SIMPLIFYING RADICALS 11-3
SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect squares left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: square root of 50 = SQRT(25 * 2)
Next, simplify the SQRT of the perfect square and leave the nonperfect factor under the radical:
SQRT(25 * 2) = SQRT(25) *SQRT(2) = 5 SQRT( 2 )
***Imagine the square root sign top extends over all numbers****
√(25 * 2) = √25 * √2 = 5√2
HELPFUL HINTS:
When you are factoring the radicand, you're looking for the LARGEST PERFECT SQUARE that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
You can even use all the perfect squares that you have memorized... it really helps// just remember when you pull out a number... you are pulling out the number which multiples by itself to get that perfect square... NOT the perfect square itself...
Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors
(remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!
Example: SQRT 250
Prime factorization = 2 * 5 * 5 * 5
Circle the first two 5's
5 * 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical
Final answer: SQRT 250 = 5 SQRT 10
√250 = 5√10
VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: SQRT (x10 ) = x5 or
√x10 = x5
We saw this already in factoring!!!
SIMPLIFYING NONPERFECT VARIABLES:
If the variable has an even power:
EXAMPLE: The square root of 75x10 = SQRT [(25)(3)(x10 )] =
SQRT [ (5)(5)](3)(x5x5)
***Imagine the square root sign top extends over all numbers****
√75x10 = √[(25)(3)(x10 )] = √[ (5)(5)](3)(x5x5)
Simplified, you can pull out a factor of 5 and x5
Final simplified radical = 5x5 SQRT(3)
or
5x5√3
If the variable has an odd power:
If you have an odd power variable, simply express it as the even power one below that odd power times that variable to the 1 power:
Example: x5 = x4 x
so if you have the SQRT( x5 ) = SQRT (x4 x) = x2SQRT(x)
√x5 =√x4 x = x2√x
FACTORING A GCF FIRST, THEN FINDING A BINOMIAL SQUARED:
Sometimes you will need to factor what's under the radical before you start to simplify
Example: SQRT(3x2 + 12x + 12) or
***Imagine the square root sign top extends over all numbers****
√3x2 + 12x + 12
First factor out a 3:
SQRT [3 (x2 + 4x + 4) ] or
***Keep imagining that line across all the numbers under the radical √ ****
√3(x2 + 4x + 4)
Now factor the trinomial: SQRT [3 (x + 2) (x + 2) ] or
√3(x + 2)2
The (x + 2)2 is a perfect square so SQRT [3 (x + 2) (x + 2) ] = (x + 2) SQRT( 3 )
or (x+2)√3
you need to include the HUGS around the (x +2) because you want that binomial to be multiplied by the √3 .. not just the 2.
Thursday, March 17, 2011
Wednesday, March 16, 2011
Algebra (Period 1)
Radical Expressions 11-2
If an expression under the square root sign is NEGATIVE, it does not exist in the REAL numbers!
There is no number that you can square and get a NEGATIVE PRODUCT
VARIABLES UNDER THE SQUARE ROOT SIGN: If you have a variable under the square root sign, you need to determine what values of the variable will keep the radicand greater than or equal zero
The square root of x then is only real when x is greater than or equal to zero
The SQRT (x + 2) is only real when x + 2 is greater than or equal to 0
Set x + 2 greater than or equal to 0 and solve as an inequality!
You will find that x must be greater than or equal to -2
SPECIAL CASE!!!! a variable squared plus a positive integer under radical: If you're trying to find the principal square root of x2 (or any variable squared) plus a positive integer, then all numbers will work because a squared number will always end up either positive or zero!
Example: √(x2 + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.
If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x2 - 10)
x2 must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)
ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x2 is the absolute value (positive) of x SHown here as: ( I x I )
Why? Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE: x = -3 √x2 = √(-3)2 = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED (see p. 489 #17-30)
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)
EXAMPLE: √( x2 + 10x + 25) factors to
√(x + 5)2 = I x + 5 I
Determine the values for the variable that will make each expression a real number
√m(m+3)
you know that m(m+3) ≥ 0 so m ≥0 OR m ≤ -3
√x2(x-3)
again set x2(x-3) ≥ 0 and you discover x = 0 or x ≥3
Given a and c, what must be true of b to make
√b2 -4ac
a real number?
a = -3 and c = 2
substitute in and we have
√b2 -4(-3)(2)
√b2 +24
b can be any real number!!
