MATHS SYMBOLS - PROPORTIONS - DEFINITION - FIRST PROPERTIES - CROSS PRODUCTS PROPERTY - EXTREMES SWITCHING PROPERTY - MEANS SWITCHING PROPERTY - UPSIDE-DOWN PROPERTY - DENOMINATOR ADDITION PROPERTY - DENOMINATOR SUBTRACTION PROPERTY - MANY OTHER PROPERTIES - PROOFS - EXAMPLES and CALCULATIONS STEP by STEP - 28 TABLES

1. ENZO EXPOSYTO
MATHS
SYMBOLS
n PROPERTIES of PROPORTIONS
Enzo Exposyto 1

2. a : b = c : d a x d = b x c
n PROPERTIES
of
PROPORTIONS
d : b = c : a a : c = b : d b : a = d : c
Enzo Exposyto 2

3.
Enzo Exposyto 3

4. PROPORTIONS
1 - Definition 7
2 - First 4 Properties From a : b = c : d 11
3 - First 4 Properties From a : b = c : d - Tables 25
4 - Other Properties From a : b = c : d 26
5 - Other Properties From a : b = c : d - Tables 35
6 - Properties From (2a) d : b = c : a 36
7 - Properties From (2a) d : b = c : a - Tables 44
Enzo Exposyto 4

5. PROPORTIONS
8 - Properties From (2b) a : c = b : d 45
9 - Properties From (2b) a : c = b : d - Tables 53
10 - Properties From (3) b : a = d : c 54
11 - Properties From (3) b : a = d : c - Tables 62
12 - Properties of the Proportion a : b = c : d - Table 63
13 - Properties of the Proportion 2 : 4 = 3 : 6 - Table 65
14 - Properties From (4a) 67
15 - Properties From (4a + 3) 70
Enzo Exposyto 5

6. PROPORTIONS
16 - Properties From (4b) 73
17 - Properties From (4b + 3) 76
18 - Properties From (5) 79
19 - Properties From (5 + 3) 82
20 - Other Properties of the Proportion a : b = c : d - Table 85
21 - Other Properties of the Proportion 2 : 4 = 3 : 6 - Table 88
22 - SitoGraphy 92
Enzo Exposyto 6

7. PROPORTIONS
-
Definition
Enzo Exposyto 7

8. Proportions - Definition - 1
a:b=c:d proportion
Since the meaning is the same, we can write:
a:b=c:d
or
a/b=c/d
or
a = c
b d
where with b ≠ 0 and d ≠ 0
Say: a is to b as c is to d
a and d are named extremes
b and c are called means
Enzo Exposyto 8

9. Proportions - Definition - 2
a:b=c:d A proportion is a statement that two ratios are equal.
For example:
2:4 = 3:6
or
2/4 = 3/6
or
2 = 3
4 6
2 = 3 is equivalent to 1 = 1
4 6 2 2
Since the two ratios are equal,
these equalities represent the same proportion.
2 is to 4 as 3 is to 6
2 and 6 are named extremes
4 and 3 are called means
Enzo Exposyto 9

10. Proportions - Definition - 3
a:b=c:d A proportion is a statement that two ratios are equal.
In other words, if, and only if, two ratios are equal,
the equality with the two ratios is a proportion.
For example, are the next equalities true?
Are they a proportion?
4:2 = 6:2
or
4/2 = 6/2
or
4 = 6
2 2
They aren't true, because
4 = 2 and 6 = 3
2 2
These equalities don’t exist and don't represent a proportion
Enzo Exposyto 10

11. PROPORTIONS
-
First 4 Properties
Enzo Exposyto 11

12. Proportions - First 4 Properties - (1) - Proof
a:b=c:d proportion - Property (1) - Cross Products
a x d = b x c
Say: the product of the extremes
is equal to the product of the means
Proof:
a : b = c : d is equivalent to a = c
b d
If we multiply both sides by b x d
a x b x d= c x b x d and we simplify,
b d
since c x b = b x c (see the right side), we get:
a x d = b x c
Enzo Exposyto 12

13. Proportions - First 4 Properties - (1) - Examples
a:b=c:d proportion - Property (1) - Cross Products
a x d = b x c
Examples:
1) From the proportion 2:4 = 3:6 we get
2 x 6 = 4 x 3
12 = 12
2) From the proportion 15:5 = 9:3
or
15 = 9 that's equivalent to 3 = 3 and is true
5 3
we have
15 x 3 = 5 x 9
45 = 45
Enzo Exposyto 13

