be in its simplest form if the greatest common factor (GCF) of its numerator and denominator is 1.
A fraction is in its simplest form when the numerator and the denominator have no common factors besides one.
Examples and Non-examples of Fractions in Simplest Form
How to Reduce Fractions to their Simplest Form
We can simplify the fractions by two methods. Let’s check them out.
Method 1 : Repeated division by Common Factors
Try to divide both the numerator and denominator by their common factors, until we cannot go any further. This method is a little bit tedious.
Example: $frac{35}{105}$.
Factors of 35 = 1, 5, 7, and 35
Factors of 105 = 1, 3, 5, 7, 15, 21, 35 and 105.
Common factors of 35 and 105 = 1, 5, 7, and 35
Divide both the numerator and denominator by 5.
$frac{35}{105} = frac{35 div 5}{105 div 5} = frac{7}{21}$
Now, divide by 7.
$frac{7}{21} = frac{7 div 7}{21 div 7} = frac{1}{3}$
We cannot divide further since the common factor of 1 and 3 is 1.
So, the simplest form of $frac{35}{105}$ is $frac{1}{3}$.
Method 2: GCD Method
Divide both the numerator and denominator by their Greatest Common Divisor (GCD) or GCF.
Example: Reduce $frac{98}{126}$ into the simplest form.
Step 1: Find the GCF of the numerator and the denominator.
Factors of 98: 1, 2, 7, 14, 49 and 98.
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Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63 and 126
Common factors of 98 and 126: 1, 2, 7, 14
GCF of 98 and 126 is 14.
Step 2: Divide the numerator and denominator by the G.C.F. The fraction obtained after dividing is in the simplest form.
$frac{98}{126} = frac{98 div 14}{126 div 14} = frac{7}{9}$
Thus, $frac{7}{9}$ is the simplest form of the fraction $frac{98}{126}$.
How to Reduce Mixed Fractions in the Simplest Form
A mixed number or a mixed fraction is a combination of a whole number and a proper fraction. To simplify a mixed fraction, we simplify the fractional part only. Reduce the fraction portion to lowest terms by finding the GCD.
Example 1: Simplify $5frac{2}{4}$.
GCD(2, 4) = 2
$frac{2}{4} = frac{2 div 2}{4 div 2} = frac{1}{2}$
Thus, $5frac{2}{4} = 5frac{1}{2}$
Example 2: Simplify $3frac{10}{15}$.
$frac{10}{15} = frac{10div 5}{15 div 5} = frac{2}{3}$
Simplest Form of Fractions with Exponents
To reduce the fractions having numerators and denominators with exponents, we expand the expressions using exponents in the numerator and denominator.
Example 1: $frac{5^{6}}{5^{3}}$
We will express the numerator and denominator as the product of numbers and then cancel out the common numbers.
$frac{5^{6}}{5^{3}} = frac{5 times 5 times 5 times 5 times 5 times 5}{5 times 5 times 5} = 125$
Example 2: $frac{3^{5}}{6^{2}} = frac{3 times 3 times 3 times 3 times 3}{6 times 6} = frac{243}{36} = frac{27}{4}$
Simplest Form of Fractions with Variables
To reduce the fractions with variables into the simplest form, use the expanded form of each expression in the numerator and denominator.
Example 1: Simplify frac{x^{5};y^{6}}{x^{3};y^{2}}$
The first step is to express the numerator and denominator as the product of variables.
$frac{x^{5};y^{6}}{x^{3};y^{2}} = frac{x times x times x times x times x times y times y times y times y times y times y}{x times x times x times y times y}$
The next step is to cancel out the common variables and write what is left.
$frac{x^{5};y^{6}}{x^{3};y^{2}} = frac{x^{2}}{y^{4}}$
Note: When dividing indices with the same base, subtract the powers and get the direct answer.
$frac{x^{m}}{x^{n}} = x^{m-n}$
Ratio in the Simplest Form
A ratio is the comparison of two quantities of the same kind represented in the form of a : b or $frac{a}{b}$, where a and b are the whole numbers.
For converting the ratio in the lowest form, we use the same GCD method.
Reduce 34 : 289 into simplest form.
GCD(34, 289) = 17
Thus, by dividing 34 and 289 by 17 we get the ratio 34:289 in the simplest form 2 : 17.
Facts about the Simplest Form of a Fraction
Solved Examples on the Simplest Form of a Fraction
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1. Write as a fraction in the simplest form:
i) $frac{4}{16}$
ii) $frac{24}{60}$
iii) $frac{10}{24}$
iv) $frac{12}{20}$
Solution:
i) Given fraction $= frac{4}{16}$
GCD of 4 and 16 is 4.
$frac{4}{16} = frac{4 div 4}{16 div 4} = frac{1}{4}$
ii) Given fraction $= frac{24}{60}$
GCD of 24 and 60 is 12.
$frac{24}{60} = frac{24 div 12}{60 div 12} = frac{2}{5}$
$frac{24}{60}$ simplified is $frac{2}{5}$.
iii) Given fraction $= frac{10}{24}$
GCD of 10 and 24 is 2.
$frac{10}{24} = frac{10 div 2}{24 div 2} = frac{5}{12}$
$frac{10}{24}$ simplified is$frac{5}{12}$.
iv) Given fraction $= frac{12}{20}$
GCD of 12 and 20 is 4.
$frac{12}{20} = frac{12 div 4}{20 div 4} = frac{3}{5}$
$frac{12}{20}$ simplified is $frac{3}{5}$.
2. What is the simplest form of 24:36?
Solution:
GCD of 24 and 36 is 12.
$frac{24}{36} = frac{24 div 12}{36 div 12} = frac{2}{3}$
So, the simplest form of 24:36 is equal to 2:3.
3. Reduce the mixed fraction $5frac{25}{75}$ in the simplest form.
Solution:
Focus on the fractional part $frac{25}{75}$.
GCD of 25 and 75 = 25
$frac{25}{75} = frac{25 div 25}{75 div 25} = frac{1}{3}$
Thus, $5frac{25}{75} = 5frac{1}{3}$
Practice Problems on the Simplest Form of a Fraction
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