But what if we have
a = 2 and c = 8
√b2 -4ac
substitute in
√b2 -4(2)(8)
√b2 -64
b is either
b≤ -8 or b ≥ 8
Determine whether each of the follow statements is sometimes, always or never true:
√a2 + b 2 is a real number ---> ALWAYS
√3 - t is a real number for t ≥ 3 ---> SOMETIMES
√a2 - b 2 is a real number ---> SOMETIMES
√a2 + 2ab + b 2 is a real number ---> ALWAYS
For a polynomial in the form ax2 + bx + c = 0 to have real solutions,
√b2 -4ac must be a real number. Which of the following polynomials have real solutions?
x2 - 12x + 3 = 0 is real because √b2 -4ac = √122 -4(1)(3) =
√144 -12 √132 and that's real
x2 + 5x + 7 = 0 is NOT REAL because √52 -4(1)(7) = √25 -28 = √-3 which is NOT REAL
If an expression under the square root sign is NEGATIVE, it does not exist in the REAL numbers!
There is no number that you can square and get a NEGATIVE PRODUCT
VARIABLES UNDER THE SQUARE ROOT SIGN: If you have a variable under the square root sign, you need to determine what values of the variable will keep the radicand greater than or equal zero
The square root of x then is only real when x is greater than or equal to zero
The SQRT (x + 2) is only real when x + 2 is greater than or equal to 0
Set x + 2 greater than or equal to 0 and solve as an inequality!
You will find that x must be greater than or equal to -2
SPECIAL CASE!!!! a variable squared plus a positive integer under radical: If you're trying to find the principal square root of x2 (or any variable squared) plus a positive integer, then all numbers will work because a squared number will always end up either positive or zero!
Example: √(x2 + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.
If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x2 - 10)
x2 must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)
ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x2 is the absolute value (positive) of x SHown here as: ( I x I )
Why? Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE: x = -3 √x2 = √(-3)2 = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED (see p. 489 #17-30)
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)
EXAMPLE: √( x2 + 10x + 25) factors to
√(x + 5)2 = I x + 5 I
Determine the values for the variable that will make each expression a real number
√m(m+3)
you know that m(m+3) ≥ 0 so m ≥0 OR m ≤ -3
√x2(x-3)
again set x2(x-3) ≥ 0 and you discover x = 0 or x ≥3
Given a and c, what must be true of b to make
√b2 -4ac
a real number?
a = -3 and c = 2
substitute in and we have
√b2 -4(-3)(2)
√b2 +24
b can be any real number!!
But what if we have
a = 2 and c = 8
√b2 -4ac
substitute in
√b2 -4(2)(8)
√b2 -64
b is either
b≤ -8 or b ≥ 8
Determine whether each of the follow statements is sometimes, always or never true:
√a2 + b 2 is a real number ---> ALWAYS
√3 - t is a real number for t ≥ 3 ---> SOMETIMES
√a2 - b 2 is a real number ---> SOMETIMES
√a2 + 2ab + b 2 is a real number ---> ALWAYS
For a polynomial in the form ax2 + bx + c = 0 to have real solutions,
√b2 -4ac must be a real number. Which of the following polynomials have real solutions?
x2 - 12x + 3 = 0 is real because √b2 -4ac = √122 -4(1)(3) =
√144 -12 √132 and that's real
x2 + 5x + 7 = 0 is NOT REAL because √52 -4(1)(7) = √25 -28 = √-3 which is NOT REAL
Tuesday, March 15, 2011
Algebra (Period 1)
Real Numbers 11-1
... as opposed to IMAGINARY numbers! : )
(Seriously!)
√ is the symbol for square root
MAIN CONCEPT: Square rooting "undoes" squaring!
It's the inverse operation!!!
Just as subtraction undoes addition
Just as division undoes multiplication
If you square a square root:
(√243)2 = 243 (what you started with)
If you square root something squared:
√2432 = 243 (what you started with)
If you multiply a square root by the same square root:
(√243)(√243) = 243 (what you started with)
IN SUMMARY: (√243)2= √2432 = (√243)(√243) = 243
1)RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example:
The square root of 64 is 8.
The cubed root of 64 is 3.
The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power) to get the RADICAND (whatever is under the radical sign)
In the example above, the roots were 8, 3, and 2.
