Transcript Chapter 4 Solving and Applying Proportionsx

Chapter 4
Example 1
Ratio
• A comparison of two
numbers by division.
• a:b, a to b, a/b
• Must be same units
• If they can be changed into
the same units, you change
them and then reduce.
• Ex. Inches to Feet
Pounds to Ounces
Example 2
• Rate:
A ratio that has
different units.
• Unit Rate:
A rate with a
denominator of 1.
Examples:
65 miles per hour.
$0.45 per pound.
Rates must include units!
Example 3
• The table gives prices for
different sizes of the same
brand of apple juice. Find
the unit rate for each.
Price
Volume
$0.72 16 oz
$1.20 32 oz
$1.60 64 oz
Unit
Rate
$0.045 oz.
$0.0375 oz.
$0.025 oz.
Example 4
Rate
1. A 10-ounce bottle of
shampoo costs $2.40. What
is the cost per ounce?
Example 5
Rate
2. A 12-ounce bottle of juice
costs $1.80. What is the
cost per ounce?
Example 6
Dimensional Analysis.
When we convert from one
unit of measure to
another, we must choose
which conversion factor
or factors will produce
the appropriate unit(s).
This is called Dimensional
or UnitAnalysis.
Conversion Factors
Standard
Standard
Length
capacity
12 inches = 1 foot
8 ounces = 1 cup
3 feet = 1 yard
2 cups = 1 pint
36 inches = 1 yard
2 pints = 1 quart
1760 yards = 1 mile
4 quarts = 1 gallon
5280 feet = 1 mile
Standard
weight
16 ounces = 1 pound
2000 pounds = 1 ton
Conversion Factors
Standard
time
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
365 days = 1 year
Length - Bridge Units
1 km
0.625mi
1 in.
2.54
Mass - Bridge Units
1 kg
2.2 lbs
Length - Metric Units
1 km
1000m
1m
1000mm
1m
100 cm
1m
10 dm
1 cm
10 mm
POD 1 –
Complete. Show your
conversion factors.
1. 6 miles/hour = ____
yards/min.
2. 2 km/15 min = ___
meters/hour
3.
5 gals. = ______ cups
Complete. Show your conversion factors.
1
176
qt
2 pt1760
cup
km 41000
60 2
min
meters
62miles
1hourm
yards
yards
8000
80
176
__________
5 gal     
cups
__________
_________
601km
min1qt 1hour
1mile
min
15hour
min 1gal
hour
1 pt
6 1
POD 2 –
Complete. Show your
conversion factors.
1. 5 km/30min = ____
meters/hour
2. 13.7 miles/gal = ____ feet/qt
3.
3 miles = ______ yards
Complete. Show your conversion factors.
135.km
7miles5280
5280
ft1yard
1m
gal
feet
60 min
1000
meters
ft
 
 10
,000 yards
18084
3miles  
 5280
301min
hr1mile 1km
hrqt
gals 11mile
34ftqts
POD 1
• A cheetah ran 300 feet in
2.92 seconds. What was
the cheetah’s speed in
miles per hour?
POD 2
Copy the conversion units and
complete worksheet for
team points.
POD 2
• A sloth travels .15 miles per
hour. Convert this speed to
feet per minute.
Notes for Quiz
I.
II.
Ratio, Rate, Unit Rate
Dimensional Analysis
EX 5. PROPORTION
• An equation that states that
two ratios are equal.
POD
• I have a leaky faucet at my
house. It drips 6 oz of
water every 10 minutes.
How many gallons of water
am I losing in 30 days?
6oz 60 min 24 hr 1cup 1 pt 1 qt 1 gal
10 min 1 hr
1 day 8 oz 2 cup 2 pt 4 qt
Example 6
Properties of Proportions
a/b = c/d
• 1) Cross Products
• 2) Means ( product of the
inside) b c
• 3) Extremes (product of the
outside) d a
• 4) The product of the means
equals the product of the
extremes! bc = da
Example 7.
a. 18/50 = x/15
b. -2.5/y = -4/3
c. (x+4)/5 = (x – 2)/7
d. (x + 2)/14 = x/10
e. (y -15)/(y + 4) = 35/7
f. 3/(w+6) = 5/(w-4)
g. (d-7)/4 = (2d + 1)/3
Example 8.
1.
2.
3.
4.
5.
6.
7.
