# Simplest Form – Definition, Methods, Facts, Examples, FAQs

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## 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.

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.

## 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.

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 $5\frac{2}{4}$.

GCD(2, 4) = 2

$\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}$

Thus, $5\frac{2}{4} = 5\frac{1}{2}$

Example 2: Simplify $3\frac{10}{15}$.

$\frac{10}{15} = \frac{10\div 5}{15 \div 5} = \frac{2}{3}$

## 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}$, where a and b are the whole numbers. For converting the ratio in the lowest form, we use the same GCD method. Reduce 34 : 289 into simplest form. GCD(34, 289) = 17 Thus, by dividing 34 and 289 by 17 we get the ratio 34:289 in the simplest form 2 : 17. ## Facts about the Simplest Form of a Fraction • The simplest form is the smallest possible equivalent fraction of the number. • If both the numerator and denominator are prime numbers, then the fraction is in its simplest form. • Different expressions that refer to the same thing: 1. To find a fraction in the simplest form or reduced form 2. To simplify a fraction 3. To find the simplest form of a fraction 4. To find the lowest form of a fraction 5. To reduce a fraction to its lowest form 6. To find fractions in lowest terms ## Solved Examples on the Simplest Form of a Fraction 1. Write as a fraction in the simplest form: i)$\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$5\frac{25}{75}$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,$5\frac{25}{75} = 5\frac{1}{3}$## Practice Problems on the Simplest Form of a Fraction 1 ### On reducing$\frac{x^{2} \times 3^{4}}{x^{3} \times 3}$into the simplest form, we get$-\;27x\frac{x}{27}\frac{27}{x}27x$CorrectIncorrect Correct answer is:$\frac{27}{x}\frac{x^{2} \times 3^{4}}{x^{3} \times 3} = \frac{x \times x \times 3 \times 3 \times 3 \times 3}{x \times x \times x \times 3} = \frac{3 \times 3 \times 3}{x} = \frac{27}{x}$2 ### Which of the following fractions are NOT in the simplest form?$\frac{1}{9}\frac{3}{19}\frac{13}{52}\frac{13}{54}$CorrectIncorrect Correct answer is:$\frac{13}{52}\frac{13}{52}$can be further reduced as$\frac{13}{52} = \frac{13 \div 13}{52 \div 13} = \frac{1}{4}$3 ### What will be the simplest form of$\frac{96}{10}$?$\frac{48}{5}\frac{5}{48}\frac{24}{5}\frac{12}{5}$CorrectIncorrect Correct answer is:$\frac{48}{5}$GCF(96, 10) = 2$\frac{96 \div 2}{10 \div 2} = \frac{48}{5}$4 ### Find the value of b if the fraction$\frac{2}{b}$is in its simplest form. 4 5 6 8 CorrectIncorrect Correct answer is: 5 If$\frac{2}{b}\$ is in the simplest form, the numbers 2 and b must have only 1 as their common factor.
So, we can eliminate the options 4, 6, and 8.
GCD(2, 5) = 1