4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.
Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared = 25
5) PRINCIPAL SQUARE ROOT: The positive square root.
Generally, the first section just asks for the principal square root unless there is a negative sign in front of the radical sign.
6) ± sign in front of the root denotes both the positive and negative roots at one time!
Example: √ 25 = ±5
7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical.
In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.
EXAMPLE: √ (36 + 64) = 10 not 14!!!!
First add 36 + 64 = 100
Then find √100 = 10
Radicals by themselves function as exponents in order of operations (that makes sense because they undo exponents).
Actually, roots are FRACTIONAL EXPONENTS!
Square roots = 1/2 power,
Cubed roots = 1/3 power,
Fourth roots = 1/4 power, etc.
So √25 = 25½ = 5
EXAMPLE: 3 + 4√25 you would do powers first...in this case square root of 25 first!
3 + 4(5) Now do the multiplication
3 + 20
Now do the addition 23
8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
EXAMPLE: √52 = 5
√(a -7)2 = a - 7
RATIONAL SQUARE ROOTS: Square roots of perfect squares are RATIONAL
REVIEW OF NUMBER SYSTEMS:
Rational numbers are decimals that either terminate or repeat which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.
Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1
Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.
Real numbers: all of these!
In Algebra II you will find out that there are Imaginary Numbers!
Square roots of NEGATIVE numbers are IMAGINARY
IRRATIONAL SQUARE ROOTS: Square roots of a nonperfect squares are IRRATIONAL - They cannot be stated as the ratio of two integers -
As decimals, they never terminate and never repeat - you round them and use approximately sign.
MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!
There is another group of irrational numbers:
Square roots of not perfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than not perfect!
Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.
PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!
EXAMPLE: √(.04) is rational because it is ±0.2
But all the square roots in between these perfect squares are IRRATIONAL
For example, the square root of 2, the square root of 3, the square root of 5, etc.
You can estimate irrational square roots.
For example, the square root of 50 is close to 7 because the square root of 49 is 7. You can estimate that the square root of 50 is 7.1 and then square 7.1 to see what you get.
If that's too much, try 7.05 and square that.
This works much better with a calculator!
And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.
Remember: These will never end or repeat (even though your calculator only shows a certain number of places physically!)
In class we went through a method of finding a good approximate to any square using the perfect square above it and below it!!
... as opposed to IMAGINARY numbers! : )
(Seriously!)
√ is the symbol for square root
MAIN CONCEPT: Square rooting "undoes" squaring!
It's the inverse operation!!!
Just as subtraction undoes addition
Just as division undoes multiplication
If you square a square root:
(√243)2 = 243 (what you started with)
If you square root something squared:
√2432 = 243 (what you started with)
If you multiply a square root by the same square root:
(√243)(√243) = 243 (what you started with)
IN SUMMARY: (√243)2= √2432 = (√243)(√243) = 243
1)RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example:
The square root of 64 is 8.
The cubed root of 64 is 3.
The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power) to get the RADICAND (whatever is under the radical sign)
In the example above, the roots were 8, 3, and 2.
4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.
Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared = 25
5) PRINCIPAL SQUARE ROOT: The positive square root.
Generally, the first section just asks for the principal square root unless there is a negative sign in front of the radical sign.
6) ± sign in front of the root denotes both the positive and negative roots at one time!
Example: √ 25 = ±5
7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical.
In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.
EXAMPLE: √ (36 + 64) = 10 not 14!!!!
First add 36 + 64 = 100
Then find √100 = 10
Radicals by themselves function as exponents in order of operations (that makes sense because they undo exponents).
Actually, roots are FRACTIONAL EXPONENTS!
Square roots = 1/2 power,
Cubed roots = 1/3 power,
Fourth roots = 1/4 power, etc.
So √25 = 25½ = 5
EXAMPLE: 3 + 4√25 you would do powers first...in this case square root of 25 first!
3 + 4(5) Now do the multiplication
3 + 20
Now do the addition 23
8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
EXAMPLE: √52 = 5
√(a -7)2 = a - 7
RATIONAL SQUARE ROOTS: Square roots of perfect squares are RATIONAL
REVIEW OF NUMBER SYSTEMS:
Rational numbers are decimals that either terminate or repeat which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.
Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1
Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1
Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.
Real numbers: all of these!
In Algebra II you will find out that there are Imaginary Numbers!
