## 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!

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## 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.

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## Examples and Non-examples of Fractions in Simplest Form

Fractions in their simplest form | Fractions NOT in their simplest form |
---|---|

$\frac{2}{7}$ | $\frac{20}{70}$ |

$\frac{1}{9}$ | $\frac{5}{45}$ |

$\frac{23}{9}$ | $\frac{46}{18}$ |

## 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:
- To find a fraction in the simplest form or reduced form
- To simplify a fraction
- To find the simplest form of a fraction
- To find the lowest form of a fraction
- To reduce a fraction to its lowest form
- 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

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

### On reducing $\frac{x^{2} \times 3^{4}}{x^{3} \times 3}$ into the simplest form, we get

$\frac{x^{2} \times 3^{4}}{x^{3} \times 3} = \frac{x \times x \times 3 \times 3 \times 3 \times 3}{x \times x \times x \times 3} = \frac{3 \times 3 \times 3}{x} = \frac{27}{x}$

### Which of the following fractions are NOT in the simplest form?

$\frac{13}{52}$ can be further reduced as

$\frac{13}{52} = \frac{13 \div 13}{52 \div 13} = \frac{1}{4}$

### What will be the simplest form of $\frac{96}{10}$?

GCF(96, 10) = 2

$\frac{96 \div 2}{10 \div 2} = \frac{48}{5}$

### Find the value of b if the fraction $\frac{2}{b}$ is in its simplest form.

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

## Frequently Asked Question about the Simplest Form of a Fraction

**Do the fractions in the lowest terms mean the same thing as the simplest form?**

Yes, the lowest terms refer to the simplest form of a fraction.

**Which arithmetic operation is used to find the simplest form?**

We divide both the numerator and the denominator by their GCD to find the simplest form.

**Is the fraction in the simplest form always equivalent to the original fraction?**

Yes, in fact it will be the smallest equivalent fraction of the original fraction. For example, 2/3 is the smallest equivalent fraction of 10/15.

**How do I know if a fraction is already in its simplest form?**

To check if a fraction is in simplest form, you can find the greatest common divisor (GCD) of the numerator and denominator. If the GCD is 1, then the fraction is already in its simplest form.