14. Proportions - First 4 Properties - (2a) - Proof
a:b=c:d proportion - Property (2a) - Extremes Switching Property
d : b = c : a with b ≠ 0 and a ≠ 0
If we switch the extremes each other,
we get two equal ratios
and, then, the new equality is a proportion
Proof:
a : b = c : d is equivalent to a = c
b d
If we multiply both sides by d
a x d = c x d a
b a d a
and simplify, we get
d = c which is equivalent to d : b = c : a
b a
Enzo Exposyto 14

15. Proportions - First 4 Properties - (2a) - Examples
a:b=c:d proportion - Property (2a) - Extremes Switching Property
d : b = c : a with b ≠ 0 and a ≠ 0
Examples:
1) From the proportion 2:4 = 3:6
if we exchange the extremes each other, we get
6 : 4 = 3 : 2 Since it means
3 : 2 = 3 : 2 the new equality is a proportion
2) From the proportion 15:5 = 9:3
if we switch the extremes each other, we have
3 : 5 = 9 : 15 Since its meaning is
3 : 5 = 3 : 5 the new equality is a proportion
Enzo Exposyto 15

16. Proportions - First 4 Properties - (2b) - Proof
a:b=c:d proportion - Property (2b) - Means Switching Property
a : c = b : d with c ≠ 0 and d ≠ 0
If we switch the means each other,
we get two equal ratios
and, then, the new equality is a proportion
Proof:
a : b = c : d is equivalent to a = c
b d
If we multiply both sides by b
a x b = c x b c
b c d c
and simplify, we get
a = b which is equivalent to a : c = b : d
c d
Enzo Exposyto 16

17. Proportions - First 4 Properties - (2b) - Examples
a:b=c:d proportion - Property (2b) - Means Switching Property
a : c = b : d with c ≠ 0 and d ≠ 0
Examples:
1) From the proportion 2:4 = 3:6
if we exchange the means each other, we get
2 : 3 = 4 : 6 Since it means
2 : 3 = 2 : 3 the new equality is a proportion
2) From the proportion 15:5 = 9:3
if we exchange the means each other, we have
15 : 9 = 5 : 3 Since its meaning is
5 : 3 = 5 : 3 the new equality is a proportion
Enzo Exposyto 17

18. Proportions - First 4 Properties - (3) - Proof
a:b=c:d proportion - Property (3) - Upside-Down Property
b : a = d : c with a ≠ 0 and c ≠ 0
If we switch extreme for mean to both sides,
we get two equal ratios
and, then, the new equality is a proportion
Proof:
a : b = c : d is equivalent to a = c
b d
If we multiply both sides by b x d
a x b x d = c x b x d a c
b a c d a c
and simplify, we get
d = b or b = d that's equivalent to b : a = d : c
c a a c
Enzo Exposyto 18

19. Proportions - First 4 Properties - (3) - Another Proof
a:b=c:d proportion - Property (3) - Upside-Down Property
b : a = d : c with a ≠ 0 and c ≠ 0
Another Proof:
a : b = c : d is equivalent to a = c
b d
If we raise both sides to -1
we get
b = d which is equivalent to b : a = d : c
a c
(
a
b )
−1
=
(
c
d)
−1
Enzo Exposyto 19

20. Proportions - First 4 Properties - (3) - Examples
a:b=c:d proportion - Property (3) - Upside-Down Property
b : a = d : c with a ≠ 0 and c ≠ 0
Examples:
1) From the proportion 2:4 = 3:6 we get
4 : 2 = 6 : 3 Since it means
2 = 2 the new equality is a proportion
2) From the proportion 15:5 = 9:3 we have
5 : 15 = 3. : 9 Since its meaning is
1 : 3 = 1 : 3 the new equality is a proportion
We can write as well:
5 = 3 that's equivalent to 1 = 1 and is true
15 9 3 3
Enzo Exposyto 20

21. Proportions - First 4 Properties - (4a) - Proof
a:b=c:d proportion - Property (4a) - Denominator Addition Property
a + b = c + d with b ≠ 0 and d ≠ 0
b d
Proof:
a : b = c : d is equivalent to a = c
b d
If we add to the left side b = 1 and to the right side d = 1
b d
a + b = c + d
b b d d
we get
a + b = c + d
b d
Enzo Exposyto 21

22. Proportions - First 4 Properties - (4a) - Examples
a:b=c:d proportion - Property (4a) - Denominator Addition Property
a + b = c + d with b ≠ 0 and d ≠ 0
b d
Examples:
1) From the proportion 2:4 = 3:6 we get
2 + 4 = 3 + 6 If we simplify, it means
4 6
6 = 9 that's equivalent to 3 = 3 and is true
4 6 2 2
2) From the proportion 15:5 = 9:3 we have
15 + 5 = 9 + 3 If we simplify, it means
5 3
20 = 12 which is equivalent to 4 = 4 and is true
5 3
Enzo Exposyto 22

