Simple & Compound Interest 7-8
Let's talk about different types of interest: credit cards, car loans, mortgages
None of them charge us simple interest, but you need to understand this first:
I = PRT
(mneumonic devices "PRINT" or "PARTY")
I = Interest
P = Principal ($ deposited in savings or $ owed to credit card or bank)
R = % (Change it to a decimal or fraction before multiplying)
T = Time (based on one year so if you have 9 months, put it over 12 months or 9/12 of a year)
Once you compute the interest, you'll need to ADD it to your cost.
In that way, it is LIKE TAX AND TIP.
You'll move it, move it, multiply, then add.
BUT WHAT IS DIFFERENT ABOUT INTEREST?
The TIME element!
WHY IS THAT NEEDED IN THIS FORMULA?
Because you may pay it back over different amounts of time.
Obviously, the sooner you pay it back, the less interest you should owe VISA.
WHY WASN'T THERE A TIME ELEMENT IN TAX AND TIP?
Because you pay tax and tip right then...not over time!
LET'S LOOK AT BUYING THE SAME IPhone, CHARGING IT, AND PAYING IT BACK OVER 4 DIFFERENT PERIODS OF TIME:
1 year, 2 years, 10 years, and 2 months
EXAMPLE 1:
You buy an Iphone for $300 on your parent's VISA at 20% and pay it back in 1 year
I = PRT
I = ($300)(.20)(1) = $60 Interest
$300 Ipod + $60 interest to VISA = $360 total cost
EXAMPLE 2:
You buy an Iphone for $300 on your parent's VISA at 20% and pay it back in 2 years
I = PRT
I = ($300)(.20)(2) = $120 Interest
$300 Iphone + $120 interest to VISA = $420 total cost
EXAMPLE 3:
You buy the same Iphone for $300 on your parent's VISA at 20%
but you pay it back in 10 years!
I = PRT
I = ($300)(.20)(10) = $600 Interest
$300 Iphone + $600 interest to VISA = $900 total cost
EXAMPLE 4:
You buy the same Iphone for $300 on your parent's VISA at 20%
but you pay it back in 10 months!
Since the interest formula is based on YEARS, you'll need to put the
MONTHS OVER 12 (there are 12 months in a year!)
I = PRT
I = ($300)(.20)(2/12) = $10 Interest
$300 Iphone + $10 interest to VISA = $310 total cost
SO FOR THE SAME IPHONE, YOU'VE SPENT A LOW OF $310 TO A HIGH OF $900!!
SO BEWARE OF BUYING THINGS ON CREDIT CARDS IF YOU CAN'T PAY THEM BACK FOR A VERY LONG TIME!!!!
INTEREST INCOME:
Interest that you receive on your money in the bank
Interest is computed the same way whether it is interest YOU PAY (credit cards)
or
interest the BANK PAYS you (savings accounts)
EXAMPLE:
You deposit $200 in your savings account at Wells Fargo for 9 months at 2%
I = PRT
I = ($200)(.02)(9/12) = $3 interest
$200 deposit + $3 = $203 in your savings account after 9 months
Thursday, March 24, 2011
Algebra (Period 1)
ADDING AND SUBTRACTING RADICALS 11-6
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2
Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.
EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)
Simplify each term first!!!!!!!!
x√y + 2x√y + 3√y - y√ y
NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2
Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.
EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)
Simplify each term first!!!!!!!!
x√y + 2x√y + 3√y - y√ y
NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)
Algebra (Period 1)
FINDING SQUARE ROOTS
Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!
Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.
Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:
1. Guess
2. Divide
3. Average.
... and then just keep repeating steps 2 and 3.
First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.
In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465
We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225
Method 2- Estimating your Square root
This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 132 and 142 So our SQ RT must also be between 13 and 14.
196
183
169
Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14
Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5
What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.
Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.
If you have any questions or comments about these two ways, post a comment here.
Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!
Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.
Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:
1. Guess
2. Divide
3. Average.
... and then just keep repeating steps 2 and 3.
First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.
In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465
We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225
Method 2- Estimating your Square root
This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 132 and 142 So our SQ RT must also be between 13 and 14.
196
183
169
Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14
Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5
What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.
Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.
If you have any questions or comments about these two ways, post a comment here.
Wednesday, March 23, 2011
Pre Algebra (Period 2 & 4)
Discount & Markups Continued 6-9
(this is similar to finding tax and tip) MARKUP = the PROFIT you make on an item that you sell
SELLING PRICE = COST + MARKUP
How do you calculate markup?
You MOVE IT, MOVE IT, MULTIPLY! (Where have we heard that before? :)
Then you ADD the markup to calculate the SELLING PRICE (the price you want to sell the item for so that you'll make the profit that you want)
EXAMPLE:
You bought a snow board for $200
You've decided to resell it on EBAY so that you can get a better one.
You want to make 25% on what you paid so you can afford a better one.
DECIDE THE MARKUP YOU WANT (the profit):
(markup %)(your cost) (25%)($200)
MOVE IT, MOVE IT, MULTIPLY: (.25)($200) = $50 markup
(you could have made the 25% into 1/4 as a fraction)
ADD the MARKUP to the original COST to find the SELLING PRICE:
Markup + Cost = Selling Price
$50 + $200 = $250
YOU CAN DO THIS IN ONE STEP!
After you MOVE IT, MOVE IT, add 1 to the decimal, then MULTIPLY:
In the problem above, add 1 to the .25 = 1.25
Now multiply (1.25)(200) = $250
You got the SELLING PRICE in 1 step!
WHAT IF YOU KNOW THE SELLING PRICE AND THE COST AND YOU WANT TO SEE WHAT THE MARKUP % IS:
MARKUP = a % of INCREASE
MARKUP % = INCREASE FROM COST (markup or profit)/ COST
In the example above, we know the markup % was 25%,
but we should be able to go backwards and prove it:
MARKUP = Selling Price - Cost =
$250 - $200 = $50
MARKUP % = Markup/Cost =
$50/$200 = 1/4 or 25%
(this is similar to finding tax and tip) MARKUP = the PROFIT you make on an item that you sell
SELLING PRICE = COST + MARKUP
How do you calculate markup?
You MOVE IT, MOVE IT, MULTIPLY! (Where have we heard that before? :)
Then you ADD the markup to calculate the SELLING PRICE (the price you want to sell the item for so that you'll make the profit that you want)
EXAMPLE:
You bought a snow board for $200
You've decided to resell it on EBAY so that you can get a better one.
You want to make 25% on what you paid so you can afford a better one.
DECIDE THE MARKUP YOU WANT (the profit):
(markup %)(your cost) (25%)($200)
MOVE IT, MOVE IT, MULTIPLY: (.25)($200) = $50 markup
(you could have made the 25% into 1/4 as a fraction)
ADD the MARKUP to the original COST to find the SELLING PRICE:
Markup + Cost = Selling Price
$50 + $200 = $250
YOU CAN DO THIS IN ONE STEP!
After you MOVE IT, MOVE IT, add 1 to the decimal, then MULTIPLY:
In the problem above, add 1 to the .25 = 1.25
Now multiply (1.25)(200) = $250
You got the SELLING PRICE in 1 step!
WHAT IF YOU KNOW THE SELLING PRICE AND THE COST AND YOU WANT TO SEE WHAT THE MARKUP % IS:
MARKUP = a % of INCREASE
MARKUP % = INCREASE FROM COST (markup or profit)/ COST
In the example above, we know the markup % was 25%,
but we should be able to go backwards and prove it:
MARKUP = Selling Price - Cost =
$250 - $200 = $50
MARKUP % = Markup/Cost =
$50/$200 = 1/4 or 25%
Pre Algebra (Period 2 & 4)
Discount & Markup 10-6
This is when we rush out to go shopping! :)
DISCOUNT = The amount OFF when something is on sale
The calculation is similar to Tax and Tip in that you still
MOVE IT, MOVE IT, MULTIPLY
The difference is that instead of adding to what you had, this time you will SUBTRACT
EXAMPLE: A jacket that was $100 is now 35% off.
