Pre Algebra (Period 2 & 4)

Area & Parallelograms 10-1 & Area of Triangles & Trapezoids 10-2
AREA-
all these formulas are related to the basic concept of A = bh
Area of Parallelograms 10-1

Area of rectangles and parallelograms = (base)(height)





Now we're starting area!
Where the perimeter/circumference fenced in my puppy, the area of the yard will tell me how much sod (grass) I should buy to stop the puppy's paws from getting muddy!

Area for me is all basically the length of the base times the height of the figure

A = bh

In a parallelogram, whether it's a rectangle, rhombus, square or other parallelogram

A = bh with the height being a line perpendicular to both bases (not the slanted side!)


You have learned the area of a rectangle as A = lw, but the l = b and the w = h

You may have learned the area of a square as A = s² , but that's because the b = h


Any parallelogram can be split into 2 triangles using a diagonal.

Triangles & Trapezoids 10-2
Area of triangle = (1/2)(base)(height or altitude)


Area of trapezoid = (average of the 2 bases)(height)
Any parallelogram can be split into 2 triangles using a diagonal. Because of this, the area of a triangle is half that of a parallelogram.

A = 1/2 bh


A trapezoid has 2 bases that ARE NOT EQUAL. So which base is THE base?

If you use the smaller base, you won't have enough sod for your yard and the puppy's paws are still getting muddy.

If you use the larger base, you'll have too much sod for your yard and the extra will rot.

Sooooooooo..... you actually need to take the average of the two bases times the height

A = (average of the 2 bases)(height)

A = (b₁ + b₂)h /2

Math 6 Honors (Period 6 and 7)

Percent of Increase or Decrease 9-4

Let's say we have an iPod that originally sold for $260. It is on sale for $208. What is the amount of change? "How much did you save?"
Just subtract
260-208 = 52
$52.
What is the percent of change?
The percent of change = amount of change/original
52/260 - x/100
or just divide 52 by 260 = .2
which is 20%
REMEMBER: The denominator in the formula is ALWAYS the ORIGINAL AMOUNT.
Amount of change = percent of change X the original amount.
Find the new number when 75 is decreased by 26%
Amount of change - 26% (75
= .26(75)
=19.5
Now take the difference ( the amount of change) and subtract THAT from 75
75- 19.5 = 55.5

Remember the circle with the various parts of this formula?

Difference or amount of change
% of change X original amount

Difficult to show here so if you missed these notes make sure to ask a fellow student to see this!! IT is a great way to remember what to do!!

State the increase or decrease. Tell what the amount of change is and the percent of change.
from:
10 to 12
increase
amount of increase: 2
% of change : 20%

4 to 3
decrease
amount of decrease:
% of change : 25%

2 to 5
increase
amount of increase: 3
% of change : 3/2 = 1.5 = 150%

12 to 6
decrease
amount of decrease: 6
% of change : 50%

6 to 12
increase
amount of increase: 6
% of change : 6/6 = 1 = 100%

Find the new number produced when the given number is increased or decrease by the given percent.

120; 20% decrease
120(.20) = 24 120 -24 = 96

30; 10% decrease
30(.10) = 3 30 -3 = 27

48: 50% increase
48(.50) = 24 48 + 24 = 72

128: decrease by 25%, then increased by 25%
What... why multiply by .25 if you can use a fraction and work smarter?
128(1/4) = 32
128 - 32 = 96
then 96 ( 1/4) = 24
96 + 24 = 120

Did you think it would be the starting number? Why wasnn't it?

Pre Algebra (Period 2 & 4)

Circles 9-6

A set of points equidistant from a center point

Radius - one endpoint on circle and one is the center (1/2 diameter)

Diameter - both endpoints on the circle and goes through the center point (twice radius)

Chord - both endpoints on the circle but not necessarily through the center

A VERY SPECIAL RELATIONSHIP WAS FOUND IN B.C.E. TIMES:
IF YOU DIVIDED THE CIRCUMFERENCE BY THE DIAMETER OF ANY SIZE CIRCLE, YOU WOULD ALWAYS GET ABOUT 3.14


Circumference = the circle fence - distance around the circle -
 
(pi)(diameter) = 2(pi)r

If you have radius, double it to get the diameter


Note: the circumference is always about triple the diameter (plus a little bit!)


Circles have 360 degrees and every circle is similar to every other circle


Central angles: vertex is the center point of a circle

Circles have 360 degrees.

