Pre Algebra (Period 2 & 4)

Adding & Subtracting Decimals Equations 3-5

Most important... LINE UP THE DECIMALS!!
We did a number of examples from the textbook... check out this section!!


Make sure you use a SIDE BAR for your calculations!!


Multiplying & Dividing Decimal Equations 3-6


For multiplication, count the decimals and place the decimal from the right!!
THese are difficult to show here as well...but

( m/7.2) = -12.5

multiply both sides by -7.2

(-7.2)(m/-7.2) = -12.5(-7.2)
m = 90

Make sure to use a side bar and multiply carefully!!

s/2.5 = 5

multiply both sides by 2.5

(2.5)(s/2.5) = 5(2.5)
3 = 12.5


4.5 = m/-3.3

multiply both sides by -3.3
Notice your answer will be a negative!!

(-3.3) (4.5) = [m/-3.3](-3.3)

-14.85 = m

... and I used a sidebar in class to show the multiplication!!

For division

0.9r = -5.4
Divide both sides by 0.9

.9r = -5.4
0.9 0.9

r = -6

use a sidebar to divide carefully

divide both sides by -0.9
-0.9n = -81.81

-0.9n/-0.9 = -81.81/-0.9

notice that your solution will be a positive!!
and use a side bar to divide carefully

n = + 90.9

Algebra (Period 1)

Scientific Notation 5-4



You've had this since 6th grade!
It is just a bit more complicated


You restate very big or very small numbers using powers of 10 in exponential form


Move the decimal so the number fits in this range:

less than 10 and greater than or equal to 1
That is 1 ≤ n < 10 Scientific notation is the product of two factors: one of the factors is a number (1)1 ≤ n < 10 and the other factor (2) is a power of ten 
Count the number of places you moved the decimal and make that your exponent 
Very big numbers - exponent is positive 
Very small numbers (decimals) - exponent is negative (just like a fraction!) 

Remember that STANDARD notation is what you expect (the normal number)
 
When you multiply or divide scientific notations, use the power rules! 
 Just be careful that if your answer does not fit the scientific notation range, that you restate it. (2.2 x 10 ^-3)(3.0 X 10 ⁵)
(2.2)(3.0) X 10 ^-310 ⁵

6.6 X 10 ²

(6.1 x 10 ⁹)(2.5 x 10^-4)
(6.1)(2.5) x 10 ⁹⁺⁴
15.25 x 10 ⁵
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 10¹
put that back in your work
1.525 x 10¹ ⋅ 10⁵
1.525 X 10⁶


(2.5 x 10 ^-3)(4.0 X 10 ^-8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 ^-3)(10 ^-8)

(2.5)(4)= 10
so initially you have 10 X 10 ^-11
but 10 does not fall into the required 1 ≤ n < 10 so changed 10 into scientific notation or 1.0 X 10¹
so you have 1.0 X 10¹ X 10 ^-11
or 1 X 10 ^-10

(5.4 X 10^-6)(5.1 X 10 ^-8)

(5.4)(5.1) = 27.54
so
27.54 X 10 ^-6+-8
27.54 X 10 ^-14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 10¹
put it back into your work
2.754 x 10¹ ⋅ 10^-14
2.754 X 10 ^{1 + -14}
2.754 X 10 ^-13


What about
9.0 X 10 ⁸
3.0 X 10 ²

do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0


and
10⁸
10²

= 10 ^8-2
=10⁶
so the answer is
3 x 10 ⁶

6.9 X 10 ⁴
3.0 X 10 ⁸

2.3 X 10 ^4-8
2.3 X 10 ^-4

2.7 X 10 ¹²
9.0 X 10 ¹²
Divide 2.7 by 9
you get
0.3
10^{12 -12} = 10 ⁰
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10^-1
so the answer is
3.0 X 10 ^-1





6.0 x 10 ⁷
3.0 X 10²




First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 ^{7 -2} or 10⁵
so 2.0 X 10⁵

4.2 X 10⁵
2.1 X 10³

= 2 X 10²

2.5 X 10 ^-7
5.0 X 10 ⁶
first 2.5/5 = 0.5 and 10^-7/10⁶ = 10^-7-6 = 10^-13
so initially you have
0.5 x 10^-13 but 0.5 is not within 1 ≤ n < 10 so change 0.5 to scientific notation 5.0 X 10 ^-1 and mult that by 10^-13
5 X 10 ^-14


How about this harder one:

(3.6 X 10⁶)(4 X 10 ^-3
(4.8 x 10 ^-2)(1.2 X 10 ⁶)

put all your factors together and see if you can simplify before you begin...

(3.6)(4)
(4.8)(1.2)

You can simplify
(3.6)
(4.8)


isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 10^6-3
10 ^{-2 + 6}
which
becomes
2.5 X 10 ^3-4 or
2.5 X 10 ^-1

Math 6 Honors (Period 6 and 7)

Multiplying Decimals 3-8

According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!

You do NOT need to line up the decimal point when you are multiplying.

11.32 X 8.73

11.32
8.73

98.8236

estimate and you get 11 X 9 = 99

What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)

Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1


How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50






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.