MOVE IT, MOVE IT, MULTIPLY Discount = (.35)($100) = $35
SUBTRACT THIS "DISCOUNT" FROM THE ORIGINAL PRICE
Sales price = $100 - $35 = $65
YOU CAN DO THIS IN ONE STEP:
In the example above, you are NOT PAYING the 35% DISCOUNT
So what % are you paying?
65% (100% - 35% discount %)
Finding the sales price in one step: ($100)(.65) = $65
Again, this is what you do to find the SALES PRICE in ONE STEP:
100% - DISCOUNT% = % you will pay
Then move it, move it, multiply
And you won't have to subtract!
You'll already have the SALES PRICE!
What if you know what the price is on sale, you know the original price, and you're trying to figure out if this is a really good % off?
Can you figure our what % off this is????
Yes you can because the...
The DISCOUNT% = a % of DECREASE!
When would a store not state the % off?
This usually happens when items are on sale in the supermarket.
They don't state the % off. They just tell you what the sales price is.
Why do you think they do that?
Usually when the Discount% is small.
It also happens whenever the % is strange amount (not a multiple of 5).
EXAMPLE at a supermarket - small and strange:
Fruit snacks are 50 cents off a box of 10
Original Price: $3.99
Sales Price: $3.49
Sales Discount % = discount/original = $.50/$3.99 = .125
or about 13% off
EXAMPLE at a store where an item is on clearance - clearance item %s off vary depending on how long an item has been on clearance:
You look at the price tag on a jacket and see that the original price = $200
and sales price = $50.
You know this is a great price, but still you're wondering what is the DISCOUNT %?
(% of decrease?)
Decrease or discount = original price - sales price
$200 - $50 = $150 discount (amount off)
What is this as a % off? Calculate the % of decrease:
$150 decrease/$200 original = 3/4 = 75% Discount &
WOW! THAT'S A GREAT SALE!!!
EXAMPLE where % is a strange amount so the store just states it as a certain number of dollars off (instead of as a %):
You look in the Best Buy weekly flyer. The sales price on a computer is $700.
The original price is $770.
What is the amount off (the discount)? $770 - 700 = $70 discount
What is this as a sale discount %?
Discount/Original = $70/$770 = .090909... or about 9%
NOT REALLY A GREAT % OFF SALE! (but you ARE still saving $70!)
ANOTHER REASON WHY SUPERMARKETS GENERALLY NOT USE % OFF:
The %s are usually small AND they are all different.
One item might be 13% off and the item next to it might be 9% off.
It would drive shoppers crazy to figure out all those different percentages that only end up being say $.50 off!
Whereas, when Macy's has a sale, clothes tend to be all 20% off (for example)
Original prices are much higher and the % off used tends to be the same for each item.
What about sales that are stated as fractions off?
1/3 off sale, 1/2 off sale
You won't have to move it, move it!
You'll just multiply and subtract!
FRACTION EXAMPLE:
A $900 LCD TV is 1/3 off
Discount = (1/3)(900) = $300 off
Sales price = $900 - 300 = $600
YOU CAN FIND THE SALES PRICE IN ONE STEP:
You're NOT PAYING 1/3 of the price.
What fraction are you PAYING?
1 (whole original price) - fraction off = fraction of the original you are paying
1 - 1/3 = 2/3
In the example above with the LCD TV: (2/3)($900) = $600 on sale
SALES THAT ARE STATED AS RATIOS:
Sales are stated like this:
Buy 2, get 1 free
Buy 1, get the 2nd one for a penny
Get 3 for $10 (but when you buy less than that, the price/item is $3.99)
Papa Johns has a sign that states: Buy 1 pizza at regular price, get the 2nd for a penny.
