What Is Ratio??(20 MARKS!!)

A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another.

Ratios are typically unitless, as they relate quantities of the same dimension. A rate is a special kind of ratio in which the two quantities being compared are of different units. The units of a rate are the units of the first quantity "per" unit of the second — for example, a rate of speed or velocity can be expressed in "miles per hour".

In algebra, two quantities having a constant ratio are in a special kind of linear relationship called proportionality.

Ratio = 比率 or 圓周率 or 比例



In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.





[edit] Definition
More formally, the variable y is said to be proportional (or sometimes directly proportional) to the variable x, if there exists a constant non-zero number k such that


The relation is often denoted


and the constant ratio


is called the proportionality constant or constant of proportionality of the proportionality relation.


[edit] Examples
If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is proportional to its diameter, with the constant of proportionality equal to π.
On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations the points represent, with the constant of proportionality being the scale of the map.
The amount of force acting on a certain object from the gravity of the Earth at sea level is proportional to the object's mass, with the gravitational constant being the constant of proportionality.

[edit] Properties
Since


is equivalent to


it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality.






Retrieved from

http://en.wikipedia.org/wiki/Proportionality_%28mathematics%29

2007-04-01 18:08:39 補充：
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2007-04-01 18:10:04 補充：
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本文涉及比例的性質，一種數學關係。關於'比例'一詞的其它應用，請見比例 (消除歧義)。
在數學中，若兩個量的變化關係符合其中一個量是另一個量乘以一個常數，或等價地表達為兩者具有一個為常數的比率，則稱兩者是成比例的。

目錄 [显示]
1 定義
2 例子
3 性質
3.1 比例關係中，位於兩端的兩數之積等於位於中間的兩數之積
3.2 比例的其它性質
4 反比關係
5 指數比例和對數比例
6 確定比例關係的實驗方法
7 參見



[編輯] 定義
更正式地，若存在一非零常數 k 使


則稱變數 y 與變數 x 成比例（有時也稱為成正比）。

該關係通常表示為：


並稱該常數比率

k = y / x
為比例常數或比例關係中的比例恆量。


[編輯] 例子
假設某人以勻速運動，則其運動的距離是和運動的時間成正比的，該速度值即是所述的比例常數。
圓的周長與其直徑成正比，其中的比例常數等於π。
在按比例尺繪製的地圖上，地圖上任意兩點間的距離是和該兩點所代表的實際地點之間的距離成比例的，其比例常數即是繪製該地圖所使用的比例尺係數。
物理學中，地球的地心引力對在海平面上的某物體的作用力的數值與該物體的質量成正比，其比例常數即萬有引力常數。[1]
^ 譯注：此中文連結定向到「萬有引力常數」。目前該詞條的內容似乎不適合直接應用在這裡，這裡的重力常數是特定於地球海平面處的值g，建議在「萬有引力常數」中加入「(地球上的) 重力常數」一節。

[編輯] 性質
因為


等價於


因此可推出，若 y 與 x 具有比例常數為 k 的比例關係，則 x 也與 y 具有比例常數為 1/k 的比例關係。

若 y 與 x 成比例，則 y 作為 x 的一個函數的函數圖像會是一條穿過原點的直線，該直線的斜率等於其比例常數。


[編輯] 比例關係中，位於兩端的兩數之積等於位於中間的兩數之積
↔


[編輯] 比例的其它性質
↔

↔

↔

↔

若有且有則有


[編輯] 反比關係
在上面定義中，我們說有時稱兩個成比例的變數成正比例，這是為了和反比例關係相對應.

