Proportion 9 PPT

1. Classroom Rules:
Queen Melvs’ Rule

2. Queen Melvs’ Rule
1. Be on time in class. I want you to be
here inside the classroom before the
discussion started.
2. Be active in class. I want everyone to
participate during our discussions.
3. Raise your hand if you want to ask
something to clarify or ask permission.
4. Respect everyone in the class.

3. Queen Melvs’ Rule
5. Avoid unnecessary noise to make sure
that everyone can listen and pay
attention to our lesson.
6. Keep your language clean and
appropriate for the classroom setting
7. And lastly, do your best. I want you to
put effort into understanding our
lessons.

4. Activity:
“Figure Me Out”

5. Mechanics of the Game:
1. The class will be divided into 4 groups
and each group has a chalk and a board.
2. The members of the group will study
the scrambled letters and try to
unscramble or rearrange the letters to
form a word.
3. Arrange the letters in the correct
sequence.

6. Mechanics of the Game:
4. Each item is given 1 minute time to
arrange the scrambled word.
5. After the time is up. One member of the
group will raise their board.
6. The group who will get the right answer
will get 1 point.
7. The group that gets the highest points
will be the winner.

7. I O R T A
R A T I O

8. P O I O T P O R R N
P R O P O R T I O N

9. T I T Y N A U Q
Q U A N T I T Y

10. P L I F I S M I O N A C T I
S I M P L I F I C A T I O N

11. S O R S C
I O N T T M U L C A P I L I
C R O S S
M U L T I P L I C A T I O N

12. M E N S A
E X E R T M E
R O P E R P Y T
M E A N S
E X T R E M E
P R O P E R T Y

13. Activity:
Study the figures and complete the table that
follows by indicating proportional quantities on
the appropriate column.

14. Figure A.
Ratios/Rate Proportional Quantities
Feet : Inches 3 ft : __ in = __ ft : 72 in
4 ft : __ in = __ ft : 108 in
36 6
48 9

15. Ratios/Rate Proportional Quantities
Minutes :
Gallons
2 min : __ gal = __ min : 30 gal
Figure B.
3 min : __ gal = __ min : 40 gal
10 6
15 8

16. PROPORTIONS
PREPARED BY: MELVIN VERDADERO

17. Ratio
• Ratio is the comparison between two
quantities of the same units.
• It is the result of comparing them by
division.
• Since the quantities are of the same kind,
ratio doesn’t make use of units.
• The comparison of quantities that
considers different units is called rate.

18. The ratio or rate can be represented in
three ways and can be written as:
“a to b” a:b
𝑎
𝑏

19. 1.Find the ratio of a to b if a = 6 cm and b =
8 cm.
2.Find the ratio of 1 meter to 20 centimeters.
3.Write the ratio of the number of female
students to the number of male students of
Grade 9 Narra in different forms.
Example:
6:8 or 3:4
100:20 or 5:1

20. Proportion
• A proportion is the equality between two
ratios.
• If ratios 𝑎: 𝑏 and 𝑐: 𝑑 are equal, the proportion
formed can be written in two ways:
𝑎: 𝑏 = 𝑐: 𝑑 or
𝒂
𝒃
=
𝒄
𝒅
,
where 𝑏 ≠ 0, 𝑑 ≠ 0
and is read as “a is to b as c is to d”.

21. Proportion
• Each quantity in a proportion is called term
of the proportion. Thus, in the proportion 𝑎:
𝑏 = 𝑐: 𝑑, a, b, c, and d are the terms.

22. Example:
In the given proportion,
15 and 6 are called means;
2 and 45 are the extremes

23. How to determine if two ratios form a proportion
• Simplification
• Product of the Mean and Extremes
or Cross Multiplication.

24. Simplification
• If two ratios can be simplified into the same ratio,
then the two ratios form a proportion.
Which of the following ratios are equal?
1. 32 : 80 =
2. 27 : 72 =
3. 18 : 63 =
4. 34 : 85 =
2 : 5
3 : 8
2 : 7
2 : 5
Since the ratios in items 1 and 4 have the same simplest
form, then 32:80 is equal to 34:85.This can be written as
a proportion, 32: 80 = 34: 85 or 32/80 = 34/85.