23. Proportions - First 4 Properties - (4b) - Proof
a:b=c:d proportion - Property (4b) - Denominator Subtraction Property
a - b = c - d with b ≠ 0 and d ≠ 0
b d
Proof:
a : b = c : d is equivalent to a = c
b d
If we subtract from the left side b = 1 and from the right side d = 1
b d
a - b = c - d
b b d d
we get
a - b = c - d
b d
Enzo Exposyto 23

24. Proportions - First 4 Properties - (4b) - Examples
a:b=c:d proportion - Property (4b) - Denominator Subtraction Property
a - b = c - d with b ≠ 0 and d ≠ 0
b d
Examples:
1) From the proportion 2:4 = 3:6 we get
2 - 4 = 3 - 6 If we simplify, it means
4 6
- 2 = - 3 that's equivalent to - 1 = - 1 and is true
4 6 2 2
2) From the proportion 15:5 = 9:3 we have
15 - 5 = 9 - 3 If we simplify, it means
5 3
10 = 6 which is equivalent to 2 = 2 and is true
5 3
Enzo Exposyto 24

25. TABLE - I
TABLE - II
FIRST 4 PROPERTIES from a : b = c : d OR a = c
b d
1 2a 2b 3 4a 4b
a x d = b x c
d = c
b a
a = b
c d
b = d
a c
a + b = c + d
b d
a - b = c - d
b d
FIRST 4 PROPERTIES from 2 : 4 = 3 : 6 OR 2 = 3
4 6
1 2a 2b 3 4a 4b
2 x 6 = 4 x 3
6 = 3
4 2
2 = 4
3 6
4 = 6
2 3
2 + 4 = 3 + 6
4 6
2 - 4 = 3 - 6
4 6
Enzo Exposyto 25

26. PROPORTIONS
-
Other Properties
From
a : b = c : d
Enzo Exposyto 26

27. Proportions - Other Properties - (4a + 3) - Proof
a:b=c:d proportion - Property (4a + 3)
b = d with a+b ≠ 0 and c+d ≠ 0
a + b c + d
Proof:
Remembering Property (4a) a + b = c + d,
b d
if we raise both sides to -1
we get
b = d
a + b c + d
(
a + b
b )
−1
=
(
c + d
d )
−1
Enzo Exposyto 27

28. Proportions - Other Properties - (4a + 3) - Examples
a:b=c:d proportion - Property (4a + 3)
b = d with a+b ≠ 0 and c+d ≠ 0
a + b c + d
Examples:
1) From the proportion 2:4 = 3:6 we get
4 = 6 If we simplify, it means
2 + 4 3 + 6
4 = 6 that's equivalent to 2 = 2 and is true
6 9 3 3
2) From the proportion 15:5 = 9:3 we have
5 = 3 If we simplify, it means
15 + 5 9 + 3
5 = 3 which is equivalent to 1 = 1 and is true
20 12 4 4
Enzo Exposyto 28

29. Proportions - Other Properties - (4b + 3) - Proof
a:b=c:d proportion - Property (4b + 3)
b = d with a-b ≠ 0 and c-d ≠ 0
a - b c - d
Proof:
Remembering Property (4b) a - b = c - d,
b d
if we raise both sides to -1
we get
b = d
a - b c - d
(
a − b
b )
−1
=
(
c − d
d )
−1
Enzo Exposyto 29

30. Proportions - Other Properties - (4b + 3) - Examples
a:b=c:d proportion - Property (4b + 3)
b = d with a-b ≠ 0 and c-d ≠ 0
a - b c - d
Examples:
1) From the proportion 2:4 = 3:6 we get
4 = 6 If we simplify, it means
2 - 4 3 - 6
- 4 = - 6 that's equivalent to -2 = -2 and is true
2 3
2) From the proportion 15:5 = 9:3 we have
5 = 3 If we simplify, it means
15 - 5 9 - 3
5 = 3 which is equivalent to 1 = 1 and is true
10 6 2 2
Enzo Exposyto 30

31. Proportions - Other Properties - (5) - Proof
a:b=c:d proportion - Property (5)
a + b = c + d with a-b ≠ 0 and c-d ≠ 0
a - b c - d
Proof:
Remembering a + b = c + d and b = d :
b d a - b c - d
a + b x b = c + d x d we multiply and simplify
b a - b d c - d
and we get
a + b = c + d
a - b c - d
Enzo Exposyto 31