5/6 = c/9
8/d = -12/30
-8/11 = 12/v
(x + 3)/4 = 7/8
8/(b+10) = 4/(2b – 7)
(k+5)/10 = (k – 12)/9
1 qt/min = ___ gal/wk
Ex. 9
Similar Figures Figures that have the same
shape but not necessarily
the same size
Symbol for Similar Figures
Means
“is similar to”
Ex. 10
Properties of Similar
Figures • Corresponding angles are
congruent.
• Corresponding sides are in
proportion.
Example 11
Properties of Similar Figures
• The order of the letters
indicates the corresponding
angles.
ABC
DEF
A
D
B
E
C
F
Example 11
B
F
E
A
D
C
ABC
DEF
Homework
Page 186
Problems 53-61
Example 11
B
E
D
A
C
ABC
DEF
AB:AC = DE:DF
AB:BC = DE:EF
AC:BC = DF:EF
F
Example 12
B
ABC
DBE
D
A
Find the value of x.
Separate the shapes
E
x
C
Example 12
DBE
ABC
B
D
A
Set up proportion
B
E
6 = 18
5
x
6x = 90
6
6
x = 15
C
Ex. 13
ABC is similar to DEF.
The length of EF is 6. The
length of ED is 8. Find the
length of AB if the length of
BC is 9.
Hint: Draw the shape
and then set up your
proportion.
Note for Exam
1. Carry out a procedure to
perform operations with
matrices (including
addition, subtraction, and
scalar multiplication).
2. Carry out procedures to
solve linear equations for
one variable algebraically.
3. Carry out a procedure to
evaluate an expression by
substituting a value for the
variable.
Note for Exam
4. Carry out a procedure using
the properties of real
numbers (including
commutative, associative,
and distributive) to simplify
expressions.
5. Carry out a procedure
(including addition,
subtraction, multiplication,
and division by a
monomial) to simplify
polynomial expressions.
Note for Exam
6. Apply proportional
reasoning to solve
problems.
7. Represent applied
problems by using
matrices. Apply algebraic
methods to solve problems
in real-world contexts.
8. Exemplify elements of the
real number system
(including integers, rational
numbers, and irrational
numbers
Note for Exam
9. Use dimensional analysis
to convert units of measure
within a system.
10. Carry out a procedure to
evaluate an expression by
substituting a value for the
variable.
1. Carry out a procedure to perform operations with
matrices (including addition, subtraction, and scalar
multiplication).
2. Carry out procedures to solve linear equations for
one variable algebraically.
3. Carry out a procedure to evaluate an expression by
substituting a value for the variable.
4. Carry out a procedure using the properties of real
numbers (including commutative, associative, and
distributive) to simplify expressions.
5. Carry out a procedure (including addition, subtraction,
multiplication, and division by a monomial) to
simplify polynomial expressions.
6. Apply proportional reasoning to solve
problems.
7. Represent applied problems by using
matrices. Apply algebraic methods to solve
problems in real-world contexts.
8. Exemplify elements of the real number system
(including integers, rational numbers, and
irrational numbers
9. Use dimensional analysis to convert units of
measure within a system.
10. Carry out a procedure to evaluate an
expression by substituting a value for the
variable.
Homework
Page 192
Problems 1-10
Check the odd answers in the
back of the book.
POD
Copy and complete.
ABC is similar to XYZ.
The length of AC is 10. The
length of BC is 16. Find the
length of XZ if the length of
YZ is 12.
POD
Prentice Hall Lesson Quiz
Section 4-1
Example 14
A flagpole has a shadow of
102 ft long. A 6ft tall man
casts a shadow 17ft.
Long. Approximately
how tall is the flagpole?
Ex. 15
A map scale is 1in. : 10 miles
The map distance from
Anderson to Greenville is
2.25 inches.
Approximately how far is
the actual distance to
Greenville?
Similar Figures and Indirect
Measurement.
Turn in your books
to page 191.
Prentice Hall Examples
Chapter 4 Section 2
Examples 1,2,3
Homework
Page 192
Problems 2 – 14 even
Page 195
Checkpoint Quiz 1
Problems 1 – 10 all
POD
Prentice Hall Lesson Quiz
Chapter 4 Section 2
Percent Problems
Turn in your books
to page 199.
Prentice Hall Examples
Chapter 4 Section 3
Examples 1,2,3,4,5,6,7
Example 20
Two ways to solve:
Proportion
Is
=
%
Of
100
or
Equation
Percent Equations- you must
turn percents into decimals
and decimals into percents.
Of = multiply, is = , what = x.
Example 21
Proportion
Percent
IS = %
Of
100
what % of what # is what?
25 % of what # is 40?
40 % of 50 is what?
what % of 80 is 20?