So 1/4 of a circle is 90 degrees and 1/2 is 180 degrees


Review central angles with fractions, percents and degrees
We used the example from the book
Lunch (l) 25%
Recreation (r) 20%
Clothes (c) 15%
Savings (s) 40%
We need to change these percents to degrees. WE know that a circle is 360˚
so set up a proportion
% = unknown
100 360

25 = l
100 360
l = 90
so Lunch portion of the circle graph is 90˚

20 = r
100 360

Solve for r
r = 72
so the Recreation portion of the circle graph is 72˚

15 = c
100 360

Solve for c
c= 54
So the clothes portion is 54˚

40 = s
100 360

solve for s
s = 144
so the Savings portion of the cirlce graph is 144˚

Area: Circles 10-3

A = pi ∙ r²

Math 6 Honors (Period 6 and 7)

Computing with Percents 9-3
The statement 20% of 300 is 60 can be translated into the following equations
20/100(300) = 60 or 0.20 •300 = 60

EQUATION METHOD:
Notice the following relationship between the words and the symbols
20% of 300 is 60
0.20 • 300 = 60

WRITE THE PROBLEM OUT AND THEN DIRECTLY UNDER THE "IS" WRITE AN EQUAL SIGN. DIRECTLY UNDER THE WORD 'OF" WRITE A MULTIPLICATION SIGN. iF YOU ARE GIVEN A % CHANGE IT FIRST TO A DECIMAL. THEN BRING DOWN ALL THE OTHER NUMBERS GIVEN IN YOUR PROBLEM. LET x OR n REPRESENT YOUR VARIABLE... THAT IS THE "WHAT " PART OF YOUR PROBLEM.

A similar relationship occurs whenever a statement or a question involves a number that is a percent of another number

What is 8% of 75?
Let n represent the number asked for
What number is 8% of 75?
n = 0.08 • 75

solve
What percent of 40 is 6?
let n represent the percent asked for.

What percent of 40 is 6?
n% • 40 = 6
n% • 40 = 6
n% (40)/40 = 6/40
n% = 6/40
n/100 = 6/40
(100) n/100 = (100) 6/40
n=15 so 15% of 40 is 6

140 is 35 % of what number?
let n represent the number asked for
140 is 35% of what number?
140 = 0.35 • n
140 = 0.35n
140/0.35 = 0.35n/0.35 divide carefully!! Watch those decimals!!
400 = n
so 140 is 35% of 400

Always check to see if your answer is logical.

PROPORTION METHOD

In these types of percent problems you are always know three parts of the following proportion

n/1oo = a/b
or better yet
n/100 = is/ of

The n represents the %

Read the problems carefully and you can easily determine which is the "is" and which represents the 'of"
For example:
What percent of 40 is 6?
What percent -- from the problem above indicates that we DO NOT know the n
of 40-- hmm... then 40 must be the 'of' and
similarly is 6 means that 6 represents the 'is'

n/100 = 6/40 solve as a proportion
and you get n= 15 but since it asked us to state the 5 your answer is 15%

140 is 35 % of what number?
In this problem I notice 35% right away so that is the n!!
Then I read the problem again and notice 140 is... hmmm.. THat says 140 must be the is

35/100 = 140/ x I do not know the 'of'

Solve again
x = 400

Math 6 Honors (Period 6 and 7)

Percent and Fractions 9-1

The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100

A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100

Our book’s example is as follows;
During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws. 17/25 or 7/10
We have calculated this type of problem before.. this time when we compare fractions use the common denominator 100, even if 100 is not the LCD of the fractions.
17/25 = 68/100 and
7/10 = 70/100

Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.
the ratio of a number to 100 is called a percent. We write percents by using the symbol %
so
17/25 =68% and
7/10= 70%

Rule
To express the fraction a/b
as a percent, solve the equation

a/b = n/100

for the variable n and write n%


Express 17/40 as a percent
n/100 = 17/40 multiply both sides by 100 100 ( n/100) = 17(100)/40
n = 17(100)/40 n = 85/2 n= 42½

Therefore, 17/40 = 42 1/2 %


Rule
To express n% as a fraction, write the fraction

n/100 in lowest terms


Express 7 ½ % as a fraction in lowest terms

7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal?
multiply the numerator and the denominator by 10
7.5(10)/100(10) simplify
Similarly, you could change a mixed numebr into an improper fraction
5 3/8% becomes 43/8 % and to change that to a fraction simple divide by 100
That looks messy but if you remember that to divide by 100 you are actually multiplying by 1/100
(43/8) (1/100) = 43/800 and you are finished with your calculations!! EASY!!

Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%

1 = 100/100 = 100%

165/100 = 165 %

Write 250% as a mixed number in simple form

250% = 250/100

250/100 = 2 50/100 = 2 1/2

The town of Wonderful spends 42% of its budget on education. What percent is used for other purposes?
the whole budget is represented by 100%. Therefore, the part used for other purposes is

100 - 42 or 58%

Percents and Decimals 9-2

By looking at the following examples, you will be able to see a general relationship between decimals and percents

57% = 57/100
0.79 = 79/100 = 79%
113% = 113/100 = 1 13/100
0.06 = 6/100 = 6%

Rules

To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign

57% = 0.57
113% = 1.13

To express a decimal as a percent, move the decimal point two places to the right and add a percent sign

0.79 = 79%
0.06 = 6%


In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method

Rule
To express a fraction as a percent, first express the fraction as a decimal
and then as a percent

Express 7/8 as a percent

Divide 7 by 8
7/8 = 0.875 = 87.5%

Express 1/3 as a percent
divide 1 by 3
0.33333….. it’s a repeating decimal
express the decimal as a percent 0.333… = 33 1/3%
so, to the nearest tenth of a percent = 33.3% but it is much more accurate to keep the 1/3 and write 33 1/3%