Ratio

In mathematics, a ratio is a relationship between two numbers of the same kind(i.e., objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two,which explicitly indicates how many times the first number contains the second.

Examples

The quantities being compared in a ratio might be physical quantities such as speed, or may simply refer to amounts of particular objects. A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made.

Older televisions have a 4:3 ratio which means that the height is 3/4 of the width. Widescreen TVs have a 16:9 ratio which means that the width is nearly double the height.

Fraction

If there are 2 oranges and 3 apples, the ratio of oranges to apples is shown as 2:3, whereas the fraction of oranges to total fruit is 2/5.

If concentrated orange is to be diluted with water in the ratio 1:4, then one part of orange is mixed with four parts of water, giving five parts total, so the fraction of orange is 1/5 and the fraction of water is 4/5.

Number of Terms

In general, a ratio of 2:3 means that the amount of the first quantity is 2/3(two thirds) of the amount of the second quantity. This pattern works with ratios with more than two terms. However, a ratio with more than two terms cannot be completely converted into a single fraction; a single fraction represents only one part of the ratio. If the ratio deals with objects or amounts of objects, this is often expressed as "for every two parts of the first quantity there are three parts of the second quantity".

If a mixture contains substances A, B, C & D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. This means that the total mixture contains 5/20 of A, 9/20 of B, 4/20 of C, and 2/20 of D. In terms of percentages, this is 25% A, 45% B, 20% C, and 10% D. (The ratio could have been written as 25:45:20:10 but this can be cancelled to the simplest form given above.)

Proportions

If the two or more ratio quantities encompass all of the quantities in a particular situation, for example two apples and three oranges in a fruit basket containing no other types of fruit, it could be said that "the whole" contains five parts, made up of two parts apples and three parts oranges. In this case, ,2/5 or 40% of the whole are apples and , 3/5 or 60% of the whole are oranges. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.

Reduction

Note that ratios can be reduced (as fractions are) by dividing each quantity by the common factors of all the quantities. This is often called "cancelling." As for fractions, the simplest form is considered to be that in which the numbers in the ratio are the smallest possible integers.

Thus the ratio 40:60 may be considered equivalent in meaning to the ratio 2:3 within contexts concerned only with relative quantities.

Mathematically, we write: "40:60" = "2:3" (dividing both quantities by 20).

Grammatically, we would say, "40 to 60 equals 2 to 3."

An alternative representation is: "40:60::2:3"

Grammatically, we would say, "40 is to 60 as 2 is to 3."

A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms.

Sometimes it is useful to write a ratio in the form 1:n or n:1 to enable comparisons of different ratios.

For example, the ratio 4:5 can be written as 1:1.25(dividing both sides by 4)

Alternatively,4:5 can be written as 08.:1 (dividing both sides by 5)

Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the colon, though, mathematically, this makes it a factor or multiplier.

Different units

Ratios are unit-less when they relate quantities which have the same or related units.

For example, the ratio 1 minute : 40 seconds can be reduced by changing the first value to 60 seconds. Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2.

Example:
Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles?
Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.

2) What is the ratio of videocassettes to the total number of items in the bag?
There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.

Comparing Ratios

To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.

Example:

Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.

Remember to be careful! Order matters!
A ratio of 1:7 is not the same as a ratio of 7:1.

Examples:

Are the ratios 7:1 and 4:81 equal? No!
7/1 > 1, but 4/81 < 1, so the ratios can't be equal.

Are 7:14 and 36:72 equal?
Notice that 7/14 and 36/72 are both equal to 1/2, so the two ratios are equal.


Reference:
http://en.wikipedia.org/wiki/Ratio
http://www.mathleague.com/help/ratio/ratio.htm