25. Product of the Mean
and Extremes
32 : 80 = 34 : 85
Cross Multiplication
32: 80 = 34: 85
(80)(34) = (32)(85)
2,720 = 2,720
32
80
=
34
85
32
80
=
34
85
(32)(85) = (80)(34)
2,720 = 2,720

26. 1. 4:12 = 9:27
2. 26:65 = 8:20
3. 85:17 = 80:16
Example
Determine if the following are proportions or not
108 = 108 proportion
520 = 520 proportion
1360 = 1360 proportion

27. Properties of Proportion
Cross Multiplication
Property / Means-Extremes
Property
If
𝑎
𝑏
=
𝑐
𝑑
then 𝑎𝑑 = 𝑐𝑏; 𝑏 ≠ 0; 𝑑 ≠ 0
Alternation Property If
𝑎
𝑏
=
𝑐
𝑑
then
𝑎
𝑐
=
𝑏
𝑑
; 𝑏 ≠ 0; 𝑐 ≠ 0; 𝑑 ≠ 0
Inverse Property /
Reciprocal Property
If
𝑎
𝑏
=
𝑐
𝑑
then
𝑏
𝑎
=
𝑑
𝑐
; 𝑎 ≠ 0; 𝑏 ≠ 0; 𝑐 ≠ 0; 𝑑
≠ 0
Addition Property If
𝑎
𝑏
=
𝑐
𝑑
then
𝑎+𝑏
𝑏
=
𝑐+𝑑
𝑑
; b≠ 0; 𝑑 ≠ 0
Subtraction Property If
𝑎
𝑏
=
𝑐
𝑑
then
𝑎−𝑏
𝑏
=
𝑐−𝑑
𝑑
; 𝑏 ≠ 0; 𝑑 ≠ 0
Sum Property of the Original
Proportion
If
𝑎
𝑏
=
𝑐
𝑑
, then
𝑎
𝑏
=
𝑐
𝑑
=
𝑎+𝑐
𝑏+𝑑
; 𝑏 ≠ 0; 𝑑 ≠ 0

28. Example:
Rewrite
𝑦
𝑥
=
3
10
using the different properties
and verify whether the ratios in the
rewritten proportions are still equal using
Cross Multiplication Property.

30. Finding the missing values in a proportion
Cross Multiplication Property or
Means-Extremes Property.
1. Solve for the value of x in
2
5
=
𝑥
30
2. Solve for the value of 𝑏 in 5: 25 = 𝑏: 150
3. Solve for the value of x in
4x−1
3
=
6x+1
5

31. 1. Solve for the value of x in
2
5
=
𝑥
30
Solution:
2 30 = 𝑥 5
60 = 5𝑥
60
5
=
5𝑥
5
𝑥 = 12

32. 1. Solve for the value of x in
2
5
=
𝑥
30
Checking:
𝑥 = 12
2 30 = 12 5
60 = 60
proportion

33. 2. Solve for the value of 𝑏 in 5: 25 = 𝑏: 150
Solution:
5 150 = 25 𝑏
750 = 25𝑏
750
25
=
25𝑏
25
𝑏 = 30

34. 2. Solve for the value of 𝑏 in 5: 25 = 𝑏: 150
Checking:
𝑏 = 30
5 150 = 25 30
750 = 750
proportion

35. 3. Solve for the value of x in
4x−1
3
=
6x+1
5
Solution:
5 4𝑥 − 1 = 3 6𝑥 + 1
20𝑥 − 5 = 18𝑥 + 3
5 + 20𝑥 − 5 = 18𝑥 + 3 + 5
20𝑥 = 18𝑥 + 8
−18𝑥 + 20𝑥 = 18𝑥 + 8 − 18𝑥
2𝑥 = 8
2𝑥
2
=
8
2
𝑥 = 4

36. 3. Solve for the value of x in
4x−1
3
=
6x+1
5
Checking:
𝑥 = 4
5 [4 4 − 1] = 3 [6 4 + 1]
5 16 − 1 = 3 (24 + 1)
(5)(15) = (3)(25)
75 = 75
proportion

37. Try this
Find the missing values of the following proportions.
1.
8
3
=
24
x
2.
2
3
=
x
9
3.
18
11x−4
=
2
x

38. 1.
8
3
=
24
x
Solution:
8 𝑥 = 24 3
8𝑥 = 72
8𝑥
8
=
72
8
𝑥 = 9

39. 1.
8
3
=
24
x
Checking:
𝑥 = 9
8 9 = 24 3
72 = 72
proportion

40. 2.
2
3
=
x
9
Solution:
2 9 = 𝑥 3
18 = 3𝑥
18
3
=
3x
3
𝑥 = 6

41. 2.
2
3
=
x
9
Checking:
𝑥 = 6
2 9 = 6 3
18 = 18
proportion

42. 3.
18
11x−4
=
2
x
Solution:
18 𝑥 = 2 11𝑥 − 4
18𝑥 = 22𝑥 − 8
−22𝑥 + 18𝑥 = −8
−4𝑥 = −8
−4𝑥
−4
=
−8
−4
𝑥 = 2

43. 3.
18
11x−4
=
2
x
Solution:
𝑥 = 2
18 2 = 2 [11 2 − 4]
36 = 2 (22 − 4)
36 = (2)(18)
36 = 36
proportion

44. 1.The ratio of the number of boys to the number of girls in the
Mathematics club is 4 to 5. If there are 25 girls in the club, how
many boys are in the club?
2. In a photograph, Jane is 9 cm tall and her brother John is 10 cm
tall. Jane’s actual height is 153 cm. What is John’s actual height?
3. Ms. Peters wants to prepare a party for 80 people. She has a
chocolate cake recipe that makes 3 small cakes that can serve 16
people. How many cakes does she need to bake for her party?

