Basics: A fraction is number of the form \(\frac{x}{y}\), where \(x\) and \(y\) are integers and y NOT EQUAL TO 0. Here, \(x\) is called numerator while \(y\) is called denominator.

Following are some properties of fractions:

Properties of Fractions:

• \(\frac{xa}{ya} = \frac{x}{y}\)
• \(\frac{x}{{-y}} = \frac{{-x}}{y} = -\frac{x}{y}\)
• \(\frac{{-x}}{{-y}} = \frac{x}{y}\)

For example,
\(\frac{3}{7} = \frac{(3)(2)}{(7)(2)} = \frac{6}{14}\)
\(\frac{15}{20} = \frac{(5)(3)}{(5)(4)} = \frac{3}{4}\), here \(5\) is called common factor

Rational numbers and Fractions

Type of Fractions

• Proper Fractions
  A fraction whose numerator is less than the denominator is called a proper fraction.
  \(\frac{2}{3}\), \(\frac{3}{5}\),... are the examples of a proper fraction
  
• Improper Fractions
  A fraction whose numerator is more than the denominator is called a proper fraction.
  \(\frac{3}{2}\), \(\frac{5}{3}\),... are the examples of a improper fraction
  
• Mixed Fractions
  A combination of whole number and a poper fraction is called mixed fraction.
  \(1\frac{2}{3}\), \(2\frac{3}{5}\), ... are the examples of a mixed fraction

Keep in mind: The value of a proper fraction is always less than 1. Whereas, the value of an improper fraction is always greater than 1.

An improper fraction can be converted into a mixed fraction by dividing the numerator by the denominator and using the quotient and the remainder.
For example, let us take an improper fraction \(\frac{7}{5}\). When \(7\) is divided by \(5\), we get \(1\) as the quotient and \(2\) as the remainder. Its equivalent mixed fraction is, \(1\frac{2}{5}\).

Hence, \(\frac{7}{5}\) = \(1\frac{2}{5}\)

A mixed fraction can also be converted into an improper fraction. This done by multiplying the denominator by the whole number and adding it to the numerator. This value is the numerator of its equivalent improper fraction and the denominator remains the same. For example, let us take a mixed fraction \(3\frac{5}{7}\). By multiplying the denominator 7 with the whole number 3 and adding it to the numerator 5 we get 26. Hence, an equivalent improper fraction of this mixed fraction is, \(\frac{26}{7}\)


Hence, \(3\frac{5}{7}\) = \(\frac{26}{7}\)

Operations with Fractions- Basics

Addition & Substraction

Following steps illustrate the steps to add or subtract two different fractions with same denominator:

• \(\frac{x}{y}\) +OR- \(\frac{z}{y}\)
• \(\frac{{x+OR-z}}{t}\)

For example:

In order to add \(\frac{2}{7}\) and \(\frac{3}{7}\), we simply add the numerators of two fractions. Hence the soluaiotn is \(\frac{5}{7}\).

In order to subtract \(\frac{7}{5}\) and \(\frac{9}{5}\), we simply subtract the numerators of two fractions. Hence the solutions is \(-\frac{2}{5}\).

If the denominators of two fractions are different, then a lengthy procedure is to be followed. This is explained in following steps:

• \(\frac{x}{y}\) +OR- \(\frac{z}{a}\)
• Find the LCM (Least common multiple) of the denominators
• Convert both the fractions to their equivalents with denominator as LCM
• Add or Subtract the numerators accordingly

For example:
In order to add \(\frac{5}{8}\) and \(\frac{7}{6}\) we will first find the LCM of the denominators, 8 and 6.

The LCM of 8 and 6 is 24. Next, we will convert each fraction into its equivalent fraction with 24 as the denominator.

\(\frac{(5)}{(8)} = \frac{(5)(3)}{(8)(3)} = \frac{15}{24}\). Similarly, \(\frac{7}{6} = \frac{(7)(4)}{(6)(4)} = \frac{28}{24}\).

Finally, we will add the numerators of the equivalent fractions. Hence, the final solution is \(\frac{43}{24}\).

In order to subtract \(\frac{2}{9}\) and \(\frac{4}{15}\) we will again find the LCM of the denominators, 9 and 15.

The LCM of 9 and 15 is 45. Next, we will convert each fraction into its equivalent with 45 as the denominator.

\(\frac{2}{9} = \frac{(2)(5)}{(9)(5)} = \frac{10}{45}\). Similarly, \(\frac{4}{15} = \frac{(4)(3)}{(15)(3)} = \frac{12}{45}\).

Finally, we will subtract the numerators of the equivalent fractions. Hence, the final solution is \(-\frac{2}{45}\).


Keep in mind:

In order to add or subtract two fractions, the denominators MUST be the same.


Multiplying and Divison
Multiplying and dividing two different fractions is simpler compared to adding or subtracting them.

In order to multiply two different fractions we just multiply numerators with numerators and denominators with denominators, irrespective of values of denominators.

For example:

Multiplication of \(\frac{2}{7}\) and \(\frac{6}{9}\) is \(\frac{(2)(6)}{(7)(9)}\). Hence, the solution of the multiplcation is, \(\frac{12}{63}\).

Procedure to divide two fractions is explained in following:

• Divide fraction1 from fraction2
• Invert the numerator and denominator of fraction1
• Multiply fraction2 and converted new fraction1

For example:
In order to divide \(\frac{3}{12}\) from \(\frac{5}{9}\). We first invert the numerators and denominators of the fraction1, \(\frac{3}{12}\). Finally we will multiply \(\frac{5}{9}\) and the new fraction \(\frac{12}{3}\). Hence, the finaly solution is \(\frac{60}{27}\).

Equivalent Fractions
Now that we have seen that a given fraction can have multiple equivalent fractions, we will see how to check if a given pair of fractions are equivalent of one other.

• Multiply the numerator of fraction1 with denominator of fraction2
• Multiply the numerator of fraction2 with denominator of fraction1
• If both values are same, then the pair of fractions are equivalent

For example:

Let us take \(\frac{2}{5}\) and \(\frac{14}{35}\). In order to see if they are equivalent we will multiply 2 with 35 and 5 with 14. We see that both values are 70, hence the fractions are equivalent.

Let us take \(\frac{2}{7}\) and \(\frac{6}{14}\). In order to see if they are equivalent we will multiply 2 with 14 and 7 with 6. We see that values are 28 and 42 respectively, hence the fractions are not equivalent.
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