Math 6 Honors Periods 6 & 7

Changing a Decimal to a Fraction 6-6

As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.

To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.

Change 0.385 to a fraction in lowest terms

.385 = 385/1000 = 77/200

Change 3.64 to a mixed number in simple form

3.64 = 3 64/100 = 3 16/25

To change a repeating decimal into a fraction follow these examples

th__
0.54

tthththth__
Let n = 0.54 = 0.54545454….

[How many numbers are under the vinculum?] 2

Multiple both sides by 10²

So then, 100n = 54.54545454…

100n = 54.54545454…
122- n = 2.54545454….

We can subtract n from 100n to get 99n

100n = 54.54545454…
-23n =thu .54545454….

99n = 54

Divide both sides by 99

99n = 54
99thit 99

n = 54/99 = 6/11

Let’s try

th___
0.243

theitheith___
Let n = 0.243 = .243243243243….

How many numbers are under the vinculum? 3

So multiply both sides by 10³

1000n = 243.243243243243….

1000n = 243.243243…
thie- n = 243.243243

999n = 243

Divide both sides by 999

n = 243/999 = 27/111 = 9/37

Let’s try one that is a bit more complicated
tethit htitii__thiethitiehtiehtitheihtii___theithihitheii_
Change 0.318 [Notice this isn’t 0.318 nor is it 0.318

the__
0.318

How many numbers are under the vinculum? 2

So, we multiply by 10²

Let n = 0.318181818…

100n = 31.818181818…
thit-n=31.318181818…

99n = 31.500000…

Divide both sides by 99
99n/99 = 31.5/99

n = 31.5/99 but that isn’t a proper fraction. What can I do to change this?

Multiply by 10

315/990 = 63/198 = 7/22

HERE ARE SOME STEPS TO FOLLOW:

Step 1	set up “ n= the repeating decimal”	n = .515151…
Step 2	determine how many numbers are under the bar	in this case = 2
Step 3	Use that number as a power of 10	102 = 100
Step 4	Multiply both sides of the equation in step 1 by that power of 10	100n = 51.515151…
Step 5	Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation	100n = 51.5151… - 00n= 51.5151…
Step 6	Solve as a 1-step equation	99n = 51 so n =51/99
Step 7	Simplify	51/99 = 17/33
****	REMEMBER- sometimes you need to get the decimal out of the numerator—so multiply by a power of 10

Math 6 Honors Periods 6 & 7

Changing a Fraction to a Decimal 6-5

There are two methods that can be used to change a fraction into a decimal.

The first one, we try to find an equivalent fraction whose denominator is a power of 10.

13/25 is a great example because we can easily change the denominator into 100 : multiplying 24 by 4.

So

13/25 ( 4/4) = 52/100 = .52

In the second method of changing a fraction into a decimal, we divide the numerator by the denominator.

Change 3/8 divide 3 by 8 carefully

When the remainder is 0, as above, the decimal is referred to as a terminating decimal. By examining the denominator of a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal. If the denominator has no prime factors of then 2 or 5, the decimal representation will terminate.. (This is so since the fraction can be written as an equivalent fraction whose denominator is a power of ten)

7/40

40 = 2³ ∙ 5; since the only prime factors of the denominator are 2 and 5, the fraction can be expressed as a terminating decimal

5/12

12 = 2² ∙ 3 since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal

9/12 = 3/4

4 = 2². Since 4 has no prime factors other than 2, this fraction can be expressed as a terminating decimal

Now, what happens if the denominator of a fraction has prime factors other than 2 or 5

Change 15/22 to a decimal I know that this cannot be expressed as a terminating decimal because the denominator (22) has the prime factorization of 2 ∙ 11.

15 divided by 22 ? divide carefully and you will get 0.6818181….

Notice the pattern of repeating remainders of 18 and 4. They produce a repeating block of digits 81, in the quotient.

we write 15/22 = 0.681818181…. or 0.681 with a bar over the 81 where the bar, also know as the vinculum, means that the block 81 repeats without ending.

_________________
a decimal such as 0.681 , in which a block of digits continues to repeat indefinitely is called a repeating decimal.

Property

Every fraction can be expressed as either a terminating decimal or a repeating decimal..

Pre Algebra Periods 1, 2, & 4

Simple & Compound Interest 7-8

When you first deposit money into a bank or savings & loan, your money is called principal.

The bank takes you money and invests it. In return, the bank pays you interest– currently NOT MUCH– based on the interest rate.

Simple interest is interest paid ONLY on the principal.

Simple Interest Formula (yes, you need to memorize this formula)

I = PRT

Where I = Interest

P = the principal ($$ in savings or $$ owed to creditors)

R= interest rate PER YEAR (this is a % you must change to a decimal or fraction)

T = the time IN YEARS ( so if you have 9 months you really have 9/12 of a year)

Suppose you deposit $400 in a saving account. The interest rate is 5% per year.

