Chapter 4 Review

1
Chapter 4 Review
Ratios and Proportions Similar Triangles Trig
Ratios Percentages Percent of Change Probability
Odds Mixture Problems
2
Solve the proportion
x - 5 9 3 - x 7
7(x - 5) 9(3 - x) 7x - 35 27 - 9x 9x 35
35 9x 16x 62 X 3.875
3
Solve the proportion
2x - 5 x 3 10 4
4(2x - 5) 10(x 3) 8x 20 10x 30 -10x 20
-10x 20 -2x 50 X -25
4
There are 142 Carats in one ounce. In 1905, a
rough diamond weighing 1306 carats was
discovered. To the nearest ounce, how much did
the diamond weigh?
Carats 142 1306 Ounces 1 x X
1306/142 X 9 ounces
5
Similar Triangles
x
15
6
y
3
5
15/5 y/3 y 9
x/5 6/3 x 10
6
Completing the TriangleIdentifying all angles
and legs
Find angle M TanM O/A 8/11 Tan-1(8/11)
36.0
Given the two legs
Find angle N 180 - 90 - 36.0 54.0
54.0
N
h
13.6
Find hypotenuse h Cos(36.0) 11/h .809
11/h h 11/(.809) h 13.6
8
M
11
36.0
7
Trigonometric Ratios

• Sine, Cosine, Tangent
• Ratios used to determine the angle measurements
  and side lengths of right triangles.

8
Definitions
Hypotenuse - the side opposite the 90 angle of a
right triangle. It is always the longest side on
a right triangle.
Opposite side for angle A
A
Adjacent side for angle A
9
Sine, Cosine, Tangent

• We use these ratios as a way to find missing
  angle measures or lengths of triangle legs.
• Sin(angle measure) leg measure
• hyp. measure

10
SOH CAH TOA

• Sine is opposite over hypotenuse
• Cosine is adjacent over hypotenuse
• Tangent is opposite over adjacent

11
Sin x
Opposite side
8/13 .615
Hypotenuse
Cos x
Adjacent side
12/13 .923
Hypotenuse
Tan x
Opposite side
8/12 .667
Adjacent side
13
8
X
12
12
Sin 33
Opposite side
Sin 33 x/13
Hypotenuse
.545 x/13 (13)(.545) x x 7.08
13
7.08
x
33
y
13
Cos 33
Adjacent side
Cos 33 y/13
Hypotenuse
.839 y/13 13(.839) y y 10.9
13
x
33
y
10.9
14
Tan 33
Opposite
Tan 33 x/13
Adjacent
.649 x/13 13(.649) x x 8.44
h
8.44
X
33
13
15
Sine A
To find the measurement of angle A
Sin A 5/13
Opposite side
Hypotenuse
Sin (5/13) 22.6
-1
13
5
A
12
16
Cosine A
To find the measurement of angle A
Adjacent side
Cos A 12/13
Hypotenuse
-1
Cos (12/13) 22.6
13
5
A
12
17
Tangent A
To find the measurement of angle A
Opposite side
Tan A 5/12
Adjacent Side
-1
Tan (5/12) 22.6
13
5
A
12
18
Completing the TriangleIdentifying all angles
and legs
Find angle M SinM o/h 8/20 Sin-1(8/20)
23.6
Given a leg and the hypotenuse
Find angle N 180 - 90 - 23.6 66.4
66.4
N
20
Find leg x Cos(23.6) a/h x/20 .916 x/20 x
(.916)20 18.3
8
23.6
M
x
18.3
19
Completing the TriangleIdentifying all angles
and legs
Find angle M TanM O/A 8/11 Tan-1(8/11)
36.0
Given the two legs
Find angle N 180 - 90 - 36.0 54.0
54.0
N
h
13.6
Find hypotenuse h Cos(36.0) 11/h .809
11/h h 11/(.809) h 13.6
8
M
11
36.0
20
Completing the TriangleIdentifying all angles
and legs
Find angle N 180 - 90 - 18 72
Given an angle and a leg
Find leg x Tan(18) x/20 .324 x/20 (.324)20
6.5
N
72
h
Find hypotenuse h Use the Pythagorean Thereom
x
6.5
18
20
21
The Pythagorean Thereom
a2 b2 c2
(6.5)2 (20)2 c2 442.25 c2 c
v442.25 21.0
c
21.0
a 6.5
b 20
22
Percents
35 of what number is 88?
88
35
.35

x
100
.35x 88 x 251.4
23
Percents
Find 112 of 500
What is
x
112
1.12

100
500
x (1.12)(500) x 560
24
Percentages

• Changing a fraction to a percentage

25

• Change fraction to decimal by completing division
  problem.
• 8/11 .727
• Multiply by 100 and add sign.
• 72.7
• Or move decimal two spaces to the right and add
  sign.

