Group 4

Ratio

In general, a ratio is a way of concisely showing the relationship between two quantities of something. The most formal way of stating a ratio is by separating the two quantities with a colon (:) although sometimes a division sign (/) is used in place of the colon.

For a ratio to have meaning, both numbers must be nonzero.

2) In mathematics, a ratio is a quotient used to compare quantities of the same units of measure.

3)Ratio is able to solve many mathematics questions.

4)Ratio can be changed to forms of fraction.

Basic ratio problems Youtube

From Youtube---Ratio


ratio2.png

I would like to see your smile when you look at our work.


Image from collaborativejourneys.com

Fraction

Fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by another whole number.

We got:Proper fraction,improper fraction,equivalent fraction and mixed fraction

Proper fraction- the numerator (top of fraction) is smaller than the denominator (bottom of fraction)

Improper fraction-the numerator (top) is equal to or larger than the denominator (bottom)

Equivalent fraction-a fraction that can be reduced to have the same value as another

Mixed fraction-a mix of a whole number and a fraction.

Fractions-Youtube

external image fractions.jpg

Image from http://ms-abuboo.com/mathwebsites.htm


The difference between a fraction and a ratio:

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, however in theory any number of quantities can be compared.However, a fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus, a ratio and a fraction is different.



The image of ratio and fraction



candy.jpeg

image from beaconlearningcenter.com

Ratio:The green sweets to the red sweets to the orange sweets is 7:6:3

Fraction:The green sweets is 7/6 of the red sweets.The red sweets is 6/3 of the orange sweets.The green sweets is 7/3 of the orange sweets.


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, when comparing the quantities of a two-quantity ratio, this can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount/size/volume/number of the first quantity will be that of the second quantity. This pattern also holds in ratios with more than two terms; however, a ratio with more than two terms cannot be converted into a single fraction, as a single fraction can represent two quantities. This pattern works with ratios with more than two terms.
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, , or 40% of the whole are apples and , 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 may be considered equivalent in meaning to the ratio within contexts concerned only with relative quantities.
Mathematically, we write: " " " " (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.
For example, the ratio can be written as (dividing both sides by 4)
Alternatively, can be written as (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.

fractions.gif

Group 2

The difference between a ratio and a fraction is:

A ratio is a relationship between two numbers of the same kind, usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the of the two, which explicitly indicates how many times the first number contains the second but a fraction is a number that can represent part of a whole.

The difference between ratio and fraction

A fraction (from Latin: fractus, "broken") is a number that can represent part of 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.[1] 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.
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 (vulgar) fractions is called the set of rational numbers, and is represented by the symbol Q.

Mixed numbers
A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; for example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other:
2+tfrac{3}{4}=2tfrac{3}{4}
2+tfrac{3}{4}=2tfrac{3}{4}
.

An improper fraction can be thought of as another way to write a mixed number; consider the
2tfrac{3}{4}
2tfrac{3}{4}
example below.

We can imagine that the two entire cakes are each divided into quarters, so that the denominator for the whole cakes is the same as the denominator for the parts. Then each whole cake contributes
tfrac{4}{4}
tfrac{4}{4}
to the total, so
tfrac{4}{4}+tfrac{4}{4}+tfrac{3}{4}=tfrac{11}{4}
tfrac{4}{4}+tfrac{4}{4}+tfrac{3}{4}=tfrac{11}{4}
is another way of writing
2tfrac{3}{4}
2tfrac{3}{4}
.

A mixed number can be converted to an improper fraction in three steps:

  1. Multiply the whole part by the denominator of the fractional part.
  2. Add the numerator of the fractional part to that product.
  3. The resulting sum is the numerator of the new (improper) fraction, with the 'new' denominator remaining precisely the same as for the original fractional part of the mixed number.
Similarly, an improper fraction can be converted to a mixed number:
  1. Divide the numerator by the denominator.
  2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
  3. The new denominator is the same as that of the original improper fraction.
Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number, the results of the new fraction is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. That is, they retain the same integrity - the same balance or proportion. This is true because for any non-zero number n,
tfrac{n}{n} = 1
tfrac{n}{n} = 1
. Therefore, multiplying by
tfrac{n}{n}
tfrac{n}{n}
is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider the fraction
tfrac{1}{2}
tfrac{1}{2}
: when the numerator and denominator are both multiplied by 2, the result is
tfrac{2}{4}
tfrac{2}{4}
, which has the same value (0.5) as
tfrac{1}{2}
tfrac{1}{2}
. To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together (
tfrac{2}{4}
tfrac{2}{4}
) make up half the cake (
tfrac{1}{2}
tfrac{1}{2}
).