What % off is this?
Well first of all, in the real world, the first pizza price will vary based on what you get on it.
Let's just assume that your first pizza is $15.
The first pizza is 100% of the original cost...it is NOT on sale.
But then you add a 2nd pizza of the same price and pay a penny.
You have paid $15.01 for 2 pizzas. You would have paid $30.00 for the 2 pizzas without this "deal"
DISCOUNT % = discount (decrease in price) / original price
DISCOUNT % = $30 - $15.01 / $30 = $14.99/$30.00 = about 50%
SO THIS IS A HALF PRICE SALE FOR THE 2 PIZZAS!
WHY NOT JUST MAKE EACH PIZZA HALF PRICE????
Because you might just buy 1 pizza for $7.50 and not buy the 2nd pizza!
When does this make sense for the company?
When they are looking for incremental (more) sales and the cost of the 2nd item is not going to be sold at a loss. (pizzas don't cost $15 to make!)
I ALWAYS BUY A LOT MORE CHEWY CHIPS AHOYS WHEN PAVILLIONS HAS THEM AS BUY 1 GET 1 FREE.
That's a half price sale! Make sure you buy an EVEN number of them or the last one is at full price!
This is when we rush out to go shopping! :)
DISCOUNT = The amount OFF when something is on sale
The calculation is similar to Tax and Tip in that you still
MOVE IT, MOVE IT, MULTIPLY
The difference is that instead of adding to what you had, this time you will SUBTRACT
EXAMPLE: A jacket that was $100 is now 35% off.
MOVE IT, MOVE IT, MULTIPLY Discount = (.35)($100) = $35
SUBTRACT THIS "DISCOUNT" FROM THE ORIGINAL PRICE
Sales price = $100 - $35 = $65
YOU CAN DO THIS IN ONE STEP:
In the example above, you are NOT PAYING the 35% DISCOUNT
So what % are you paying?
65% (100% - 35% discount %)
Finding the sales price in one step: ($100)(.65) = $65
Again, this is what you do to find the SALES PRICE in ONE STEP:
100% - DISCOUNT% = % you will pay
Then move it, move it, multiply
And you won't have to subtract!
You'll already have the SALES PRICE!
What if you know what the price is on sale, you know the original price, and you're trying to figure out if this is a really good % off?
Can you figure our what % off this is????
Yes you can because the...
The DISCOUNT% = a % of DECREASE!
When would a store not state the % off?
This usually happens when items are on sale in the supermarket.
They don't state the % off. They just tell you what the sales price is.
Why do you think they do that?
Usually when the Discount% is small.
It also happens whenever the % is strange amount (not a multiple of 5).
EXAMPLE at a supermarket - small and strange:
Fruit snacks are 50 cents off a box of 10
Original Price: $3.99
Sales Price: $3.49
Sales Discount % = discount/original = $.50/$3.99 = .125
or about 13% off
EXAMPLE at a store where an item is on clearance - clearance item %s off vary depending on how long an item has been on clearance:
You look at the price tag on a jacket and see that the original price = $200
and sales price = $50.
You know this is a great price, but still you're wondering what is the DISCOUNT %?
(% of decrease?)
Decrease or discount = original price - sales price
$200 - $50 = $150 discount (amount off)
What is this as a % off? Calculate the % of decrease:
$150 decrease/$200 original = 3/4 = 75% Discount &
WOW! THAT'S A GREAT SALE!!!
EXAMPLE where % is a strange amount so the store just states it as a certain number of dollars off (instead of as a %):
You look in the Best Buy weekly flyer. The sales price on a computer is $700.
The original price is $770.
What is the amount off (the discount)? $770 - 700 = $70 discount
What is this as a sale discount %?
Discount/Original = $70/$770 = .090909... or about 9%
NOT REALLY A GREAT % OFF SALE! (but you ARE still saving $70!)
