Mathematics play a large role in our lives

“All things are numbers,” said Pythagoras, the Greek mathematician, philosopher and mystic 2600 years ago. To him, numbers brought order and harmony to everything, from cosmos to life to music.

He was, of course, right. We see numbers all around us, and we live our daily lives in numbers: Our medical reports and vital signs come in numbers, as do our businesses and finances. We check our calories, weights, driving speed, time, temperature and weather in numbers.

The West learned algebra and algorithm from the 1200-year old work of al-Khwarizmi, a Muslim mathematician. (Algebra comes from al-jabr in Arabic, and algorithm is the misspelled name of Khwarizmi.)

What Is Infinity?

1/0 = ∞ or 1/∞ = 0?

 Read 

Ratio

Ratio

A ratio is a comparison of two numbers. 

Examples:

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

1) What is the ratio of books to marbles?

A ratio is a comparison between two or more like quantities in the same units.

Note:

The ratio 1 : 2 is read as '1 to 2' or '1 is to 2'.

Scale Factor

If the ratio is expressed in the form 1 : n, then n is called the scale factor.
E.g.  5 : 20 = 1 : 4
So, 4 is the scale factor.

Note:

We can use ratios to compare more than two quantities conveniently.  Fractions are not usually suitable for this.

 Ratios are used in areas including concentration of solutions, drug dosages, financial mathematics and gears.

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.

proportion (Meaning)

pro·por·tion  (pr-pôrshn, -pr-)

n.

1. A part considered in relation to the whole.

2. A relationship between things or parts of things with respect to comparative magnitude, quantity, or degree: the proper proportion between oil and vinegar in the dressing.

3. A relationship between quantities such that if one varies then another varies in a manner dependent on the first: "We do not always find visible happiness in proportion to visible virtue" (Samuel Johnson).

4. Agreeable or harmonious relation of parts within a whole; balance or symmetry.

5. Dimensions; size. Often used in the plural.

6. Mathematics A statement of equality between two ratios. Four quantities, a, b, c, d, are said to be in proportion if ^a/_b = ^c/_d .

tr.v. pro·por·tioned, pro·por·tion·ing, pro·por·tions

1. To adjust so that proper relations between parts are attained.

2. To form the parts of with balance or symmetry.

Proportion

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 

When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion.

We compare rates just as we compare ratios, by cross multiplying

Rate

A rate is a ratio

Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.


When comparing rates, always check to see which units of measurement are being used. For instance, 3 kilometers per hour is very different from 3 meters per hour! 

Important:

One of the most useful tips in solving any math or science problem is to always write out the units when multiplying, dividing, or converting from one unit to another.

Average Rate of Speed

The average rate of speed for a trip is the total distance traveled divided by the total time of the trip.

Increasing or Decreasing a Quantity in a Given Ratio

If the ratio of a new quantity to an old quantity can be expressed as an improper fraction, then the new quantity is greater than the old quantity.  Applying this ratio to the old quantity is known as increasing the old quantity in a given ratio. If the ratio of a new quantity to an old quantity can be expressed as a proper fraction, then the new quantity is less than the old quantity.  Applying this ratio to the old quantity is known as decreasing the old quantity in a given ratio.

Example 8

Increase 20 in the ratio 3 : 2.

Solution:

Example 9

Decrease 32 in the ratio 3 : 4.

Using a Ratio to Solve Problems

If the ratio of two quantities is known and one of the quantities is given, then the other quantity can be calculated. Example 10 A company wants to reduce its operating cost in the ratio 2 : 3. If the operating cost was $51000 last year, what would be its target cost? to be continue............. http://www.mathsteacher.com.au/year8/ch06_ratios/05_divide/ratio.htm Golden_ratio

Percentages (%)

When you say "Percent" you are really saying "per 100"

And 25% means 25 per 100

Formula for percentage

Examples #1:

25 % of 200 is____ 

In this problem, of = 200, is = ?, and % = 25

We get:

is/200 = 25/100

Since is in an unknown, you can replace it by y to make the problem more familiar

y/200 = 25/100

Cross multiply to get y × 100 = 200 × 25

y × 100 = 5000

Divide 5000 by 100 to get y

Since 5000/100 = 50, y = 50

So, 25 % of 200 is 50




Now, we will take examples to illustrate how to use the formula for percentage on the right

Examples #4:

To use the other formula that says part and whole, just remember the following:

The number after of is always the whole

The number after is is always the part

If I say 25 % of___ is 60, we know that the whole is missing and part = 60

Your proportion will will like this:

60/whole = 25/100

After cross multiplying, we get:

whole × 25 = 60 × 100

whole × 25 = 6000

Divide 6000 by 25 to get whole

6000/25 = 240, so whole = 240

Therefore, 25 % of 240 is 60














Examples #2:

What number is 2% of 50 ?