26
Percents___ of _____ is _____
35 of what number is 88?
88
35
.35

x
100
.35x 88 x 251.4
27
Percents
What percent of 17 is 34?
34
P

100
17
P (34)(100)/17 x 200
28
Percents
What number is 0.3 of 62.7?
x
.3

100
62.7
x (.3)(62.7)/100 x 0.1881
29
Percents
Find 112 of 500
What is
x
112
1.12

100
500
x (1.12)(500) x 560
30
33 is what percent of 45? 33/45 P/100 P
73.3 18 is 44 of what number? 18/x 44/100 x
18/.44 40.9 What number is 45 of 66? X/66
45/100 x 66(.45) 29.7
31
123 is what of 75? 123/75 P/100 P 164 11
is 62 of what number? 11/x 62/100 x 11/.62
17.7 What number is 7 of 23? X/23 7/100 x
23(.07) 1.61
32
I prt Interest principleratetime Amount
earned or paid amount borrowed or
invested times rate of
interest times time of loan or investment
33
Tom invested 10,000 in an account that paid 5
annual interest for 2 years. How much interest
did he earn? I prt I (10,000)(.05)(2) 1000
34
Percent Change
change New - Old Old
The average cost of new house has risen from
85,000 to 325,000 over the past thirty year.
What is the percent change during this time?
change (325,000-85,000) 85,000 2.83 or
283
35
The price of a winter coat was decreased from
89.95 to 71.96. What was the percent of
decrease?
New-old 71.96 - 89.95 Old 89.95 X
20
If the sale price was reduced an additional 25
and then bought, what was the final amount paid
if the sales tax was 6.
71.96 (.75) 53.97 53.97(1.06) 57.21
36
Start Value Change Final Value
Increase means 100 increase Final
change Decrease means 100 decrease Final
Change Example A 40 increase means there is now
100 40 percent of the original, or 140, which
as a decimal is 1.4 A 30 decrease means there
is now 100 30 percent of the original, or 70,
which as a decimal is .7 Example A number
increased by 20 is now 48. Find the number. X
1.2 48 x 48 1.2 40 90 decreased by 25 is
what? 90 .75 67.5
37
Start Value Change Final Value
What number decreased by 35 is 4875? X .65
4875 x 7500 A number increased by 40 is 112.
Find the number. X 1.4 112 x 80 70
increased by 30 and then decreased by 30 is
what? 70 1.3 91 91 .7 63.7
38
Mixture Problems
39
An advertisement for an orange drink that claims
that the drink contains 10 orange juice. How
much pure orange juice would have to be added to
5 quarts of the drink to obtain a mixture 40
orange juice?
Concentration

Volume x
.5
.10
5
10 drink
y
1.00
y
Pure OJ
.4(5 y)
.4
5 y
Final
.5 y .4(5 y) .5 y 2 .4y -.5 -.4y
-.5 -.4y .6y 1.5
y 2.5
40
Sam is investing 6000 in two accounts, part 4.5
and the remainder at 6. If the total annual
interest earned from the two accounts is 279,
how much did Jane deposit at each rate?
Interest
Rate
Principal x
.045x
.045
x
4.5
6000 - x
6
.06
.06(6000 - x)
6000
total
___
279
.045x .06(6000 - x) 279 .045x 360 - .06x
279 -360 -360
-.015x -81
x 5400 _at_ 4.5 600 _at_ 6
41
A train travels a city heading west and travels
at 45 mph. Six hours later, a second train
leaves from the same place and travels in the
same direction at 75 mph. How long will it take
for the second train to overtake the first train?
Time
Distance
Rate x
45
1st train
45(t 6)
t 6
75t
t
75
2nd train
45(t 6) 75t 45t 270 75t -45t
-45t 270 30t
t 9 hours
42
At the Golden Oldies Theater, tickets for adults
cost 5.50 and tickets for children cost 3.50.
How many of each kind of ticket was purchased if
21 tickets were bought for 83.50?
value
cost
tickets x
5.5a
5.5
a
Adults
3.5(21 - a)
3.5
21 - a
Children
83.50
21
Combined
5.5a 3.5(21 - a) 83.50 5.5a 73.5 - 3.5a
83.50 -73.5 -73.5 2a
10
a 5 adult 16 Child
43
A liter of cream has 9.2 butterfat. How much
skim milk containing 2 butterfat should be added
to the cream to obtain a mixture with 6.4
butterfat?
Concentration