(a) Find the interest earned in six years. Find the total of principal PLUS interest.

What’s the formula?

I = PRT

What parts of the formula do we know?

PLUG THEM IN

I = (400)(.05)(6)

Solve for I

Interest = $120.00 Now what is left to do?

(b) Find the interest earned in 3 months. Find the total of principal PLUS interest

What’s the formula?

I = PRT

What parts of the formula do we know? PLUG THEM IN

I = (400)(.05)(.25)

Solve for I

Interest = $5.00 Now what is left to do?

You just bought an ipod for $300 on your parent’s credit card (and you promised to pay them back– (“really, Mom”) in 2 years. Your parents are charging you the rate on their credit card, which is

20% interest.

How much interest will you pay and what is the total you will have to pay your parents so they can pay VISA?

What’s the formula?

I = PRT and what information do we have?

Plug it in, plug it in… plug it in….. I =300(.20)(2)

I = $120.00

How much will VISA get on your purchase?

Compound Interest is very similar to simple but you get

charged INTEREST on the INTEREST . So…

COMPOUND > SIMPLE

always

The ipod you bought and are paying over two years on the VISA but this time interest is COMPOUNDED (added to the principal)

ANNUALLY ( each year)

After one year I = ($300)(.20)(1) = $60

$300 Ipod + $60 interest = $360

Now in year 2, you are charged interest on the

$360 ( not the originally $300)

Now in year 2, you are charged interest on the

$360 ( not the originally $300)

Year 2: ( $360)(.20)(1) = $72

$360 owed at the end of year 1 + $72 = $432 owed at the end of year 2.

Remember simple interest was $ 420.

Interest can be compounded

annually ( once a year),

semiannually ( twice a year),

quarterly ( 4 times a year),

monthly ( 12 times a year), or even

daily ( 365 times a year)

Which would give you the most interest on your savings deposit?

Which would give the VISA company the most?

Guess what they use… DAILY

There is a formula for compound interest:

Balance owed =

principal (1 + percent rate[time])number of compoundings

So if I owe VISA $500 at 6% compounded quarterly( 4 times a year) and I want to know how much I will owe after 1 year.. I can use this formula: A = P(1 + rt)n

A is the amount I am looking for

P = $500 r = 6% or 0.06 t = ¼ of a year or .25

n = 4 because I want to know after a year and the interest will be compounded 4 times

A = 500[1 + (.06)(.25)]4 where is that calculator

A = 500(1.015)4

A = 500 (1.06136)

A = $530.68 rounded to the nearest cent at this point.

The other way is to organize a table and calculate the interest each time 500(.06)(.25) = 7.50 interest 1st quarter

500 + 7.50 = $507.50

507.50(.06)(.25) = $7.61 interest 2nd quarter

507.50 + 7.61 = $515.11

515.11(.06)(.25) = $ 7.73 interest 3rd quarter

$515.11 + 7.73 = $ 522.84

522.84(.06)(.25) = $7.84 interest 4th quarter

$522.84 + 7.84 = $ 530.68

You wouldn’t want to use the table approach more than one year because it just takes too much time!!

Algebra Period 3 (Wednesday)

ADDING AND SUBTRACTING RADICALS 11-6:

Again, radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add SQRT 2 to SQRT 3!!!!

However, you may add (3 SQRT 2) to (5 SQRT 2) and get 8 SQRT 2:
(3 + 5) SQRT 2 = 8 SQRT 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:
SQRT( x²y) + SQRT( 4x²y) + SQRT(9y) - SQRT( y³)

Simplify each term first!!!!!!!!

x SQRT( y) + 2x SQRT( y) + 3 SQRT( y) - y SQRT( y)

NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y
(x + 2x + 3 - y) SQRT y
(3x - y + 3) SQRT y (final simplified answer)

PYTHAGOREAN THEOREM 11-7 (an old friend)

FOR RIGHT TRIANGLES ONLY!

2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)

EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

DISTANCE FORMULA
(based on the Pythagorean Theorem): see p. 513 in book

The distance between any two points on the coordinate plane (x y plane)
The distance is the hypotenuse of a right triangle that you can draw using any two points on the coordinate plane (I'll show you how to draw it in class).
The formula is:
distance = SQRT[( difference of the two x's)² + (difference of the two y's)²]
The difference between the 2x’s is the length of the leg parallel to the y axis and
the difference between the 2y’s is the length of the leg parallel to the x axis

EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = SQRT[(3 - -7)² + (-10 - -2)²]
d = SQRT[10² +( -8²)]
d = SQRT(164)
Simplifying: 2SQRT41

USING THE PYTHAGOREAN THEOREM - WORD PROBLEMS 11-8
There are many real life examples where you can use the Pythagorean Theorem to find a length.

EXAMPLE: HOW HIGH A 10 FOOT LADDER REACHES ON A HOUSE?