ANOTHER REASON WHY SUPERMARKETS GENERALLY NOT USE % OFF:
The %s are usually small AND they are all different.
One item might be 13% off and the item next to it might be 9% off.
It would drive shoppers crazy to figure out all those different percentages that only end up being say $.50 off!
Whereas, when Macy's has a sale, clothes tend to be all 20% off (for example)
Original prices are much higher and the % off used tends to be the same for each item.
What about sales that are stated as fractions off?
1/3 off sale, 1/2 off sale
You won't have to move it, move it!
You'll just multiply and subtract!
FRACTION EXAMPLE:
A $900 LCD TV is 1/3 off
Discount = (1/3)(900) = $300 off
Sales price = $900 - 300 = $600
YOU CAN FIND THE SALES PRICE IN ONE STEP:
You're NOT PAYING 1/3 of the price.
What fraction are you PAYING?
1 (whole original price) - fraction off = fraction of the original you are paying
1 - 1/3 = 2/3
In the example above with the LCD TV: (2/3)($900) = $600 on sale
SALES THAT ARE STATED AS RATIOS:
Sales are stated like this:
Buy 2, get 1 free
Buy 1, get the 2nd one for a penny
Get 3 for $10 (but when you buy less than that, the price/item is $3.99)
Papa Johns has a sign that states: Buy 1 pizza at regular price, get the 2nd for a penny.
What % off is this?
Well first of all, in the real world, the first pizza price will vary based on what you get on it.
Let's just assume that your first pizza is $15.
The first pizza is 100% of the original cost...it is NOT on sale.
But then you add a 2nd pizza of the same price and pay a penny.
You have paid $15.01 for 2 pizzas. You would have paid $30.00 for the 2 pizzas without this "deal"
DISCOUNT % = discount (decrease in price) / original price
DISCOUNT % = $30 - $15.01 / $30 = $14.99/$30.00 = about 50%
SO THIS IS A HALF PRICE SALE FOR THE 2 PIZZAS!
WHY NOT JUST MAKE EACH PIZZA HALF PRICE????
Because you might just buy 1 pizza for $7.50 and not buy the 2nd pizza!
When does this make sense for the company?
When they are looking for incremental (more) sales and the cost of the 2nd item is not going to be sold at a loss. (pizzas don't cost $15 to make!)
I ALWAYS BUY A LOT MORE CHEWY CHIPS AHOYS WHEN PAVILLIONS HAS THEM AS BUY 1 GET 1 FREE.
That's a half price sale! Make sure you buy an EVEN number of them or the last one is at full price!
Algebra (Period 1)
DIVIDING RADICALS 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Monday, March 21, 2011
Pre Algebra (Period 2 & 4)
Percent of Change 6-8
% OF INCREASE OR DECREASE:
You compute these the same way, only one is an increase and the other is a decrease
% of increase or decrease =
Increase or decrease
Original amount
HOW DO YOU FIND THE INCREASE OR DECREASE FOR THE NUMERATOR?
Think about it!
If you had $10 and now you have $15, how much did your money change?
$5
Was it an increase or decrease?
Increase!
You had a 98 in the class and now you have a 92.
How much did your grade change?
6 points
Was it an increase or decrease?
Decrease!
SO HOW DO YOU FIND THE INCREASE OR DECREASE FOR THE NUMERATOR?
SUBTRACT!
NOTICE THAT THE DENOMINATOR IS ALWAYS THE ORIGINAL
WHY?
You want to know what your number CHANGED FROM
EXAMPLE:
Original grade = 80 and new grade = 90
for an increase of 10 points
% of increase =
10/80 = 1/8 = .125
= 12.5%
EXAMPLE:
Original grade = 90 and new grade = 80
for a decrease of 10 points
% of decrease = 10/90 = 1/9 = .111. . . or about 11%
WHY ARE THE %s DIFFERENT WHEN THE CHANGES ARE THE SAME (10 points)?
