Math 6A (Periods 2 & 4)

Least Common Multiple 5-6
Here is a review of that lesson...
Let’s look at the nonzero multiples of 8 and 12—listed in order
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…

Multiples of 12: 12, 24, 36, 48, 60, 72, ….

The numbers 24, 48, and 72, ... are called common multiples of 8 and 12. The least of these multiples is 24 and is therefore called the least common multiple.

LCM(8, 12) = 24

To find the LCM of two whole numbers, we can write out lists of multiples of the two numbers.

Or, we can use prime factorization

Lets find LCM(12, 15)

12 = 2²∙3
15 = 3∙5

The LCM will be made up of the greatest power of each factor

LCM will be 2²∙3∙5 = 60
The book has a third option or method
you can check out, if you’d like

Let’s find LCM (54, 60)


54= 2∙3∙3∙3 = 2∙3³
60 = 2∙2∙3∙5 = 2²∙3∙5

The greatest power of 2 that occurs in either prime factorization is 2²
The greatest power of 3 that occurs in either prime factorization is 3³
The greatest power of 5 that occurs in either prime factorization is 5
Therefore, LCM(54,60) is 2²∙3³∙5 = 540


REMEMBER:
The GCF (greatest common factor) is a factor. The GCF of two numbers will be either the smaller of the two or smaller than both

The LCM (least common multiple) is a multiple. The LCM of the two numbers will be the largest of the two or larger than both.



To find the LCM of two whole numbers you could write out the lists of multiples-- and that works relatively easily with small numbers... but there are more efficient ways to find the least common multiple of two whole numbers.

1. Write out the first few multiples of the larger of the two numbers and test each multiple for divisibility by the smaller number. The first multiple of the larger number that is divisible by the smaller number is the LCM

2. You can use prime factorization to find the LCM. The LCM is EVERY factor to its GREATEST power!!
LCM(54, 60)
54 = 2⋅ 3⋅ 3⋅ 3 = 2⋅ 3³
60 = 2⋅ 2⋅ 3⋅ 5 = 2²⋅ 3⋅ 5
So the greatest power of 2 is 2²
The greatest power of 3 is just 3
and the greatest pwoer of 5 is just 5
so the product of 2²⋅ 3⋅ 5 will be the LCM
LCM(54, 60) = 540

3. You may use the BOX method as shown in class... unfortunately it does not show well here. Remember you need to create a L. The numbers on the side of the box represent the GCF!! You need to multiple them with the last row of factors.
See me before or after class if you want any review!!


We reviewed the concept of relatively prime and noticed that any two prime numbers are relatively prime. We also noticed that if two numbers are relatively prime-- neither of them must be prime....

We also found out that if one number is a factor of a second number, the GCF of the two numbers is the first number AND... if one whole number is a factor of a second whole number the LCM of the two numbers is the second number!!
GCF(12,24) = 12
LCM(12,24) = 24

WOW!!

If two whole numbers are relatively prime---
their GCF = 1
and their LCM is their product!!
GCF(8,9) =1
GCF(8,9) = 72

WOW!!

LCM & GCF Story PRoblems
1) Read the problem
2) Re-read the problem!!
3) Figure out what is being asked for!!
4) find the "magic " word... to help you determine if you are finding GCF or LCM
5) When in doubt... draw it out!!

Math 6 High (Period 3)

Integers & The Number Line 4.1

Integers are the Counting Numbers (also known as Natural Numbers), their opposites, and ZERO.

We have Negative Integers, Positive Integers and Zero. 

The symbol used to represent negative integers is a “negative sign.” 

-4 is normally read as negative 4 

BUT it is also understood to mean “the opposite of 4”

On a number line,
the negative numbers are to the LEFT of ZERO and
the positive numbers are to the RIGHT of ZERO.

Please turn to Page 167 in our textbook and look at the examples of graphing various negative integers on a number line. 

You will notice that -4 is to the left of -1… Therefore -4 < -1  That is -4 IS LESS THAN -1. 

You could also state that -1> -4. That is, -1 IS GREATER THAN -4.

Negative numbers are used in real-life situations to represent sub-zero temperatures as well as losses that occur in business.