Volume x
.092
.092
1
cream
.02y
.02
y
Skim milk
.064(1 y)
.064
1 y
Final
.092 .02y .064(1 y) .092 .02y .064
.064y -.092 - .064y -.092 - .064y -.044y
-.028
y .64 litter
44
Joe and Janna leave home at the same time,
traveling in opposite directions. Joe drives 45
mph and Janna drives 40mph. In how many hours
will they be 510 miles apart?
Distance
Time
Rate x
45t
t
45
Joe
40t
t
40
Janna
510
Combined
45t 40t 510 85t 510 t 6 hours
45
How many pounds of candy worth 3.60 per pound
must be added to 10 pounds of chocolate worth
2.40 per pound to form a mixture worth 3.30?
Value
/ lb
Volume x
3.6x
3.60
x
Candy
24
2.40
10
Chocolate
3.3(10 x)
10 x
Combined
3.30
3.6x 24 3.3(10 x) 3.6x 24 33
3.3x -3.3x - 24 -24 -3.3x .3x 9
x 30
46
Mrs.Smith invested a portion of her 15,000 at
10 annual interest and the rest at 6 annual
interest. If her investments earned 1260 for
the year, how much did she invest at 10?
Interest
Rate
Principal x
.1x
.10
x
10
15000 - x
6
.06
.06(15000 - x)
15000
total
___
1260
.1x .06(15000 - x) 1260 .1x 900 - .06x
1260 -900 -900
.04x 360
x 9000
47
A car radiator has a capacity of 14 quarts is
filled with a 20 antifreeze solution. How much
must be drained off and replaced with pure
antifreeze to obtain a 40 antifreeze solution?
Concentration

Volume x
.2(14 - y)
.20
14 - y
20 anit-freeze
y
1.00
y
Pure antifreeze
5.6
.4
14
Final
.2(14 - y) y 5.6 2.8 - .2y y 5.6 -.2.8
-2.8 .8y 2.8
y 3.5 quarts
48
A train travels a city heading west and travels
at 50 mph. Three hours later, a second train
leaves from the same place and travels in the
same direction at 65 mph. How long will it take
for the second train to overtake the first train?
Time
Distance
Rate x
50
1st train
50t
t
65(t - 3)
t - 3
65
2nd train
t 13 2nd train took 10 hours
50t 65(t - 3) 50t 65t - 195 -65t
-65t -15t -195
49
Mrs.Smith invested a portion of her 15,000 at
10 annual interest and the rest at 6 annual
interest. If her investments earned 1260 for
the year, how much did she invest at 10?
Interest
Rate
Principal x
.1x
.10
x
10
15000 - x
6
.06
.06(15000 - x)
15000
total
___
1260
.1x .06(15000 - x) 1260 .1x 900 - .06x
1260 -900 -900
.04x 360
x 9000
50
Probability Odds

• Probability - The Desired / All possible
• Winners All Possible
• Odds - The Desired / The Undesired
• Winners LosersThese are odds for something to
  happen
• Most odds in sports are odds against something
  happening.

51
Probability and odds
What is probability of pulling a green
marble? Winners/ all possible 3/16 What are the
odds of pulling a green marble? Winners to
Losers 313
52
Probability and odds
1) What is probability of pulling a red
marble? 2) What are the odds of pulling a green
marble? 3) What is the probability of pulling a
red or a blue marble? 4) What are the odds of
pulling an orange marble? 5) What is the
probability of pulling a black marble?