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7.6 Addition and Subtraction of all fractions > No. 5

Complete the addition and subtraction box using sum and difference of fraction.

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Fraction (mathematics)

Fractions (from Latin: fractus, "broken") are numbers expressed as the ratio of two numbers, and are used primarily to express a comparison between parts and a whole.

The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on.A much later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator, the numerator representing a number of equal parts and the denominator telling how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

A still later development was the decimal fraction, now called simply a decimal, in which the denominator is a power of ten, determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus for 0.75 the numerator is 75 and the denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator.

A third kind of fraction still in common use is the percentage, in which the denominator is always 100. Thus 75% means 75/100.

Other uses for fractions are to represent ratios, and to represent division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (three to four) and the division 3 ÷ 4 (three divided by four).

Addition

The first rule of addition is that only like quantities can be added (Our solved example in mathguru.com uses this concept)

12:22 PM 8/5/2011Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

$Description: \tfrac24+\tfrac34=\tfrac54=1\tfrac14$ .

If $Description: \tfrac12$ of a cake is to be added to $Description: \tfrac14$ of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

$Description: http://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cake_fractions.svg/270px-Cake_fractions.svg.png$

Adding unlike quantities

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction.

For adding quarters to thirds, both types of fraction are converted to $Description: \tfrac14\times\tfrac13=\tfrac1{12}$ (twelfths).

Consider adding the following two quantities:

$Description: \tfrac34+\tfrac23$

First, convert $Description: \tfrac34$ into twelfths by multiplying both the numerator and denominator by three:

$Description: \tfrac34\times\tfrac33=\tfrac9{12}$ .

Note that $Description: \tfrac33$ is equivalent to 1, which shows that, $Description: \tfrac34$ is equivalent to the resulting $Description: \tfrac9{12}$ .

Secondly, convert $Description: \tfrac23$ into twelfths by multiplying both the numerator and denominator by four: $Description: \tfrac23\times\tfrac44=\tfrac8{12}$ . Note that $Description: \tfrac44$ is equivalent to 1, which shows that, $Description: \tfrac23$ is equivalent to the resulting $Description: \tfrac8{12}$ .

Now it can be seen that:

$Description: \tfrac34+\tfrac23$

is equivalent to:

$Description: \tfrac9{12}+\tfrac8{12}=\tfrac{17}{12}=1\tfrac5{12}$

(Our solved example in mathguru.com uses this concept)

This method can be expressed algebraically:

$Description: \tfrac{a}{b} + \tfrac {c}{d} = \tfrac{ad+cb}{bd}$

And for expressions consisting of the addition of three fractions:

$Description: \tfrac{a}{b} + \tfrac {c}{d} + \tfrac{e}{f} = \tfrac{a(df)+c(bf)+e(bd)}{bdf}$

Subtraction

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

$Description: \tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16$

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Fraction_(mathematics)

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.