external image images?q=tbn:ANd9GcQ9ixTvMMNXF3Vxa2ixu5Z3IUUWDeI62rZqXvGaPNOpn0vtmDqcIw

The ratio of the green area to the white area to the red area is 1 to 1 to1. This is because all of the three areas are the same.

Image from http://www.google.com.sg/images?um=1&hl=en&biw=1024&bih=499&tbs=isch:1&sa=1&q=flag+of+italy&aq=f&aqi=g4g-m6&aql=&oq=

Arithmethic with fractions

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

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=1tfrac14
tfrac24+tfrac34=tfrac54=1tfrac14
.
Adding unlike quantities

If
tfrac12
tfrac12
of a cake is to be added to
tfrac14
tfrac14
of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.
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
tfrac14timestfrac13=tfrac1{12}
tfrac14timestfrac13=tfrac1{12}
(twelfths).

Consider adding the following two quantities:
tfrac34+tfrac23
tfrac34+tfrac23

First, convert
tfrac34
tfrac34
into twelfths by multiplying both the numerator and denominator by three:
tfrac34timestfrac33=tfrac9{12}
tfrac34timestfrac33=tfrac9{12}
. Note that
tfrac33
tfrac33
is equivalent to 1, which shows that
tfrac34
tfrac34
is equivalent to the resulting
tfrac9{12}
tfrac9{12}
.

Secondly, convert
tfrac23
tfrac23
into twelfths by multiplying both the numerator and denominator by four:
tfrac23timestfrac44=tfrac8{12}
tfrac23timestfrac44=tfrac8{12}
. Note that
tfrac44
tfrac44
is equivalent to 1, which shows that
tfrac23
tfrac23
is equivalent to the resulting
tfrac8{12}
tfrac8{12}
.

Now it can be seen that:
tfrac34+tfrac23
tfrac34+tfrac23

is equivalent to:
tfrac9{12}+tfrac8{12}=tfrac{17}{12}=1tfrac5{12}
tfrac9{12}+tfrac8{12}=tfrac{17}{12}=1tfrac5{12}

This method can be expressed algebraically:
tfrac{a}{b} + tfrac {c}{d} = tfrac{ad+cb}{bd}
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}
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,

tfrac23-tfrac12=tfrac46-tfrac36=tfrac16
tfrac23-tfrac12=tfrac46-tfrac36=tfrac16

Multiplication
Multiplying by a whole number
Considering the cake example above, if you have a quarter of the cake and you multiply the amount by three, then you end up with three quarters. We can write this numerically as follows:
textstyle{3 times {1 over 4} = {3 times 1 over 4} = {3 over 4}},!
textstyle{3 times {1 over 4} = {3 times 1 over 4} = {3 over 4}},!

As another example, suppose that five people work for three hours out of a seven hour day (i.e. for three sevenths of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 sevenths of a day is a whole day and 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of a day. Numerically:
textstyle{5 times {3 over 7} = {15 over 7} = 2{1 over 7}} ,!
textstyle{5 times {3 over 7} = {15 over 7} = 2{1 over 7}} ,!

Mixed numbers
When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. For example:
textstyle{3 times 2{3 over 4} = 3 times left ({{8 over 4} + {3 over 4}} right ) = 3 times {11 over 4} = {33 over 4} = 8{1 over 4}},!
textstyle{3 times 2{3 over 4} = 3 times left ({{8 over 4} + {3 over 4}} right ) = 3 times {11 over 4} = {33 over 4} = 8{1 over 4}},!