如果兩變數中，一個變數和另外一個變數的倒數成正比，或等價地，若這兩變數的乘積是一個常數，則稱這兩個變數是成反比例（或相反地變化）的。從而可繼續推出，若存在一非零常數 k 使


則變數 y 和變數 x 成反比。

反比例關係的概念基本上說明的是這樣一種關係，即當一個變數的值變大時，另一變數的值相應變小，而兩者之積總是保持為一常數（即比例常數）。

舉例來說，運動中的車輛走完一段路程所花費的時間是和這輛車運動的速度成反比的；在地上挖個坑所花的時間也（大致地）和雇來挖坑的人數成反比的。

在笛卡爾坐標平面上，兩個具有反比例關係的變數的圖形是一對雙曲線。該圖線上的每一點的 X 和 Y 坐標值之積總是等於比例常數 (k)。由於 k 非零，所以圖線不會與坐標軸相交。


[編輯] 指數比例和對數比例
若變數 y 與變數 x 的指數函數成正比，即：若存在非零常數 k 使

y = kax,
則稱 y 與 x 成指數比例。

類似地，若變數 y 與變數 x 的對數函數成正比，即：若存在非零常數 k 使


則稱 y 與 x 成對數比例。


[編輯] 確定比例關係的實驗方法
用實驗方法確定兩個物理量是否具有正比關係，可採用這樣的辦法，即進行多次測量並在笛卡爾坐標系中將這些測量結果用多個點來表示，而繪製出這些點的分佈圖形；如果所有點完全（或接近）地落在一條穿過原點 (0, 0) 的直線上，則這兩個變數（很有可能）具有比例常數等於該直線斜率的正比關係。

Ratio=比;比率;【數】比例
The ratio of students to teachers is 35:1.
學生和老師的比率是三十五比一。
The ratio of 15 to 5 is 3 to 1.
十五與五的比率是三比一。
We divided it in the ratio 3:1.
我們將它分為三與一之比。

比率,55

Ratios are typically unitless, as they relate quantities of the same dimension. A rate is a special kind of ratio in which the two quantities being compared are of different units. The units of a rate are the units of the first quantity "per" unit of the second — for example, a rate of speed or velocity can be expressed in "miles per hour".

Fractions and percentages are both specific applications of ratios. Fractions relate the part (the numerator) to the whole (the denominator) while percentages indicate parts per hundred.

A ratio can be written as two numbers separated by a colon (:) which is read as the word "to". For example, a ratio of 2:3 ("two to three") means that the whole is made up of 2 parts of one thing and 3 parts of another — thus, the whole contains five parts in all. To be specific, if a basket contains 2 apples and 3 oranges, then the ratio of apples to oranges is 2:3. If another 2 apples and 3 oranges are added to the basket, then it will contain 4 apples and 6 oranges, resulting in a ratio of 4:6, which is equivalent to a ratio of 2:3 (thus ratios "reduce" like regular fractions). In this case, 2/5 or 40% of the fruit are apples and 3/5 or 60% are oranges in the basket.

Note that in the previous example the proportion of apples in the basket is 2/5 ("two of five" fruits, "two out of five" fruits, "two fifths" of the fruits, or 40% of the fruits). Thus a proportion compares part to whole instead of part to part.

Throughout the physical sciences, ratios of physical quantities are treated as real numbers. For example, the ratio of 2π metres to 1 metre (say, the ratio of the circumference of a certain circle to its radius) is the real number 2π. That is, 2πm/1m = 2π. Accordingly, the classical definition of measurement is the estimation of a ratio between a quantity and a unit of the same kind of quantity. (See also the article on commensurability in mathematics.)

In algebra, two quantities having a constant ratio are in a special kind of linear relationship called proportionality.

More colloquially, a ratio is a value calculated by dividing one number by another. Five divided by two gives a "ratio" of 2.5. (More accurately, this gives a ratio of 2.5:1, but this shortcut disregards the latter half of the expression in favor of simpler notation.)

In the business world it is typical to use ratios to analyze financial statements. For example, the current ratio assesses liquidity, or time required for some asset to be converted to cash. The current ratio looks at current assets relative to current liabilities.

One indicator, or ratio, for strength or stability of revenue in government is own source revenues (property taxes, for example) divided by total revenues (property tax and outside grants). In some respects, a high ratio suggests safety and stability. Grants or intergovernmental revenues can be taken away and heavy reliance on these outside sources, which would produce a low ratio, can spell trouble for a state or local government.