Example 22
Solving Percents
Problems using
equations.
OF means times and IS means =
25 % of what # is 40?
40 % of 50 is what?
20 is what % of 80?
Homework
Page 200
Problems 2 – 30 even
POD
Prentice Hall Lesson Quiz
Chapter 4 Section 3
Problems 1-5 only
Examples
23, 24, 25, 26
Percent of Change.
Turn in your books
to page 204.
Prentice Hall Examples
Chapter 4 Section 4
Examples 1,2
Page 201
Check for Understanding
1-2
Example 27
Percent of Change
Amount of Change
Original Amount
Two Types
Percent of Increase
Percent of Decrease
Example 28
Find the percent of change if
the CD is on sale and its
price decreases from
$13.99 to $12.99. Round
to the nearest percent.
Percent of decrease = 7%
Example 29
In 1990, there were 1330
registered alpacas in the
US. By the summer of
2000, there were 29,856.
What was the percent of
increase in registered
alpacas.
Percent of increase = 2145%
Example 30
The price of a sweater
decreased from $29.99 to
$24.49. Find the percent
of decrease.
Change
Change
Change
$5.50
$29.99 $5.50
– $24.99
Original $29.99
Rounding to the nearest %
Percent of decrease = 18%
Example 31
Find the percent of change if
the price of a CD
increases from $12.99 to
$13.99. Round to the
nearest tenth.
Percent of increase = 7.7%
Homework
Page 207
Problems 30 – 40
Example 32
Juan earns a 5.5%
commission on his bicycle
sales. In September, he
earned $214.28 in
commissions. What were
his sales for the month?
Example 33
Pablo has a goal to lose 25
lb. He has lost 16 lb.
What percent of his goal
has he reached?
Example 34
You spent 16% of your
vacation money on food.
If you spent $48 on food,
how much money did you
spend on your vacation?
Example 35
A writer earns $3400 a
month. Last month she
spent $204 on food.
What percent of her
income was spent on
food?
POD
Kiko spends 30% of her
monthly income on rent.
If she pays $810 for rent
each month, what is her
monthly income?
Notes
Add indirect measurement
to test and percent word
problems
Example 36
Suppose that 62.5% of
freshmen entering a
college graduate from it.
If there are 2680
freshmen, how many will
graduate from that
college?
Example 37
Solving Percents Problems
with percents greater than
100% and less than 1%.
105 is 125% of what number?
what is 48 % of 250?
what % of 90 is 135?
Example 38
Estimating percents using
compatible numbers.
Compatible numbers are
numbers that are easy to
divide mentally.
Estimate 32% of 241
32% is close to 33 1/3 % = 1/3
240 is divisible by 3.
240 divided by 3 is 80
Example 39
Estimating percents using
compatible numbers.
Compatible numbers.
Estimate 65 % of 334
65% is close to 66 2/3 %
66 2/3 % = 2/3
2/3 means to divide by 3 then
multiply by 2.
333 is divisible by 3.
333 divided by 3 is 111
111 multiplied by 2 is 222
Fraction
Decimal
Percent
1/10
0.1
10%
2/10 = 1/5
0.2
20%
3/10
0.3
30%
4/10 = 2/5
0.4
40%
5/10 = ½
0.5
50%
6/10=3/5
0.6
60%
7/10
0.7
70%
Fraction
8/10= 4/5
9/10
1/8
2/8=1/4
3/8
4/8 = 1/2
5/8
Decimal
0.8
0.9
0.125
0.25
0.375
0.5
0.625
Percent
80%
90%
12.5%
25%
37.5%
50%
62.5%
Fraction
6/8= ¾
7/8
1/6
2/6 =1/3
3/6 = ½
4/6 = 2/3
5/6
Decimal
0.75
0.875
0.16
0.3
0.5
0.6
0.83
Percent
75%
87.5%
16 2/3%
33 1/3%
50%
66 2/3%
83 1/3%
Fraction
1/9
2/9
3/9=1/3
4/9
5/9
6/9=2/3
7/9
8/9
Decimal
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percent
11 1/9%
22 2/9%
33 1/3%
44 4/9%
55 5/9%
66 2/3%
77 7/9%
88 8/9%
Example 40
Number your paper (4 min)
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
Fraction
1
7/8
1/10
6
3/8
9
11
Decimal
0.75
3
4
7
8
0.6
12
Percent
2
87.5%
5
33 1/3%
37.5%
10
83 1/3%
Example 40
POD – Number your paper
1.3/4
7. 0.3
2. 75%
8. 0.375
3. 0.875
9. 3/5
4. 0.1
10. 60%
5. 10%
11. 5/6
6. 1/3
12. 0.83
Example 41
POD – Number your paper
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
Fraction
1
5/8
3/10
6
1/8
9
11
Decimal
0.25
3
4
7
8
0.8
12
Percent
2
62.5%
5
66 2/3%
12.5%
10
16 2/3%
Example 41
POD – Number your paper
1.1/4
7. 0.6
2. 25%
8. 0.125
3. 0.625
9. 4/5
4. 0.3
10. 80%
5. 30%
11. 1/6
6. 2/3
12. 0.16
Example 42
POD – Number your paper
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
Fraction
1
1/3
2/5
6
8
9
5/6
Decimal
0.2
3
4
7
0.125
0.9
11
Percent
2
33 1/3%
5
25%
12.5%
10
12
Example 42
POD – Number your paper
1.1/5
7. 0.25
2. 20%
8. 1/8
3. 0.3
9. 9/10
4. 0.4
10. 90%
5. 40%
11. 0.83
6. 1/4
12. 83 1/3%
Example 43
Probability
Probability – Tells how likely
an event is to occur.