In other words,
textstyle{2{3 over 4}}
textstyle{2{3 over 4}}
is the same as
textstyle{({8 over 4} + {3 over 4})}
textstyle{({8 over 4} + {3 over 4})}
, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is
textstyle{8{1 over 4}}
textstyle{8{1 over 4}}
, since 8 cakes, each made of quarters, is 32 quarters in total.

Division
Division by a fraction is done by multiplying the dividend by the reciprocal of the divisor, in accordance with the identity
m div frac{a}{b} = m times frac{b}{a}.
m div frac{a}{b} = m times frac{b}{a}.


A proof for the identity, from fundamental principles, can be given as follows:
m ; div  ; frac{a}{b}  = frac{m}{frac{a}{b}} = m ; times ; frac{1}{frac{a}{b}} = m ; times ; left (frac{a}{b} right )^{-1} = m ; times ; frac{frac{1}{a}}{frac{1}{b}}  = m ; times ; frac{1}{a} ; times ; frac{1}{frac{1}{b}} = m ; times ; frac{1}{a} ; times ; b = m ; times ; frac{b}{a}.
m ; div ; frac{a}{b} = frac{m}{frac{a}{b}} = m ; times ; frac{1}{frac{a}{b}} = m ; times ; left (frac{a}{b} right )^{-1} = m ; times ; frac{frac{1}{a}}{frac{1}{b}} = m ; times ; frac{1}{a} ; times ; frac{1}{frac{1}{b}} = m ; times ; frac{1}{a} ; times ; b = m ; times ; frac{b}{a}.


For the ratio,
In mathematics, a ratio is a relationship between two numbers of the same kind[1] (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,[2] which explicitly indicates how many times the first number contains the second.
[3]
Notation and terminology
The ratio of numbers A and B can be expressed as:[4]
-the ratio of A to B
-A is to B
-A:B
The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.
The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. this latter form, when spoken or written in the English language, is often expressed as
A is to B as C is to D.
Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.[5]
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, when comparing the quantities of a two-quantity ratio, this can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount/size/volume/number of the first quantity will be
tfrac{2}{3}
tfrac{2}{3}
that of the second quantity. This pattern also holds in ratios with more than two terms; however, a ratio with more than two terms cannot be converted into a single fraction, as a single fraction can represent two quantities. 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,
tfrac{2}{5}
tfrac{2}{5}
, or 40% of the whole are apples and
tfrac{3}{5}
tfrac{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.

Reductions
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.
Grammatically, we would say, "40 to 60 equals 2 to 3."
Alternatively,
 4:5
4:5
can be written as
 0.8:1
0.8:1
(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.
All extracted from Wikipedia by Dominic Tey

external image cupcakes.jpg
look at the picture above!!
what is the ratio of the pink butterfly to the total number of butterfly?? ans:2:7
external image smsmeafl.gif what is the ratio of the yellow strips to the total number of strips? 5:10
GROUP 2

Fractions --x/y Ratio -- x:y A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, however in theory any number of quantities can be compared. Mathematically they are represented by separating each quantity with a colon, for example the ratio 2:3, which is read as the ratio "two to three". A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus 3⁄4 represents three divided by four, in decimals 0.75, as a percentage 75%.

from yahoo.com

GROUP 10

Fraction


    • An expression that indicates the quotient of two quantities, such as 1/3.
    • A disconnected piece; a fragment.
    • A small part; a bit: moved a fraction of a step.
    • A chemical component separated by fractionation.

Read more: http://www.answers.com/topic/fraction#ixzz1DjHGxzcO

Ratio


A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.

Difference between a ratio and fraction


A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another.
A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus 3⁄4 represents three divided by four, in decimals 0.75, in percents 75%. The three equal parts of the cake are 75% of the whole cake.

The picture below has a ratio of squares to circles which is 2:5 respectively . The fraction is 2/7 of it is square and 5/7 of it is circle .
D1_ratio_example.JPG

Group 6

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.
  • Relation in degree or number between two similar things.
  • The relative value of silver and gold in a currency system that is bimetallic.
  • Mathematics. A relationship between two quantities, normally expressed as the quotient of one divided by the other: The ratio of 7 to 4 is written 7:4 or 7/4.

external image _wsb_298x255_Valentines+Day+cupcakes.JPGThe ratio of the red roses cupcake is to the total cupcakes is 36:18.

colour-stripes-symmetrical-design.jpgThe ratio of the orange stripes to the total stripes is 1:3.

