Fractions and Their Operations

A fraction corresponds to a number of the form

\[ \displaystyle{\frac{a}{b}}\]

where \(a\) and \(b\) are integer numbers , and it can be thought as "\(a\) divided by \(b\)". For example, the numbers

\[ \displaystyle{\frac{3}{4}}, \displaystyle{\frac{8}{9}}, \displaystyle{\frac{-3}{4}}\]

are fractions. The only restriction for the fraction \( \displaystyle{\frac{a}{b}}\) is that \(b = \not 0\), because in that case the fraction is undefined .

Sum of Fractions

The easiest case is when the denominators coincide. In fact, in that case, we find that:

\[ \displaystyle{\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} }\]

This makes sense because \( \frac{a}{b} \) can be interpreted as "\(a\) times \(\frac{1}{b}\)", and hence, "\(a\) times \(\frac{1}{b}\)" plus "\(c\) times \(\frac{1}{b}\)" must be "\(a + c\) times \(\frac{1}{b}\)"

Example: The the sum

\[ \displaystyle{\frac{2}{3} + \frac{4}{3}}\]

is computed as

\[ \displaystyle{\frac{2}{3} + \frac{4}{3} = \frac{2+4}{3} = \frac{6}{3} = 2}\]

This shows that a fraction can become simply a number, in the way that \(6/3\) is simply 2.

Sum of Fractions with different numerator

This case is more difficult than the other one, because we cannot sum the numerators. What we need to do is to amplify the fractions (multiply both numerator and denominator by the same number) in such a way that they have the same denominator. In fact, consider the fraction

\[ \displaystyle{\frac{2}{3} }\]

We can amplify this fraction by 2:

\[ \displaystyle{\frac{2*2}{2*3} = \frac{4}{6}} \]

The resulting fraction is completely equivalent to the original one. How do we use this to add fractions?

Example: The the sum

\( \displaystyle{\frac{2}{3} + \frac{5}{6}}\)

is computed by first amplifying the first fraction by 2, which leads to \(4/6\), and then

\[ \displaystyle{\frac{2}{3} + \frac{5}{6} = \frac{4}{6} + \frac{5}{6} = \frac{4+5}{6} = \frac{9}{6}}\]

This last fraction can be simplified by dividing both numerator and denominator by 3, so the final answer is \(3/2\)

In general: The the sum of fractions is computed

\[ \displaystyle{\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}}\]

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