2.Fractions

Introduction

Fractions are commonly used to discuss portions or proportions of multiple objects. It consists of the numerator and the denominator e.g. In \tfrac{1}{2}, 1 is the numerator and 2 is the denominator. \tfrac{1}{2} simply means a half of something.  When using a fraction as a final answer, it must always be in proper form and in its simplest form. We can also say that \tfrac{1}{2} is the same as \tfrac{5}{10}.  This is because \tfrac{5}{10} can be simplified to \tfrac{1}{2}, which is its simplest form. Thus, these two fractions are also called equivalent fractions.

Fractions can also be seen as division, thus making it easy to convert fractions into decimals/integers.

\tfrac{1}{2} = 1 ÷ 2 = 0.5 \tfrac{5}{10} = 5 ÷ 10 = 0.5

Addition

When adding and subtracting fractions with each other, it is necessary for them to have the same denominator. \tfrac{1}{2} + \tfrac{1}{2} = (1+1)/2 = 2/2 = 1 / is used to represent division or in this case, fraction.

When we have a fraction with equal nominator and denominator, we must simplify them. 2 ÷ 2 = 1

Similarly: \tfrac{1}{2}+\tfrac{1}{2}+\tfrac{1}{2}+\tfrac{1}{2} = (1+1+1+1)/2 = 4/2 = 2

4/2 is an improper fraction, therefore we must simplify it. 4÷2 = 2 Fractions with different denominators cannot be added to each other. As such, we must convert them to another fraction so that they have equal denominators.

We do this by finding a common multiple. \tfrac{1}{2} + \tfrac{5}{10} The common multiple of 2 and 10 is 10.

2*5 = 10, 10*1 = 10

\tfrac{1}{2} = (1*5)/2*5 = \tfrac{5}{10}

\tfrac{5}{10}+\tfrac{5}{10} = 10/10 = 1

One way to find a common multiple is by multiplying denominators with each other.

1/9 + 1/8
The common denominator is 9*8 = 72
(1*8)/(9*8) + (1*9)/(8*9) = 8/72 + 9/72 = 17/72

Subtraction

When subtracting, the same rules apply as with addition. The denominators must be the same. However, note that it is possible for negative numbers and zero to be the outcome

\tfrac{1}{2} – Untitled = 1*2 / 2* 2   – Untitled = 2/4 – Untitled = Untitled

Another example \tfrac{1}{2} – \tfrac{1}{2} = (1-1)/2 = 0/2 = 0

Multiplication and Division

Multiplication and division are the simplest of fraction operations. \tfrac{1}{2}\tfrac{5}{10} = (1*5)/ (2*10) = 5 / 20 = 1/4There is no need for the denominator to be equal thus making it very simple. As for division, there is only one additional step. \tfrac{5}{10} ÷ \tfrac{1}{2} = \tfrac{5}{10} * 2/1 = (5*2)/(10*1) = 10/10 = 1 The fraction following the ÷ sign must be flipped upside down. Then the ÷ sign becomes multiplication.

Here is a video on addition and subtraction of fractions by a very reputable man.

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