Fractions are a way of representing a part of a whole. It can be defined as a ratio between two numbers: the numerator and denominator. For example, if we have a whole pizza and cut it into eight equal pieces, each piece will represent 1/8th of the pizza. In this case, 1 is the numerator, and 8 is the denominator.

## Types of fractions

There are three types of fractions – proper, improper, and mixed fractions.

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Proper fractions are those in which the numerator is smaller than the denominator. For example, $$\frac{2}{3}$$, $$\frac{3}{4}$$, $$\frac{4}{5}$$, etc.
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Improper fractions are those in which the numerator is greater than or equal to the denominator. For example, $$\frac{5}{4}$$, $$\frac{7}{3}$$, $$\frac{8}{5}$$, etc.
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Mixed fractions are a combination of a whole number and a fraction. For example, $$1\frac{1}{2}$$, $$2\frac{3}{4}$$, $$3\frac{2}{5}$$, etc.

A fraction consists of two parts – the numerator and the denominator. The numerator is the number on the top of the fraction, representing the number of equal parts we have. The denominator is the number on the bottom of the fraction, which means the total number of equal parts the whole is divided into.

For example, in the fraction $$\frac{3}{4}$$, 3 is the numerator, representing three out of four equal parts. 4 is the denominator, and it means the total number of equal parts that the whole is divided into.

## Equivalent fractions

Equivalent fractions represent the same value but have different numerators and denominators. To find equivalent fractions, we can multiply or divide both the numerator and the denominator by the same number.

For example, $$\frac{1}{2}$$ is equivalent to $$\frac{2}{3}$$, which is equivalent to $$\frac{3}{6}$$, which is equivalent to $$\frac{4}{8}$$, and so on.

## Simplifying fractions

To simplify or reduce a fraction, we need to find the Highest Common Factor (HCF) of the numerator and the denominator and divide both of them by it.

For example, to simplify $$\frac{6}{8}$$, we need to find the GCF of 6 and 8, which is 2. Then, we divide the numerator and denominator by 2, giving us $$\frac{3}{4}$$.

To add or subtract fractions, we need to find a common denominator. We can find the denominators' least common multiple (LCM) to do this. Once we have a common denominator, we can add or subtract the numerators.

For example, to add $$\frac{1}{4}$$ and $$\frac{2}{5}$$, we need to find a common denominator, which is 20 (4 × 5). Then, we convert both fractions to have a denominator of 20:

$$\frac{1}{4}$$ = $$\frac{5}{20}$$ $$\frac{2}{5}$$ = $$\frac{8}{20}$$

Now, we can add the numerators:
$$\frac{5}{20}$$ + $$\frac{8}{20}$$ = $$\frac{5}{20}$$

## Multiplying fractions

To multiply fractions, we multiply the numerators and multiply the denominators. For example, to multiply 2/3 and 3/4, we multiply the numerators:

To multiply fractions, we multiply the numerators and multiply the denominators.

For example, to multiply $$\frac{2}{3}$$ and $$\frac{3}{4}$$, we multiply the numerators:

$$\frac{2}{3}$$ × $$\frac{3}{4}$$ = $$\frac{6}{12}$$

Then, we can simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 6:
$$\frac{6}{12}$$ = $$\frac{1}{2}$$

## Dividing fractions

To divide fractions, we must invert the second fraction and then multiply.

To divide $$\frac{2}{3}$$ by $$\frac{4}{5}$$, we need to invert the second fraction, which gives us the following:

$$\frac{2}{3}$$ ÷ $$\frac{4}{5}$$ = $$\frac{2}{3}$$ × $$\frac{5}{4}$$

Then, we can simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 1:
$$\frac{2}{3}$$ × $$\frac{5}{4}$$ = $$\frac{10}{12}$$ = $$\frac{5}{6}$$

Fractions are an important concept in mathematics, and understanding them is crucial for success in many areas of math. Knowing how to find equivalent fractions, simplify fractions, add and subtract fractions, multiply fractions, and divide fractions are fundamental skills that students should master. With practice and persistence, anyone can become proficient in working with fractions.

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