This is just another way of saying 2% of 50 is___

So, set up the proportion as example #1




Examples #3:

24% of___ is 36

This time, notice that is = 36, but of is missing

After you set up the formula, you get:

36/of = 24/100

Replace of by y and cross multiply to get:

36/y = 24/100

Percentage increase and decrease

Sometimes due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of percent changes:

• An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.
• An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
• A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).
• A decrease of 100% means the final amount is zero (100% − 100% = 0%).

http://en.wikipedia.org/wiki/Percentage

(New Value / Old Value)  * 100   =  % Value

Percent Increase (U Tube)

http://www.basic-mathematics.com/formula-for-percentage.html

Difference Between Rate and Ratio

A ratio is a comparison of two numbers and can be written multiple ways (like 1/6 or 1:6). You typically do not use units, but if you do, they are often the same. If you have 4 oranges and your friend as 6 oranges, the ratio of your oranges to his is 4/6, which simplifies to 2/3 or 0.666.

A rate is typically distance per unit time, such as 20 miles/hour. This rate is speed, in which you travel 20 miles per 1 hour. For rates, your units are different and often distance and time. Another rate could be some other number per unit time, such as 100kb/sec. In this case, you could say "for every one second, my computer can download 100kb of data." In three seconds, your computer would download 300kb given this rate.

Rate pertains to fixed quantity between 2 things while a ratio is the relationship between lots of things. 

A unit rate can be written as 12 kms per hour or 10km/1hr; a unit ratio can be written in this manner 10:1 or is read as 10 is to 1. 

A rate usually pertains to a certain change while a ratio is the difference of something.

.

Rate and ratios are very important in explaining the equivalence from one and the other. A rate cannot be one if ratio does not exist.

You don’t even notice that these two are still being used in our day to day living like calculating bank interest, product cost and many more. Life has been made easier because of these two.

Read more: http://www.differencebetween.com/difference-between-rate-and-vs-ratio/#ixzz1diyYi5tz




The P/E ratio (price-to-earnings ratio)

Significant Figures in Measurements and Calculations

Significant Figures

http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
There are two kinds of numbers in the world:


exact:

example: There are exactly 12 eggs in a dozen.
example: Most people have exactly 10 fingers and 10 toes.

inexact numbers:
example: any measurement.


If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures).

PRECISION VERSUS ACCURACY

Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.

accurate

(the average is accurate)

not precise

precise

not accurate

accurate

and

precise

The number of significant figures is the number of digits believed to be correct by the person doing the measuring.  It includes one estimated digit.

So, does the concept of significant figures deal with precision or accuracy?

Conclusion: The number of significant figures is directly linked to a measurement.

So, does the concept of significant figures deal with precision or accuracy? Hopefully, you can see that it really deals with precision only. Consider measuring the length of a metal rod several times with a ruler. You will get essentially the same measurement over and over again with a small reading error equal to about 1/10 of the smallest division on the ruler. You have determined the length with high precision. However, you don't know if the ruler was accurate to begin with. Perhaps it was a plastic ruler left in the hot Texas sun and was stretched. You don't know the accuracy of your measuring device unless you calibrate it, i.e. compare it against a ruler you knew was accurate. Note: in the laboratory, a good analytical chemist always calibrates her volumetric glassware before using it by weighing a known volume of liquid dispensed from the glassware. By dividing the mass of the liquid by its density, she can determine the actual volume and hence the accuracy of the glassware.

Rules for Working with Significant Figures:
Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see.

The significant figures (also called significant digits) of a number are those digits that carry meaning contributing to its precision.

This includes all digits except:

.leading and trailing zeros where they serve merely as placeholders to indicate the scale of the number.

.spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.


The concept of significant digits is often used in connection with rounding. Rounding to n significant digits is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant digits (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.

Computer representations of floating point numbers typically use a form of rounding to significant digits, but with binary numbers.

The term "significant digits" can also refer to a crude form of error representation based around significant-digit rounding; for this use, see significance arithmetic.

Identifying significant digits


The rules for identifying significant digits when writing or interpreting numbers are as follows:
All non-zero digits are considered significant. For example, 91 has two significant digits (9 and 1), while 123.45 has five significant digits (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant digits: 1, 0, 1, 1 and 2.
Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant digits (the zeros before the 1 are not significant). In addition, 130.00 has five significant digits. This convention clarifies the accuracy of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.
The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, has three significant digits (and hence indicates that the number is accurate to the nearest ten).
The last significant digit of a number may be underlined; for example, "2000" has one significant digit.
A decimal point may be placed after the number; for example "100." indicates specifically that three significant digits are meant.[1]

However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.

Scientific notation

Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant digits) becomes 1.2×10−4, and 0.00122300 (six significant digits) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant digits is written as 1.300×103, while 1300 to two significant digits is written as 1.3×103.

Rounding

To round to n significant digits:
If the first non-significant digit is a 5 followed by other non-zero digits, round up the last significant digit (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant digits should be written 1.25.
If the first non-significant digit is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant digits, Round half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.
Replace any non-significant digits by zeros.

Arithmetic
Main article: Significance arithmetic
For multiplication and division, the result should have as many significant digits as the measured number with the smallest number of significant digits.

For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.

When performing a calculation, do not follow these guidelines for intermediate results; keep as many digits as is practical to avoid rounding errors.[2]








Significant Figures and Rounding Rules

http://www.angelfire.com/oh/cmulliss/