Square roots of NEGATIVE numbers are IMAGINARY
IRRATIONAL SQUARE ROOTS: Square roots of a nonperfect squares are IRRATIONAL - They cannot be stated as the ratio of two integers -
As decimals, they never terminate and never repeat - you round them and use approximately sign.
MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!
There is another group of irrational numbers:
Square roots of not perfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than not perfect!
Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.
PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!
EXAMPLE: √(.04) is rational because it is ±0.2
But all the square roots in between these perfect squares are IRRATIONAL
For example, the square root of 2, the square root of 3, the square root of 5, etc.
You can estimate irrational square roots.
For example, the square root of 50 is close to 7 because the square root of 49 is 7. You can estimate that the square root of 50 is 7.1 and then square 7.1 to see what you get.
If that's too much, try 7.05 and square that.
This works much better with a calculator!
And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.
Remember: These will never end or repeat (even though your calculator only shows a certain number of places physically!)
In class we went through a method of finding a good approximate to any square using the perfect square above it and below it!!
Monday, March 14, 2011
Pi Day
HAPPY PI DAY
HAPPY PI DAY! WHAT AN IRRATIONAL DAY!!!!
BE IRRATIONAL.. AND TRANSCENDENTAL
... here are the facts:
Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle.
Pi is an IRRATIONAL number. That means it has no pattern and never terminates.
It CANNOT be written as a fraction with an integer in the numerator and denominator.
We use 3.14 and 22/7 as APPROXIMATIONS of pi.
These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers.
You will be able to discover many interesting facts about pi—even finding it on your own.
Here's some links to click on if you're interested in the mystery of Pi:
History of pi
Find your birthday in pi
The first ten thousand digits of pi
Here is a rap I really like...
Lose Yourself in the Digits of pi
... and of course... here are the songs we sang
Happy Pi Day (to the tune of “Happy Birthday”)
Happy PI day to you
Happy PI day to you
Happy PI day everybody
Happy PI day to you
Oh, Number Pi (to the tune of “Oh, Christmas Tree”)
Oh, Number Pi
Oh, Number Pi
Your digits are unending.
Oh Number Pi
Oh Number Pi
No pattern are you sending
You’re three point one four one five nine
And even more if we had time,
Oh, number Pi
Oh, number Pi
For circle lengths unbending.
Oh, number Pi
Oh, number Pi
You are a number very sweet
Oh, number Pi
Oh, number Pi
Your uses are so very neat.
There’s 2 Pi r and Pi r squared
A half a circle and you’re there,
Oh, number Pi
Oh, number Pi
We know that Pi’s a tasty treat
Pi Day Song (to the tune of “Jingle Bells”)
Pi day songs
All day long
Oh what fun it is
To sing a jolly pi day song
In a fun math class
Like this
Verse;
Circles in the snow
Around and round we go
How far did we have to run?
Diameter times pi! (Refrain)
We wish you a Happy Pi Day (to the tune of “We wish you a Merry Xmas”)
We wish you a happy Pi Day
We wish you a happy Pi Day
We wish you a happy Pi Day
To you and to all
Pi numbers for you
For you and for all
Pi numbers in the month of March
So three point one four!!
HAPPY PI DAY! WHAT AN IRRATIONAL DAY!!!!
BE IRRATIONAL.. AND TRANSCENDENTAL
... here are the facts:
Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle.
Pi is an IRRATIONAL number. That means it has no pattern and never terminates.
It CANNOT be written as a fraction with an integer in the numerator and denominator.
We use 3.14 and 22/7 as APPROXIMATIONS of pi.
These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers.
You will be able to discover many interesting facts about pi—even finding it on your own.
Here's some links to click on if you're interested in the mystery of Pi:
History of pi
Find your birthday in pi
The first ten thousand digits of pi
Here is a rap I really like...
Lose Yourself in the Digits of pi
... and of course... here are the songs we sang
Happy Pi Day (to the tune of “Happy Birthday”)
Happy PI day to you
Happy PI day to you
Happy PI day everybody
Happy PI day to you
Oh, Number Pi (to the tune of “Oh, Christmas Tree”)
Oh, Number Pi
Oh, Number Pi
Your digits are unending.
Oh Number Pi
Oh Number Pi
No pattern are you sending
You’re three point one four one five nine
And even more if we had time,
Oh, number Pi
Oh, number Pi
For circle lengths unbending.
