Kamis, 06 Desember 2012

Arithmetic with fractions (Mathematics)

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number n, the fraction \tfrac{n}{n} = 1. Therefore, multiplying by \tfrac{n}{n} is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction \tfrac{1}{2}.
When the numerator and denominator are both multiplied by 2, the result is \tfrac{2}{4}, which has the same value (0.5) as \tfrac{1}{2}. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (\tfrac{2}{4}) make up half the cake (\tfrac{1}{2}).
Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime [that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible, in lowest terms, or in simplest terms. For example, \tfrac{3}{9} is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, \tfrac{3}{8} is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.
Using these rules, we can show that \tfrac{5}{10} = \tfrac{1}{2} = \tfrac{10}{20} = \tfrac{50}{100}.
A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, as the greatest common divisor of 63 and 462 is 21, the fraction \tfrac{63}{462} can be reduced to lowest terms by dividing the numerator and denominator by 21:
\tfrac{63}{462} = \tfrac{63 \div 21}{462 \div 21}= \tfrac{3}{22}
The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers.

Comparing fractions

Comparing fractions with the same denominator only requires comparing the numerators.
\tfrac{3}{4}>\tfrac{2}{4} because 3>2.
If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.
One way to compare fractions with different numerators and denominators is to find a common denominator. To compare \tfrac{a}{b} and \tfrac{c}{d}, these are converted to \tfrac{ad}{bd} and \tfrac{bc}{bd}. Then bd is a common denominator and the numerators ad and bc can be compared.
\tfrac{2}{3} ? \tfrac{1}{2} gives \tfrac{4}{6}>\tfrac{3}{6}
It is not necessary to determine the value of the common denominator to compare fractions. This short cut is known as "cross multiplying" – you can just compare ad and bc, without computing the denominator.
\tfrac{5}{18} ? \tfrac{4}{17}
Multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator:
\tfrac{5 \times 17}{18 \times 17} ? \tfrac{4 \times 18}{17 \times 18}
The denominators are now the same, but it is not necessary to calculate their value – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), \tfrac{5}{18}>\tfrac{4}{17}.
Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction.

Addition

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine 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:
\tfrac24+\tfrac34=\tfrac54=1\tfrac14.
If \tfrac12 of a cake is to be added to \tfrac14 of a cake, the pieces need to be
converted into comparable quantities, such as cake-eighths or cake-quarters.

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 twelfths, thus: \tfrac14\ + \tfrac13=\tfrac{1*3}{4*3}\ + \tfrac{1*4}{3*4}=\tfrac3{12}\ + \tfrac4{12}=\tfrac7{12}.
Consider adding the following two quantities:
\tfrac34+\tfrac23
First, convert \tfrac34 into twelfths by multiplying both the numerator and denominator by three: \tfrac34\times\tfrac33=\tfrac9{12}. Since \tfrac33 equals 1, multiplication by \tfrac33 does not change the value of the fraction.
Second, convert \tfrac23 into twelfths by multiplying both the numerator and denominator by four: \tfrac23\times\tfrac44=\tfrac8{12}.
Now it can be seen that:
\tfrac34+\tfrac23
is equivalent to:
\tfrac9{12}+\tfrac8{12}=\tfrac{17}{12}=1\tfrac5{12}
This method can be expressed algebraically:
\tfrac{a}{b} + \tfrac {c}{d} = \tfrac{ad+cb}{bd}
And for expressions consisting of the addition of three fractions:
\tfrac{a}{b} + \tfrac {c}{d} + \tfrac{e}{f} = \tfrac{a(df)+c(bf)+e(bd)}{bdf}
This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add \tfrac{3}{4} and \tfrac{5}{12} the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12.
\tfrac34+\tfrac{5}{12}=\tfrac{9}{12}+\tfrac{5}{12}=\tfrac{14}{12}=\tfrac76=1\tfrac16

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,
\tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16

Multiplication

Multiplying a fraction by another fraction

To multiply fractions, multiply the numerators and multiply the denominators. Thus:
\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12}
Why does this work? First, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths.
A short cut for multiplying fractions is called "cancellation". In effect, we reduce the answer to lowest terms during multiplication. For example:
\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{\cancel{2} ^{~1}}{\cancel{3} ^{~1}} \times \tfrac{\cancel{3} ^{~1}}{\cancel{4} ^{~2}} = \tfrac{1}{1} \times \tfrac{1}{2} = \tfrac{1}{2}
A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.

Multiplying a fraction by a whole number

Place the whole number over one and multiply.
6 \times \tfrac{3}{4} = \tfrac{6}{1} \times \tfrac{3}{4} = \tfrac{18}{4}
This method works because the fraction 6/1 means six equal parts, each one of which is a whole.

Mixed numbers

When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. For example:
3 \times 2\tfrac{3}{4} = 3 \times \left (\tfrac{8}{4} + \tfrac{3}{4} \right ) = 3 \times \tfrac{11}{4} = \tfrac{33}{4} = 8\tfrac{1}{4}
In other words, 2\tfrac{3}{4} is the same as \tfrac{8}{4} + \tfrac{3}{4}, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is 8\tfrac{1}{4}, since 8 cakes, each made of quarters, is 32 quarters in total.

Division

To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, \tfrac{10}{3} \div 5 equals \tfrac{2}{3} and also equals \tfrac{10}{3 \cdot 5} = \tfrac{10}{15}, which reduces to \tfrac{2}{3}. To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, \tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{1}{2} \times \tfrac{4}{3} = \tfrac{1 \cdot 4}{2 \cdot 3} = \tfrac{2}{3}.

Converting between decimals and fractions

To change a common fraction to a decimal, divide the denominator into the numerator. Round the answer to the desired accuracy. For example, to change 1/4 to a decimal, divide 4 into 1.00, to obtain 0.25. To change 1/3 to a decimal, divide 3 into 1.0000..., and stop when the desired accuracy is obtained. Note that 1/4 can be written exactly with two decimal digits, while 1/3 cannot be written exactly with any finite number of decimal digits.
To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits in the original decimal, omitting the decimal point. Thus 12.3456 = 123456/10000.

Converting repeating decimals to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite number of repeating decimals is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions.
The preferred way to indicate a repeating decimal is to place a bar over the digits that repeat, for example 0.789 = 0.789789789… For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. For example:
0.5 = 5/9
0.62 = 62/99
0.264 = 264/999
0.6291 = 6291/9999
In case leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:
0.05 = 5/90
0.000392 = 392/999000
0.0012 = 12/9900
In case a non-repeating set of decimals precede the pattern (such as 0.1523987), we can write it as the sum of the non-repeating and repeating parts, respectively:
0.1523 + 0.0000987
Then, convert the repeating part to a fraction:
0.1523 + 987/9990000
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