Event –
- any outcome or
group of outcomes.
Outcome – the result of a
single trial.
Sample Space – all possible
outcomes.
Example 44
Probability of rolling an even
number on a number cube.
Event: Rolling an even number
Sample Space: 1,2,3,4,5,6
Favorable Outcome: 2,4,6
Probability of rolling an even
number on a number cube is
3 = 1
6
2
Example 45
Theoretical Probability
• P(event) = # of favorable outcomes
# of possible outcome P(event) = #
of good outcomes
total
# of possible outcome P(event) = good
total
• What is expected to happen.
Example 46
A bowl contains 12 slips of
paper, each with a different
name of a month. Find the
theoretical probability that a
slip selected at random from
the bowl has a name of a
month that starts with the letter
J.
Example 46
January
February
March
April
May
June
July
August
September
October
November
December
Example 46
P(J) = Number of “J” months
Total number of months
P(J) = 3 = 1
12 4
Example 47
Experimental Probability
•
# of times an event occurs
times experiment is done
What actually happens
# of
Example 48
After receiving complaints, a
skateboard manufacturer inspected
1000 skateboards at random. The
manufacturer found no defects in
992 skateboards. What is the
probability that a skateboard
selected at random had no defects.
Write the probability as a percent.
Round to the nearest tenth of a
percent.
Example 48
P=
# without defects
Total number inspected
P = 992
1000
P = 0.992
P = 99.2%
Example 49
If you ordered 2450 skateboards
from the same manufacturer,
how many skateboards would
you expect to have no defects?
Example 50
A manufacturer inspects 700
light bulbs. She finds that
the probability that a light
bulb works is 99.6%. There
are 35,400 light bulbs in the
warehouse. Predict how
many light bulbs are likely to
be defective.
Example 51
Independent Events
Events that do not influence
each other.
• Example
–
spinner and a die
coin toss and drawing a card
Example 52
Suppose you roll a red
number cube and a blue
number cube. What is the
probability that you will roll a
3 on the red cube and an
even number on the blue
cube?
Example 53
With Replacement
(Independent Events)
In a word game, you choose a
tile from a bag containing the
letter tiles shown:
A,E,I,O,U,A,A,A,E,E,O,U,U,0,0
What is the probability that you
will choose an A and then an
E?
Example 54
Dependent Events
Events that influence
each other.
Example 55
Without Replacement
Suppose you have a bag of
marbles. There are 4 red, 5
blue, and 8 yellow marbles.
What is the probability of
selecting a red marble, then
not replacing it, and then
selecting a blue marble next?
Example 56
Suppose a teacher must select 2
high school students to represent
their school at a conference. The
teacher randomly picks names
from a hat that contains the names
of 3 freshmen, 2 sophomores, 4
juniors and 4 seniors. What is the
probability that a sophomore and
then a freshman are chosen?
Deck of Cards:
52 Cards
Hearts Diamonds
Clubs Spades
A 2 3 4 5 6 7 8 9 10 J Q K
Example 57
A coin is flipped 4 times. What is
the probability of getting 3 heads?
TTTT
TTHH HHHH HHTT
TTTH
TTHT
THTT
THHT HHHT
THTH HHTH
THHH HTHH
HTTH
HTHT
HTTT
P (3 heads) = 4 = 1
16
4
Ex 58 - Notes for test
I. Proportions
Pg 195
II.