A 10 ft ladder is placed on a house 5 ft away from the base of the house.
Find how high up the house the ladder reaches.
The ladder makes a right triangle with the ground being one leg, the house the other, and the ladder is the hypotenuse ( see drawing in #1 on p. 515) You need to find the distance on the house, so you're finding one leg.

FLYING A KITE? HOW LONG MUST THE STRING BE?

You're flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you're standing 9 feet away from where the kite is in the air. The string represents the hypotenuse. You know one leg is the height in the air (13 ft) and the other leg is how far on the ground you are standing away from where the kite is flying (9 ft)
You need to find the hypotenuse.

Algebra Period 3 (Tuesday)

PRODUCT OF RADICALS 11-4
Basically, a radical is similar to variable in that you can multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
So the product of the square root of 2 and the square root of 14 is the square root of 28

The square root of 28 can be simplified to 2 square root 7, read 2 "rad" 7

HELPFUL HINT:
If you are multiplying SQRT 250 times SQRT 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:
SQRT 250 times SQRT 50
SQRT(2 x 5 x 5 x 5) times SQRT(2 x 5 x 5) = SQRT(2 x 2 x 5 x 5 x 5 x 5 x 5)

(2 x 5 x 5) SQRT 5 = 50 SQRT 5

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!

SQRT (3/16)= SQRT (3) = SQRT (3)

thiehtiehtiehtSQRT (16) thithii 4

SQRT (25/36) = SQRT (25) = 5

thisisthishtithithSQRT (36) th6

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: SQRT (27/3)

SQRT (27/3) = SQRT (9) = 3

If you took this fraction apart first and then tried to find the square root of each part, it's much more complicated:

SQRT (27/3) = SQRT (27) = SQRT (3 x 3 x 3) = 3 SQRT 3 = 3

thiththiththithtiSQRT 3 thithiSQRT 3thithithithiSQRT 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:
SQRT 7
SQRT 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
SQRT(3) in the denominator.
You need to multiply both numerator and denominator by SQRT (3) to get it out of there:

SQRT 7 = SQRT 7 * SQRT 3 = SQRT (7 * 3) = SQRT 21
SQRT 3 thiSQRT 3 thiSQRT 3 thithithi3thithithithi3

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!)
For example, if you were to simplify SQRT(9)/3, you would get 3/3 = 1

If you had simplified the 3 with the 9 first, you would have incorrectly gotten SQRT (3)

Pre Algebra Periods 1, 2, & 4

Mark Up and Discount 6-9

MARKUP = how much you want to make on selling something Then add the markup to get what you want to sell that item for.
EX: you want to make 75% of what you paid (.75)(your cost)
You paid $100 for an ipod. (.75)($100)=$75 markup. So you want to sell it for $175!

DISCOUNT = sales price This is similar to above, but this time you will SUBTRACT (not add)
A jacket that was $100 is 35% off. (.35)($100) = $35. Sales price = $100 - $35 = $65

MARKUP = a % of INCREASE and
DISCOUNT = a % of DECREASE!

MARKUP = % OF INCREASE
EX: Your cost = $50 and your markup = $100. What is your MARKUP %? (% of increase?)
markup = increase
$100 (increase on cost)/$50 (original) = 2 = 200% markup (selling price = $50 + $100 = $150)

DISCOUNT = % OF DECREASE
EX: original price = $200 and sales price = $50. What is the DISCOUNT %? (% of decrease?)
Decrease or discount = $200 - $50 = $150
$150 decrease/$200 original = 3/4 = 75% Discount %

Algebra Period 3

Radical Expressions 11-1 and 11-2:
I will use SQRT as the symbol for square root for my web site
(since the real radical sign will not print correctly on the site!)

Square rooting "undoes" squaring!

It's the inverse operation!!!
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)

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

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: SQRT 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: SQRT (36 + 64) = 10 not 14!!!!
First add 36 + 64 = 100
Then find SQRT 100 = 10
- SQRT 25 = -25 First take the squre root and then multiply by the negative sign

8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!
EXAMPLE: SQRT 5² = 5
SQRT (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!

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!
But there is another group of irrational numbers: Square roots of nonperfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than nonperfect!
Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.
PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!
EXAMPLE: SQRT( .04) is rational because it is ± .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!)

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 square root of (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! For example, if you have 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: SQRT (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 ( IxI )

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!)
x² + 10x + 25 factors to (x + 5)² so the square root of (x + 5)² = I x + 5I

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 )

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?
etc.

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 x 5 x 5 x 5
Circle the first two 5's
5 x 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

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
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 )] =[ (5)(5)](3)(x⁵x⁵)
Simplified, you can pull out a factor of 5 and x⁵
Final simplified radical = 5x⁵ SQRT(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)

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)
First factor out a 3: SQRT [3 (x² + 4x + 4) ]
Now factor the trinomial: SQRT [3 (x + 2) (x + 2) ]
The (x + 2)² is a perfect square so SQRT [3 (x + 2) (x + 2) ] = (x + 2) SQRT( 3 )