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7.4 Comparing like and unlike fractions > No. 10

A problem on comparing unlike fractions.

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Explanation:

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).

In mathematics, the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero is called the set of rational numbers. This set is represented by the symbol Q.

Comparing fractions

Comparing fractions with the same denominator only requires comparing the numerators.

$Description: \tfrac{3}{4}>\tfrac{2}{4}$ Because 3>2.

One way to compare fractions with different denominators is to find a common denominator. To compare $Description: \tfrac{a}{b}$ and $Description: \tfrac{c}{d}$ , these are converted to $Description: \tfrac{ad}{bd}$ and $Description: \tfrac{bc}{bd}$ Then bd is a common denominator and the numerators ad and bc can be compared.

$Description: \tfrac{2}{3}$ ? $Description: \tfrac{1}{2}$ gives $Description: \tfrac{4}{6}>\tfrac{3}{6}$

As a short cut, known as "cross multiplying", you can just compare ad and bc, without computing the denominator.

$Description: \tfrac{5}{18}$ ? $Description: \tfrac{4}{17}$

Multiply 17 by 5 and multiply 18 by 4. Since 85 is greater than 72, $Description: \tfrac{5}{18}>\tfrac{4}{17}$ .

(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.