Fraction

external image fraction_strips.jpgThis is a picture of fractions.

  • A fraction (from Latin: fractus, "broken") is a number that can represent part of a whole.

  • 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.

Video from www.youtube.com



Differences between a ratio and a fraction


A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, however in theory any number of quantities can be compared. Mathematically they are represented by separating each quantity with a colon, for example the ratio 2:3, which is read as the ratio "two to three".

A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus 3⁄4 represents three divided by four, in decimals 0.75, as a percentage 75%.
Video from: www.youtube.com
Sometimes we use

Mixed numbers

A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; for example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: 2+3/4=
2tfrac{3}{4}
2tfrac{3}{4}

An improper fraction can be thought of as another way to write a mixed number; consider the
2tfrac{3}{4}
2tfrac{3}{4}
example below.
We can imagine that the two entire cakes are each divided into quarters, so that the denominator for the whole cakes is the same as the denominator for the parts. Then each whole cake contributes 4/4 to the total, so 4/4+4/4+3/4=11/4 is another way of writing
2tfrac{3}{4}
2tfrac{3}{4}
.
A mixed number can be converted to an improper fraction in three steps:
  1. Multiply the whole part by the denominator of the fractional part.
  2. Add the numerator of the fractional part to that product.
  3. The resulting sum is the numerator of the new (improper) fraction, with the 'new' denominator remaining precisely the same as for the original fractional part of the mixed number.
Similarly, an improper fraction can be converted to a mixed number:
  1. Divide the numerator by the denominator.
  2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
  3. The new denominator is the same as that of the original improper fraction.

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number, the results of the new fraction is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. That is, they retain the same integrity - the same balance or proportion. This is true because for any non-zero number n,
tfrac{n}{n} = 1
tfrac{n}{n} = 1
. Therefore, multiplying by
tfrac{n}{n}
tfrac{n}{n}
is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider the fraction
tfrac{1}{2}
tfrac{1}{2}
: when the numerator and denominator are both multiplied by 2, the result is
tfrac{2}{4}
tfrac{2}{4}
, which has the same value (0.5) as external image 3a96e71dfcb71605c92ac270bfd4d4cc.png. To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together (
tfrac{2}{4}
tfrac{2}{4}
) make up half the cake (
tfrac{1}{2}
tfrac{1}{2}
).
For example:
tfrac{1}{3}
tfrac{1}{3}
,
tfrac{2}{6}
tfrac{2}{6}
,
tfrac{3}{9}
tfrac{3}{9}
and
tfrac{100}{300}
tfrac{100}{300}
are all equivalent fractions.
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 fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in its lowest or simplest terms. For instance,
tfrac{3}{9}
tfrac{3}{9}
is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast,
tfrac{3}{8}
tfrac{3}{8}
is in lowest terms—the only number that is a factor of both 3 and 8 is 1.
Any fraction can be fully reduced to its lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, the greatest common divisor of 63 and 462 is 21, therefore, the fraction
tfrac{63}{462}
tfrac{63}{462}
can be fully reduced by dividing the numerator and denominator by 21:
tfrac{63}{462} = tfrac{63 div 21}{462 div 21}= tfrac{3}{22}
tfrac{63}{462} = tfrac{63 div 21}{462 div 21}= tfrac{3}{22}

In order to find the greatest common divisor, the Euclidean algorithm may be used.

Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of
tfrac{3}{7}
tfrac{3}{7}
, for instance, is
tfrac{7}{3}
tfrac{7}{3}
.
Because any number divided by 1 results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 =
tfrac{17}{1}
tfrac{17}{1}
(1 is sometimes referred to as the "invisible denominator"). Therefore, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be
tfrac{1}{17}
tfrac{1}{17}
.