The Percent of Change is based on WHAT YOU STARTED WITH.
In the first case, the original grade was 80, but in the second example, the original grade was 90.
10 points is a bigger percent of 80 than it is of 90!
STILL DON'T GET THAT?
LET'S DO AN EXAMPLE THAT IS VERY DIFFERENT!
Original grade: 20 (yes, I said 20% in the class :(
New grade: 30
Increase: 30 - 20 = 10
What did it increase from? 20
% PERCENT OF INCREASE: 10/20 = 50%
INCREASE!!!!
Now I know this student is still failing, but he(she) has increased the grade 50% of the prior grade!
FROM THE ABOVE EXAMPLE, YOU CAN SEE HOW PERCENT OF CHANGE CAN BE MISUSED BY PEOPLE LIKE ADVERTISERS!
The student above can tell his parents: "I increased my grade 50%!!! "
That's true! But his parents should immediately say 50% increase FROM WHAT?
An advertisement says: Your teeth will be 50% whiter!
50% whiter FROM WHAT?
You will get 50% better gas mileage if you use our gas.
50% better than what?
% OF INCREASE OR DECREASE:
You compute these the same way, only one is an increase and the other is a decrease
% of increase or decrease =
Increase or decrease
Original amount
HOW DO YOU FIND THE INCREASE OR DECREASE FOR THE NUMERATOR?
Think about it!
If you had $10 and now you have $15, how much did your money change?
$5
Was it an increase or decrease?
Increase!
You had a 98 in the class and now you have a 92.
How much did your grade change?
6 points
Was it an increase or decrease?
Decrease!
SO HOW DO YOU FIND THE INCREASE OR DECREASE FOR THE NUMERATOR?
SUBTRACT!
NOTICE THAT THE DENOMINATOR IS ALWAYS THE ORIGINAL
WHY?
You want to know what your number CHANGED FROM
EXAMPLE:
Original grade = 80 and new grade = 90
for an increase of 10 points
% of increase =
10/80 = 1/8 = .125
= 12.5%
EXAMPLE:
Original grade = 90 and new grade = 80
for a decrease of 10 points
% of decrease = 10/90 = 1/9 = .111. . . or about 11%
WHY ARE THE %s DIFFERENT WHEN THE CHANGES ARE THE SAME (10 points)?
The Percent of Change is based on WHAT YOU STARTED WITH.
In the first case, the original grade was 80, but in the second example, the original grade was 90.
10 points is a bigger percent of 80 than it is of 90!
STILL DON'T GET THAT?
LET'S DO AN EXAMPLE THAT IS VERY DIFFERENT!
Original grade: 20 (yes, I said 20% in the class :(
New grade: 30
Increase: 30 - 20 = 10
What did it increase from? 20
% PERCENT OF INCREASE: 10/20 = 50%
INCREASE!!!!
Now I know this student is still failing, but he(she) has increased the grade 50% of the prior grade!
FROM THE ABOVE EXAMPLE, YOU CAN SEE HOW PERCENT OF CHANGE CAN BE MISUSED BY PEOPLE LIKE ADVERTISERS!
The student above can tell his parents: "I increased my grade 50%!!! "
That's true! But his parents should immediately say 50% increase FROM WHAT?
An advertisement says: Your teeth will be 50% whiter!
50% whiter FROM WHAT?
You will get 50% better gas mileage if you use our gas.
50% better than what?
Algebra (Period 1)
PRODUCT OF RADICALS 11-4
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7
HELPFUL HINT:
If you are multiplying √250 • √50,
I would suggest that you don't multiply 250 x 50 too quickly!
Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!
So why not factor each factor first?!!!
Using my example above:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5
Dividing & Simplifying 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7/√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7
HELPFUL HINT:
If you are multiplying √250 • √50,
I would suggest that you don't multiply 250 x 50 too quickly!
Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!
So why not factor each factor first?!!!
Using my example above:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5
Dividing & Simplifying 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7/√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
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