### Theory:

Equivalent fractions
A set of fractions have a different numerator and denominator, but their values equal the same.
Example:
The fractions like $\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8}$ have a different numerator and different denominator but represent the same value. These are called equivalent fractions.
Obtaining the equivalent fractions:
To find the equivalent fraction of a given fraction, multiply/divide the numerator and denominator by the same number.
Example:
1. Equivalent fraction by multiplication:
$$1/5$$ can be converted into its equivalent fraction as, $\frac{1×2}{5×2}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{2}{10};\phantom{\rule{0.147em}{0ex}}\frac{1×3}{5×3}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{3}{15\phantom{\rule{0.147em}{0ex}}}\phantom{\rule{0.147em}{0ex}}$

2. Equivalent fraction by division:

The equivalent fractions of $$frac{10}{20}$$, $\frac{10÷2}{20÷2}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{5}{10};\phantom{\rule{0.147em}{0ex}}\frac{10÷5}{20÷5}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{2}{4};\frac{10÷10}{20÷10}=\frac{1}{2}\phantom{\rule{0.147em}{0ex}}$
Steps to find the equivalent fraction if denominator is given:
1. Observe the relation of the two given denominators.

2. Divide the $$1$$st denominator by the $$2$$nd denominator.

3. Divide this product with the given fraction.
Let us an example to understand this concept.
Example:
Find the equivalent fraction of $$\frac{54}{30}$$ with denominator $$10$$.

As per the above theory, divide the $$1$$st denominator by the $$2$$nd denominator.

That is, $$30 ÷ 10 = 3$$.

No we divide this product with the given fraction.

$\frac{54}{30}=\frac{54÷3}{30÷3}=\frac{18}{10}$
Simplest Form of a Fraction
If the numerator and denominator of a fraction have no common factor other than $$1$$, it is said to be in its simplest (or lowest) form of a fraction.
Steps to convert a fraction into the simplest form:
1. Find the HCF (Highest Common Factor) of the numerator and denominator of a fraction.

2. Then divide them by their HCF.
Example:
Write in the simplest form $$\frac{12}{8}$$

1. The HCF of numerator $$12$$ and denominator $$8$$ is $$4$$.

2. Divide the numerator $$(12)$$ and the denominator $$(8)$$ by $$4$$ which is equal to $\frac{3}{2}$ that is $\frac{12÷4}{8÷4}=\frac{3}{2}$.

[Note: Here we cannot reduce the fraction $\frac{3}{2}$ further, since the common factor of numerator $$(3)$$ and the denominator $$(2)$$ is $$1$$].

Thus$\frac{3}{2}$ will be the simplest form of $$\frac{12}{8}$$.