Complex fractions

A complex fraction (or compound fraction) is a fraction in which the numerator or denominator contains a fraction. For example,
cfrac{tfrac{1}{2}}{tfrac{1}{3}}
cfrac{tfrac{1}{2}}{tfrac{1}{3}}
and
frac{12frac{3}{4}}{26}
frac{12frac{3}{4}}{26}
are complex fractions. To simplify a complex fraction, divide the numerator by the denominator, as with any other fraction (see the section on division for more details):
cfrac{tfrac{1}{2}}{tfrac{1}{3}}=tfrac{1}{2}timestfrac{3}{1}=tfrac{3}{2}=1frac{1}{2}.
cfrac{tfrac{1}{2}}{tfrac{1}{3}}=tfrac{1}{2}timestfrac{3}{1}=tfrac{3}{2}=1frac{1}{2}.

frac{12frac{3}{4}}{26} = 12tfrac{3}{4} cdot tfrac{1}{26} = tfrac{12 cdot 4 + 3}{4} cdot tfrac{1}{26} = tfrac{51}{4} cdot tfrac{1}{26} = tfrac{51}{104}
frac{12frac{3}{4}}{26} = 12tfrac{3}{4} cdot tfrac{1}{26} = tfrac{12 cdot 4 + 3}{4} cdot tfrac{1}{26} = tfrac{51}{4} cdot tfrac{1}{26} = tfrac{51}{104}
cfrac{tfrac{3}{2}}5=tfrac{3}{2}timestfrac{1}{5}=tfrac{3}{10}.
cfrac{tfrac{3}{2}}5=tfrac{3}{2}timestfrac{1}{5}=tfrac{3}{10}.

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=1tfrac14
tfrac24+tfrac34=tfrac54=1tfrac14
.

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
tfrac14timestfrac13=tfrac1{12}
tfrac14timestfrac13=tfrac1{12}
(twelfths).
Consider adding the following two quantities:
tfrac34+tfrac23
tfrac34+tfrac23

First, convert
tfrac34
tfrac34
into twelfths by multiplying both the numerator and denominator by three:
tfrac34timestfrac33=tfrac9{12}
tfrac34timestfrac33=tfrac9{12}
. Note that
tfrac33
tfrac33
is equivalent to 1, which shows that
tfrac34
tfrac34
is equivalent to the resulting
tfrac9{12}
tfrac9{12}
.
Secondly, convert
tfrac23
tfrac23
into twelfths by multiplying both the numerator and denominator by four:
tfrac23timestfrac44=tfrac8{12}
tfrac23timestfrac44=tfrac8{12}
. Note that
tfrac44
tfrac44
is equivalent to 1, which shows that
tfrac23
tfrac23
is equivalent to the resulting
tfrac8{12}
tfrac8{12}
.
Now it can be seen that:
tfrac34+tfrac23
tfrac34+tfrac23

is equivalent to:
tfrac9{12}+tfrac8{12}=tfrac{17}{12}=1tfrac5{12}
tfrac9{12}+tfrac8{12}=tfrac{17}{12}=1tfrac5{12}

This method can be expressed algebraically:
tfrac{a}{b} + tfrac {c}{d} = tfrac{ad+cb}{bd}
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}
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}
tfrac{3}{4}
and
tfrac{5}{12}
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=1tfrac16
tfrac34+tfrac{5}{12}=tfrac{9}{12}+tfrac{5}{12}=tfrac{14}{12}=tfrac76=1tfrac16


Group 8


Fractions -- x/y

Ratio -- x:y

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, however in theory any number of quantities can be compared. Mathematically they are represented by separating each quantity with a colon, for example the ratio 2:3, which is read as the ratio "two to three".

A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus 3⁄4 represents three divided by four, in decimals 0.75, as a percentage 75%.
http://answers.yahoo.com/question/index?qid=20081104103634AAxgRH2

external image images?q=tbn:ANd9GcRvqOm_VVg-5oAyyClOzdihVaX8hA6qX7P8XmUfStAjg9jVWj6w

The ratio of white portion of the circle to red portion of the circle is 2:3.
1/3 of the circle is the white portion of the circle. 2/3 of the circle is the red portion of the circle.