45. 1. The ratio of the number of boys to the number of girls in the
Mathematics club is 4 to 5. If there are 25 girls in the club, how many
boys are in the club?
Solution:
Let 𝑥 be the number of boys in the club
𝑛(𝑏𝑜𝑦𝑠)
𝑛(𝑔𝑖𝑟𝑙𝑠)
4
5
=
𝑥
25
(25)(4) =(x)(5)
100 = 5x
100
5
=
5𝑥
5
𝒙 = 𝟐0
Therefore, there are 20 boys in the Mathematics club.

46. 1.The ratio of the number of boys to the number of girls in the
Mathematics club is 4 to 5. If there are 25 girls in the club,
how many boys are in the club?
Checking:
𝒙 = 𝟐0
4
5
=
20
25
(25)(4) =(20)(5)
100 = 100
proportion

47. 2. In a photograph, Jane is 9 cm tall and her brother John is 10 cm
tall. Jane’s actual height is 153 cm. What is John’s actual
height?
Solution:
Let 𝑥 be the actual height of John in cm
9
10
=
153
𝑥
(9)(x) = (153)(10)
9x = 1,530
9𝑥
9
=
1530
9
𝑥 = 170
Therefore, John’s actual height is 170 𝑐𝑚.

48. 2. In a photograph, Jane is 9 cm tall and her brother John is 10
cm tall. Jane’s actual height is 153 cm. What is John’s actual
height?
Checking:
𝑥 = 170
9
10
=
153
170
(9)(170) = (153)(10)
1,530 = 1,530
proportion

49. 3. Ms. Peters wants to prepare a party for 80 people. She has a
chocolate cake recipe that makes 3 small cakes that can serve 16
people. How many cakes does she need to bake for her party?
Solution:
Let 𝑥 be the number of cakes to be baked for the party
3𝑐𝑎𝑘𝑒𝑠
16𝑝𝑒𝑜𝑝𝑙𝑒
=
𝑥
80 𝑝𝑒𝑜𝑝𝑙𝑒
(3)(80) = (x)(16)
240 = 16x
240
16
=
16𝑥
16
𝑥 = 15
Therefore, she needs to make 15 cakes for her party.

50. 3. Ms. Peters wants to prepare a party for 80 people. She has a
chocolate cake recipe that makes 3 small cakes that can
serve 16 people. How many cakes does she need to bake for
her party?
Checking:
𝑥 = 15
3𝑐𝑎𝑘𝑒𝑠
16𝑝𝑒𝑜𝑝𝑙𝑒
=
𝑥
80 𝑝𝑒𝑜𝑝𝑙𝑒
(3)(80) = (15)(16)
240 = 240
proportion

51. Triangle
Proportionality
Theorem

52. Triangle Proportionality Theorem
A line drawn to any sides of the
triangle divides the other two sides
proportionally. This also means, if a line is
drawn parallel to one side of the triangle
and intersects the other two sides of the
triangle, then this line divides the two sides
proportionally.

53. The line drawn is 𝐶𝐷 parallel
to one side of the triangle 𝑁𝐴 ,
𝐶𝐷 intersects 𝑀𝑁 and 𝑀𝐴.
Since, 𝐶𝐷||𝑁𝐴 then,
𝑀𝐶
𝐶𝑁
=
𝑀𝐷
𝐷𝐴

54. Examples:
1. Using the figure above, if MC= 6, CN = 4 and DA= 3, find MD.
2. Using the figure above, if CN = 5, DA= 4, MD = 8, find MC.
Since, 𝐶𝐷||𝑁𝐴 then,
𝑀𝐶
𝐶𝑁
=
𝑀𝐷
𝐷𝐴

55. 1. Using the figure above, if MC= 6, CN = 4
and DA= 3, find MD.
Solution:
𝑀𝐶
𝐶𝑁
=
𝑀𝐷
𝐷𝐴
6
4
=
𝑀𝐷
3
(6)(3) = (MD)(4)
12 = 4MD
12
4
=
4𝑀𝐷
4
MD = 3

56. 1. Using the figure above, if MC= 6, CN = 4
and DA= 3, find MD.
Checking:
MD = 3
6
4
=
3
3
(6)(3) = (3)(4)
12 = 12
proportion