Algebra Honors (Periods 5 & 6)

Dividing Fractions 6-3

Use the same rule for dividing fraction that you use for dividing real numbers—multiply by its reciprocal.    
a/b  ÷ c/d = a/b ∙ d/c

Divide           x/2y  ÷ xy/4   so 

  x/2y  ∙ 4/xy

Simplify to 2/y²       

Divide  18/(x²-25) ÷   [24/(x+5)]

That becomes

18/(x²-25) ∙   [(x+5)/24]

     =  [3 ∙ 6/(x+5)(x-5)] ∙   [(x+5)/4 ∙ 6]

 Simplify

= 3/4(x+5)

Divide:

[x²+3x-10/(2x+6)]    ÷ [(x²-4)/ (x²-x-12)]

Which becomes

[x²+3x-10/(2x+6)]  ∙ [(x²-x-12)/ (x²-4)]

Factor

 [(x+5)(x-2)/2(x+3)]  ∙ [(x+3)(x-4)/ (x+2)(x-2)]

 [(x+5)(x-4)]/[2(x+2)]   

Make sure to use the O³ when simplifying an expression that involves more than one operation.   For Example:

   (2x/y)³÷(y²/x) ∙  x/4

= (8x³/y³⁾∙ (x/y²) ∙  x/4

2x³ /4

Math 6High (Period 3)

Rates 3.8 

If two quantities a and b have different units of measure, then the rate of a per b is a/b. The units for a rate tell you which numbers goes in the numerator and which units goes in the denominator .

For example

Miles per hour = miles/hour

miles

hour

When a rate is simplified so that it has a denominator of 1, it is a unit rate.

You are traveling from San Jose,  CA to Santa Rosa, CA You travel a distance of 101 miles in 2 hours. What is your average speed in miles per hour?

Speed is one type of a unit rate. 
To find the average speed in miles per hourà divide the distance traveled by the time

Average speed = distance/time

101miles/2hours =   50.5miles/hour

You can always estimate to check that your result is reasonable.

100miles in 2 hours is 50 miles/hour so your answer makes sense

In order to compare prices at the supermarket, you can calculate unit prices. A 12 ounce box of cereal costs $ 3.72  What is the unit price of the cereal?

A unit price is another type of unit rate. Divide the cost by the weight

Unit price = Cost/weight

$3.72/12 oz

DIVIDE CAREFULLY

$0.31/1 oz

The unit price of the cereal is $0.31 per ounce.

Then we talked about a scenario involving babysitting. Suppose you babysat for one family on Saturday for 2 ½ hours and were paid $10. Then on Sunday, you babysit for another family for 3 hours and were paid $10.50. 

Find the hourly rate that you were paid on each day. Which family do you want to work for again? (That is which one paid your more per hour?)

We used a table to organize the information

Day	Total	Time	Rate
Saturday	$10	2  ½ hours	$__/hour
Sunday	$10.50	3 hours	$__/hour

Saturday:  Amount paid/time = 10/2.5 = $ 4/hour

Sunday: Amount paid/ time = 10/3 = $3.50/hour

Your hourly rate was greater on Saturday.

Two rates for answering questions on a test are given below. 
We wrote each as a unit rate

60 minutes/ 75 questions  and   75 questions/ 60 minutes

for  60 minutes/ 75 questions  we found it was 0.8 min/question

and

for  75 questions/ 60 minutes , we found it was 1.25 questions/ min

Both rates are equivalent and would allow you to complete the test on time. However, it would be easier to use the rate of 0.8 min/question to figure out how long it took you per question!

You drive 1350 miles in 3 days Each day you drive for 9 hours. What is your speed in miles per hour?  

First you need to calculate how many hours you drove. 3(9)= 27à so you drove 27 hours

1350miles/27 hours = 50miles/hour

Algebra Honors ( Periods 5 & 6)

Multiplying Fractions 6-2
You know from previous years that
ac/bd = a/b ⋅ c/d

and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15

6x/y³⋅y²/15 = 2x/5y where y ≠0

Which simplifies to

This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!

Rule of Exponents for a Power of a Quotient
(a/b)^m = a^m/b^m
(x/3)³ = x³/27

(-c/2)²⋅4/3c
you must do the exponent portion first!!
c²/4⋅(4/3c)
c/3

Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)³ = 216n³/343 inches cubed

If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles

Math 6A (Periods 2 & 4)

Prime Numbers & Composite Numbers 5-4

A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...

Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)

Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.

Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7

When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.

Two is the only even prime number because all the other even numbers have two as a factor.

Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.

List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.

Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.

Math 6A (Periods 2 & 4)

Square Numbers and Square Roots 5-3

Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.

One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.

Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol

so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.

Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)

Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40

Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...

In this case, 9 + 16 = 25... hmmm... what's 25???

Algebra Honors (Periods 5 & 6)

Simplifying Fractions 6-1

When the numerator and the denominator of an algebraic fraction have no common factor other than 1 or -1, the fraction is said to be in simplest form. To simplify a fraction, first factor the numerator and the denominator.

Simplify:

(3a + 6)/ (3a + 3b)

3(a + 2)/3(a + b)

=(a  + 2)/(a +b)    (where a ≠ -b)

REMEMBER : YOU CANNOT DIVIDE BY ZERO. You must restrict the variables in a denominator by excluding any values that would make the denominator equal to ZERO.