1-5
Percents
Pg 217 1- 4
III.
IV.
V.
VI.
Pg 230 1- 4
Pg 230 5 – 7
Similar Figures
Percent of Change
Probability
Probability of
Compound Events
VII. Dimensional Analysis
Homework
Page 207
Problems 2 – 12 even
Page 214
Problems 2-32 even
POD
Team Sheet from Homework
Page 222
Problems 2-32 even
POD
Team Sheet from Homework
Page 222
Problems 2-32 even
POD
Page 228
Problems 14 – 44 evens
In pairs.
14.
16.
18.
20.
22.
24.
26.
28.
30.
2
-6
6
7.5 m
36 ft
2.5
850
$220
25% dec.
32. It costs a dinner $.11 to a cup
of tea. They sell it for $.75.
Find the percent change.
34. 0
36. 1/3
38. a)1/4 b) P(0,1,2or4)
40. Dependent, 2/45
42. If one person is dependent on
another, the 1st person relies
on the 2nd for support.
44. Once you select one sock,
there are fewer socks for the
second choice.
Homework
Page 230
Problems 1-7,11-24,27
15
2.
7.5
3.
2.4
4.
20
5.
40
6.
64%
7.
20
11. 11.1% Increase
12. 25% Decrease
13. 10% Decrease
14. 33.3% Increase
1.
15. 3/5
16. 1/5
17. 2.24
18. 1/6
19. 3080
20. 1
21. 162.5 miles
22. 12.5 ft
23. a)9/14 b) ½
c) 0
24. 12 for $6.99
27. a)¼ b)4/15 c)¼
15
2.
7.5
3.
2.4
4.
20
5.
40
6.
64%
7.
20
11. 11.1% Increase
12. 25% Decrease
13. 10% Decrease
14. 33.3% Increase
1.
15. 3/5
16. 1/5
17. 2.24
18. 1/6
19. 3080
20. 1
21. 162.5 miles
22. 12.5 ft
23. a)9/14 b) ½
c) 0
24. 12 for $6.99
27. a)¼ b)4/15 c)¼
15
2.
7.5
3.
2.4
4.
20
5.
40
6.
64%
7.
20
11. 11.1% Increase
12. 25% Decrease
13. 10% Decrease
14. 33.3% Increase
1.
15. 3/5
16. 1/5
17. 2.24
18. 1/6
19. 3080
20. 1
21. 162.5 miles
22. 12.5 ft
23. a)9/14 b) ½
c) 0
24. 12 for $6.99
27. a)¼ b)4/15 c)¼
15
2.
7.5
3.
2.4
4.
20
5.
40
6.
64%
7.
20
11. 11.1% Increase
12. 25% Decrease
13. 10% Decrease
14. 33.3% Increase
1.
15. 3/5
16. 1/5
17. 2.24
18. 1/6
19. 3080
20. 1
21. 162.5 miles
22. 12.5 ft
23. a)9/14 b) ½
c) 0
24. 12 for $6.99
27. a)¼ b)4/15 c)¼
15
2.
7.5
3.
2.4
4.
20
5.
40
6.
64%
7.
20
11. 11.1% Increase
12. 25% Decrease
13. 10% Decrease
14. 33.3% Increase
1.
15. 3/5
16. 1/5
17. 2.24
18. 1/6
19. 3080
20. 1
21. 162.5 miles
22. 12.5 ft
23. a)9/14 b) ½
c) 0
24. 12 for $6.99
27. a)¼ b)4/15 c)¼
POD
Textbook Page 194
Problems 36 and 38
Fractions
D
P
1/3
1
2
3
.875
4
5
6
12.5%
9/10
7
8
9
10
37.5%
Homework
Textbook Page 192
Problems 9,(10 -22 evens)
for team points.
Name
11/25/08
Lesson 114
TEST
Turn in funsheets
after test.
1 Kg = 2.2 lbs
1 L = 1.06 qts
1 Km = 0.625 mi.
2.54 cm = 1 in.
You must use all 4
numbers along
with any operation
signs to make 24.
You can only use
each number 1
time.
Your expression must follow
the order of operations!
POD 24 GAME
9
8
6
1
You must use all
4 numbers along
with any
operation sign to
make 24. You
can only use
each number 1
time.
24 GAME
4
8
7
5





Team Time Monday
Graphing Calculators - Matrices
Fractions – Check page 726 2 – 30 evens on
9/29/08
Square roots and Cubed Roots – quiz on
10/6/08
Transformations
Bonus
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