Oh, number Pi
Oh, number Pi
You are a number very sweet
Oh, number Pi
Oh, number Pi
Your uses are so very neat.
There’s 2 Pi r and Pi r squared
A half a circle and you’re there,
Oh, number Pi
Oh, number Pi
We know that Pi’s a tasty treat
Pi Day Song (to the tune of “Jingle Bells”)
Pi day songs
All day long
Oh what fun it is
To sing a jolly pi day song
In a fun math class
Like this
Verse;
Circles in the snow
Around and round we go
How far did we have to run?
Diameter times pi! (Refrain)
We wish you a Happy Pi Day (to the tune of “We wish you a Merry Xmas”)
We wish you a happy Pi Day
We wish you a happy Pi Day
We wish you a happy Pi Day
To you and to all
Pi numbers for you
For you and for all
Pi numbers in the month of March
So three point one four!!
Pre Algebra (Period 2 & 4)
Fractions, Decimals & Percents 6-5
Fractions = Decimals = Percents!!
A percent is a ratio that compares a number to 100.
Therefore you can write a percent as a fraction with a denominator of 100
Percent means "per hundred" or "out of a hundred"
THe symbol % comes from the fraction bar and a denominator of 100
TO CHANGE ANY % TO A FRACTION:
simply get rid of the % sign and put the number over a denominator of 100
Simplify
5% = 5/100 = 1/20
TO CHANGE ANY % TO A DECIMAL
Simply get rid of the % and move the decimal point over to the LEFT-TWO PLACES
( its running away from the %... it is so sad that the % sign went away!!)
5% = 0.05
TO CHANGE ANY DECIMAL TO A %
Simply put the % to the right of the number and move the decimal point to the RIGHT TWO PLACES ( the decimal is so happy to see the percent sign -- it runs toward it!!)
0.245 = 24.5%
TO CHANGE ANY FRACTION TO A %
Simply change it to a decimal first, then follow as above
20/50 = 2/5 = 0.4 = 40%
But this one is a good one to think of changing 20/50 = 40/100 and then easily seeing 40%
If the denominator will go into 100 easily, you can use equivalent fractions!!(as in this case)
If you ever forget how many places to move the decimal, just look at the % sign-- It tells you 2 places ( 2 zeros)
You need to know a few percentages
10%--> just move the decimal point one place to the LEFT. That's it
$75.00 with a 10% discount
The discount would be $7.50
On $80.00 the 10% discount would be $8.00 and a 5% would be half of that or $4.00
You can find %20 easily as well-- just double 10%
Know
50% --> 1/2
25% --. 1/4
75% --> 3/4
Fractions = Decimals = Percents!!
A percent is a ratio that compares a number to 100.
Therefore you can write a percent as a fraction with a denominator of 100
Percent means "per hundred" or "out of a hundred"
THe symbol % comes from the fraction bar and a denominator of 100
TO CHANGE ANY % TO A FRACTION:
simply get rid of the % sign and put the number over a denominator of 100
Simplify
5% = 5/100 = 1/20
TO CHANGE ANY % TO A DECIMAL
Simply get rid of the % and move the decimal point over to the LEFT-TWO PLACES
( its running away from the %... it is so sad that the % sign went away!!)
5% = 0.05
TO CHANGE ANY DECIMAL TO A %
Simply put the % to the right of the number and move the decimal point to the RIGHT TWO PLACES ( the decimal is so happy to see the percent sign -- it runs toward it!!)
0.245 = 24.5%
TO CHANGE ANY FRACTION TO A %
Simply change it to a decimal first, then follow as above
20/50 = 2/5 = 0.4 = 40%
But this one is a good one to think of changing 20/50 = 40/100 and then easily seeing 40%
If the denominator will go into 100 easily, you can use equivalent fractions!!(as in this case)
If you ever forget how many places to move the decimal, just look at the % sign-- It tells you 2 places ( 2 zeros)
You need to know a few percentages
10%--> just move the decimal point one place to the LEFT. That's it
$75.00 with a 10% discount
The discount would be $7.50
On $80.00 the 10% discount would be $8.00 and a 5% would be half of that or $4.00
You can find %20 easily as well-- just double 10%
Know
50% --> 1/2
25% --. 1/4
75% --> 3/4
Subscribe to:
Posts (Atom)