57. 2. Using the figure above, if CN = 5, DA= 4, MD =
8, find MC.
Solution:
𝑀𝐶
𝐶𝑁
=
𝑀𝐷
𝐷𝐴
𝑀𝐶
5
=
8
4
(MC)(4) = (8)(5)
4MC = 40
4𝑀𝐶
4
=
40
4
MC = 10

58. 2. Using the figure above, if CN = 5, DA= 4, MD =
8, find MC.
Checking:
MC = 10
10
5
=
8
4
(10)(4) = (8)(5)
40 = 40
proportion

59. Try this!
Apply the Triangle Proportionality Theorem to
find x.
Since, 𝑃𝑅||𝑄𝑂 then,
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆

60. 1.If OP= 4, QR= 3, PS= 7, find RS
2.If PS= 10, RS= 8, OP= 5, find QR.
3.If RS= 9, QR= 2, OP = 4, find PS.
4.If QR= 4, RS = 12, PS = 14, find OP.
5.If OP= 6, PS= 9, RS= 7, find QR.
Since, 𝑃𝑅||𝑄𝑂 then,
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆

61. 1. If OP= 4, QR= 3, PS= 7, find RS
Solution:
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆
4
7
=
3
𝑅𝑆
(4)(RS) = (3)(7)
4RS = 21
4𝑅𝑆
4
=
21
4
RS =
21
4
= 5.25

62. 1. If OP= 4, QR= 3, PS= 7, find RS
Checking:
RS =
21
4
= 5.25
4
7
=
3
5.25
(4)(5.25) = (3)(7)
21 = 21
proportion

63. 2. If PS= 10, RS= 8, OP= 5, find QR.
Solution:
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆
5
10
=
𝑄𝑅
8
(5)(8) = (QR)(10)
40 = 10QR
40
10
=
10𝑄𝑅
10
QR = 4

64. 2. If PS= 10, RS= 8, OP= 5, find QR.
Checking:
QR = 4
5
10
=
4
8
(5)(8) = (4)(10)
40 = 40
proportion

65. 3. If RS= 9, QR= 2, OP = 4, find PS.
Solution:
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆
4
𝑃𝑆
=
2
9
(4)(9) = (2)(PS)
36 = 2PS
36
2
=
2𝑃𝑆
2
PS = 18

66. 3. If RS= 9, QR= 2, OP = 4, find PS.
Checking:
PS = 18
4
18
=
2
9
(4)(9) = (2)(18)
36 = 36
proportion

67. 4. If QR= 4, RS = 12, PS = 14, find OP.
Solution:
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆
𝑂𝑃
14
=
4
12
(OP)(12) = (4)(14)
12OP = 56
12𝑂𝑃
12
=
56
12
OP =
14
3
≈ 4.6667

68. 4. If QR= 4, RS = 12, PS = 14, find OP.
Checking:
OP =
14
3
≈ 4.6667
4.6667
14
=
4
12
(4.6667)(12) = (4)(14)
56 = 56
proportion

69. 5. If OP= 6, PS= 9, RS= 7, find QR.
Solution:
𝑂𝑃
𝑃𝑆
=
𝑄𝑅
𝑅𝑆
6
9
=
𝑄𝑅
7
(6)(7) = (QR)(9)
42 = 9QR
42
9
=
9𝑄𝑅
9
OP =
14
3
≈ 4.6667

70. 5. If OP= 6, PS= 9, RS= 7, find QR.
Checking:
OP =
14
3
≈ 4.6667
6
9
=
4.6667
7
(6)(7) = (4.6667)(9)
42 = 42
proportion

71. ASSIGNMENT:
Think about a situation wherein you can
apply proportions. What are you going to do
is to:
1. Create a word problem about the
situation you think.
2. Answer your word problem and
present it in class.

72. I. Can the given pair of ratios form a proportion?
Write Yes or No. Show how you found out.
1. 4: 5, 24: 30
2. 2: 7, 20: 56

73. 1. 3: 14 = m: 42
2. 5: 2𝑥 − 3 = 3: 𝑥 + 5
3.
𝑥
4
=
18
24
4.
2𝑦−3
4
, =
𝑦+5
3
5.
𝑏−3
𝑏+4
, =
2
5
II. Find the missing term in each proportion.

74. III. Apply the Triangle Proportionality Theorem to find x.
A
Q
P
C B
Since, 𝑃𝑄||𝐶𝐵 then,
𝐴𝑃
𝑃𝐶
=
𝐴𝑄
𝑄𝐵
2. If AP= 7, AQ = 6, QB =12, find PC
3. If QB= 15, AQ= 6, PC= 60, find AP
1. If PC= 6, QB= 18, AP= 4, find AQ