So with the above example, a  CANNOT EQUAL –b

Simplify

(x²-9)/(2x+1)(3+x)

(x+3)(x-3)

(2x+1)(3+x)

since x + 3 = 3 +x

you can simplify both the numerator and denominator  to

x-3

2x+1

  (where x ≠-1/2, x≠ -3)

To see which values of the variable to exclude look at the denominator of the original fraction as well. Neither 2x + 1 nor 3 +x can be equal to zero.  Since 2x + 1 ≠ 0  x ≠ =1/2  and since 3 + x≠ 0  x ≠ -3

Simplify

2x² + x - 3

2- x- x²

First factor the numerator and the denominator, using  the skills you developed from the last chapter. If you do not see any common factors, look for opposites—as in this case

(x - 1)(2x+3)

(1- x)(2 + x)

Notice that (x-1) and (1-x) are opposites.

(1 -  x) = -( x - 1)

So change the sign on the fractions and use the opposites.

That is 

(x - 1)(2x+3)

-(x- 1)(2 + x)

and that can simplify to

(2x+3)

-(2 + x)

  or

-    (2x+3)

        x+2

 (where  x≠1, x≠-2)

Solve for x

ax - a²=bx –b²  

Collect all terms with x on one side of the equation

ax – bx = a² –b²

Factor BOTH sides of the equation

x(a –b) = (a+b)(a –b)

Divide BOTH sides of the equation by the coefficient of x ( which is    a-b)

x = a + b     ( where a≠ b)

Math 6A (Periods 2 & 4)

Tests for Divisibility 5-2

It is important to learn the following divisibility rules:
A number is divisibility by:

2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.

You will not need to know the divisibility rules for 7 or 11 but they are interesting...

You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.


To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20

Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.


A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.

A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?

Algebra Honors ( Period 5 & 6)

Using Factoring to Solve Problems  5-13

Example 1

A decorator plans to place a rug in a 8 m by 12 m room so that a uniform strip of wood flooring around the rug will remain uncovered. How wide will this strip be if the area of the rug is to be half the area of the room?

Let x = the width of the strip

Then 12-2x is the length of the rug and 9 – 2x is the width of the rug

(12-2x)(9-2x) = the area of the rug.

Area of the rug = ½ ( area of the room)

(12-2x)(9-2x) = ½(9∙12)

108 – 42x + 4x² = 54

4x²- 42 + 108 = 54

4x²- 42 -54 = 0

2(2x²- 21 + 27) =0

2(2x -3)(x -9) = 0

using the ZERO Products Property

2x -3 = 0 or x= 3/2

x -9 = 0  x = 9

CHECK: x = 1.5

works but when x = 9

the length 12 – 2x and the width 9-2x are negative! Since a negative length or width is meaningless reject x = 9 as an answer

This show that also the equation has a root that does not check because this equation does not  meet the hidden requirements that the rug have a positive length.  

Example 2
The FORMULA h = rt – 4.9t² is a good approximation of the height (h) in meters of an object t seconds after it is projected upward with an initial speed of r meters per second.

An arrow is shot upward with an initial speed of 34.3 m/s. When will it be at a height of 49 m?

let t = the  number of seconds  after being shot that the arrow is 49 m high.

Let h = the height of the arrow= 49 m

Let r = the initial speed = 34.3 m/s

Substitute in the formula

h = rt – 4.9t²

49 = 34.3t – 4.9t²

4.9t² – 34.3t + 49 = 0  THINK—GCF???

4.9(t² – 7t + 10 ) = 0

4.9(t -2)(t -5) = 0

Using the ZERO PRODUCTS PROPERTY

t = 2 and t = 5

Therefore the arrow is 49 m high both 2 seconds and 5 seconds after being shot… on its way up and on its way down!

Example 3

If a number is added to its square, the results is 56. Find the number

 Let x = the number

x² + x = 56

(x +8)(x - 7) = 0

 x = -8 and x = 7

Find two consecutive positive odd integers whose product is 143

Let x = the first positive odd integer

Let x + 2 = the 2^nd positive odd integer

x(x + 2) = 143

x² + 2x = 143

x² + 2x – 143 = 0

(x + 13)(x – 11) = 0

x = -13 and x = 11

But ask only for the positive integers so reject -13

The two integers are 11 and  13

Example 4

The sum of the squares of two consecutive negative odd integers is 290. Find the integers

let x = the 1^st negative odd integer

let x + 2 = the 2^nd negative odd integer

x² + (x + 2)² = 290

x² + x² + 4x + 4 = 290

2x² + 4x – 286 = 0  THINK  GCF!!!

2(x²+ 2x -143) = 0

2(x + 13)(x -11) = 0

x = -13 and x = 11

But it ask for the negative integers  so reject x = 11

The two negative integers are -13 and -11

Math 6A (Periods 2 & 4)

Finding Factors and Multiples 5-1


You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 60. A number is said to be divisible by its whole numbered factors.

To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.

We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6

When you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!

A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first five multiples of 7 are
0, 7, 14, 21, 28
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21; 4(7) = 28
... and put in set notation it would be
{0, 7, 14 ,21, 28}



If you were to ask for the first four NON-ZERO Multiples of 6
the answer would be 6, 12, 18, 24.. and in set notation
{ 6, 12, 18, 24}
Whereas the first four multiples of 6 ( you would need to include 0)
{0, 6, 12, 18}



Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!

Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number

What number is a factor of every number? ONE
Is every number a factor of itself? YES
What is ( are) the only multilpe(s) of 0? 0
How many numbers have 0 as a factor? only one number What number(s)? ZERO



The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.

Algebra Honors (periods 5 & 6)

Solving Equations by Factoring Section 5-12

a⋅0 = 0
and if a = 0 or b = 0
then we know that ab= 0
This is an if, then statement
conversely
if ab = 0 then either a= 0 or b = 0
THis Zero Products Property helps us solve equations.

(x +2)(x -5) = 0
either x + 2 must equal zero or x - 5 must equal zero
so set each expression equal to zero and solve
x + 2 = 0
x= -2
and x-5 = 0
x = 5

{-2. 5}

5m(m-3)(m-4) = 0
now you have three expressions so set each of them to zero
5m = 0 so m = 0
m-3 = 0 so m=3
m-4 = 0 so m=4
{0,3,4}

What happens with
3x²+ x = 2
It isn't the 2 products property but the ZERO products property so set the expression equal to ZERO
3x²+ x -2 = 0
Now factor
(x+1)(3x -2) = 0
set each of these equal to zero
x + 1 = 0 x = -1
3x -2 = 0 so x = 2/3

{-1, 2/3}

10x³ - 15x² = 0
factor
5x²(2x -3) = 0
again set each equal to zero
5x² = 0 so x = 0
and
2x -3 = 0 so x = 3/2

{0, 3/2}

polynomial equation named by the term of highes degree

ax + b = 0 linear equation

ax² + bx + c = 0 quadratic equations

ax³ + bx² + cx + d = 0 cubic equation

2x² + 5x = 12
becomes
2x² +5x - 12 = 0
(x + 4)(2x-3) = 0
so x = -4 and x = 3/2
{-4, 3/2}

18y³ + 8y + 24y² = 0
Rearrange first
18y³+ 24y ² + 8y = 0
Then factor the GCF
2y(9y² +12y +4) = 0
WAIT--> its a PERFECT trinomial SQ
2y(3y +2)² = 0
2y = 0 so y = 0
and 3y + 2 = 0 so y = -2/3

-2/3 is a double or multiple root but you only list it once in solution set.
That is,
{-2/3, 0}

y = x² + x - 12
solve for the roots means you set this quadratic equal to ZERO
so
x² + x - 12 = 0
(x+4)(x -3) = 0

Math 6A ( Periods 2 & 4)

Dividing Decimals 3-9 cont'd

For word Problems use the 5 step plan found on Page 18 of our textbook

296.06 ÷ (18.7 + 3.9)
Following Aunt Sally ( or PEMDAS... remember our singing...
we do the operation inside the hugs!! ( )using a sidebar
18.7 + 3.9 make sure to stack them lining up the decimals and you will get 22.6

296.06 ÷ 22.6

When dividing by a decimal remember the rule from yesterday, multiply the divisor ( 22.6) by a power of ten which makes it a natural number

then use that same power of ten and multiply the dividend,
WHen you divide you have
2960.6 ÷ 226
Please do that problem and your quotient should be 13.1

(47.1 - 16.9) ÷ (21.9 -6.8)
Again you need to do the operations inside the ( ) first. Using a side bar and lining up the decimals
47.1 - 16.9 = 30.2
and 21.9 - 6.8 = 15.1
Just take a look at those two numbers and you will notice a relationship!!
30.2 ÷ 15.1
BUT... practice your division skills and confirm what you can tell...
30.2 ÷ 15.1 = 2



At an average rate of 55 km/hour how long will it take to drive 225 km to the nearest tenth of an hour?

d = rt
What must we find and what are the clues? Well, how long... is usually time and the fact that we need to round to the nearest tenth of an hour indicates we are finding TIME as well.
So what is the distance? 225 km and what is the rate? 55 km/h
so plug into the formula
225= 55t
Now, how do we solve this one step problem?

divide both sides by 55
225/55 = 55t/55

do the division as a side bar

225/55 ≈ 4.09 so
t ≈ 4.1
and the answer is 4.1 hour

Math 6A (Periods 2 &4)

Dividing Decimals 3-9

According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.

Check out our textbook for some examples!!

When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.

Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!

the quotient is 0.3431 which rounds to 0.343


To divide one decimal by another

Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number


Divide the new dividend by the new divisor

Check by multiplying the quotient and the divisor.