32. Proportions - Other Properties - (5) - Examples
a:b=c:d proportion - Property (5)
a + b = c + d with a-b ≠ 0 and c-d ≠ 0
a - b c - d
Examples:
1) From the proportion 2:4 = 3:6 we get
2 + 4 = 3 + 6 If we simplify, it means
2 - 4 3 - 6
6 = 9 that's equivalent to -3 = -3 and is true
-2 -3
2) From the proportion 15:5 = 9:3 we have
15 + 5 = 9 + 3 If we simplify, it means
15 - 5 9 - 3
20 = 12 which is equivalent to 2 = 2 and is true
10 6
Enzo Exposyto 32

33. Proportions - Other Properties - (5 + 3) - Proof
a:b=c:d proportion - Property (5 + 3)
a - b = c - d with a+b ≠ 0 and c+d ≠ 0
a + b c + d
Proof:
Remembering Property (5) a + b = c + d
a - b c - d
if we raise both sides to -1
we get
a - b = c - d
a + b c + d
(
a + b
a − b )
−1
=
(
c + d
c − d )
−1
Enzo Exposyto 33

34. Proportions - Other Properties - (5 + 3) - Examples
a:b=c:d proportion - Property (5 + 3)
a - b = c - d with a+b ≠ 0 and c+d ≠ 0
a + b c + d
Examples:
1) From the proportion 2:4 = 3:6 we get
2 - 4 = 3 - 6 If we simplify, it means
2 + 4 3 + 6
-2 = -3 that's equivalent to -1 = -1 and is true
6 9 3 3
2) From the proportion 15:5 = 9:3 we have
15 - 5 = 9 - 3 If we simplify, it means
15 + 5 9 + 3
10 = 6 which is equivalent to 1 = 1 and is true
20 12 2 2
Enzo Exposyto 34

35. TABLE - III
TABLE - IV
OTHER PROPERTIES from a : b = c : d OR a = c
b d
4a + 3 4b + 3 5 5 + 3
b = d
a + b c + d
b = d
a - b c - d
a + b = c + d
a - b c - d
a - b = c - d
a + b c + d
OTHER PROPERTIES from 2 : 4 = 3 : 6 OR 2 = 3
4 6
4a + 3 4b + 3 5 5 + 3
4 = 6
2 + 4 3 + 6
4 = 6
2 - 4 3 - 6
2 + 4 = 3 + 6
2 - 4 3 - 6
2 - 4 = 3 - 6
2 + 4 3 + 6
Enzo Exposyto 35

36. PROPORTIONS
-
Properties
From (2a)
d : b = c : a
Enzo Exposyto 36

37. Proportions - Properties From (2a) - d : b = c : a
a:b=c:d proportion - Property (2a) - Extremes Switching Property
d : b = c : a
FROM THIS PROPORTION
WE CAN GET
OTHERS PROPERTIES …
Enzo Exposyto 37

38. Proportions - Properties From (2a) - d : b = c : a - (4c)
d:b=c:a proportion - Property (4c) - Denominator Addition Property
d + b = c + a with b ≠ 0 and a ≠ 0
b a
1) From the proportion 6:4 = 3:2 we get
6 + 4 = 3 + 2 If we simplify, it means
4 2
10 = 5 that's equivalent to 5 = 5 and is true
4 2 2 2
2) From the proportion 3:5 = 9:15 we have
3 + 5 = 9 + 15 If we simplify, it means
5 15
8 = 24 which is equivalent to 8 = 8 and is true
5 15 5 5
Enzo Exposyto 38

39. Proportions - Properties From (2a) - d : b = c : a - (4c + 3)
d:b=c:a proportion - Property (4c + 3)
b = a with d+b ≠ 0 and c+a ≠ 0
d + b c + a
1) From the proportion 6:4 = 3:2 we get
4 = 2 If we simplify, it means
6 + 4 3 + 2
4 = 2 that's equivalent to 2 = 2 and is true
10 5 5 5
2) From the proportion 3:5 = 9:15 we have
5 = 15 If we simplify, it means
3 + 5 9 + 15
5 = 15 which is equivalent to 5 = 5 and is true
8 24 8 8
Enzo Exposyto 39

40. Proportions - Properties From (2a) - d : b = c : a - (4d)
d:b=c:a proportion - Property (4d) - Denominator Subtraction Property
d - b = c - a with b ≠ 0 and a ≠ 0
b a
1) From the proportion 6:4 = 3:2 we get
6 - 4 = 3 - 2 If we simplify, it means
4 2
2 = 1 that's equivalent to 1 = 1 and is true
4 2 2 2
2) From the proportion 3:5 = 9:15 we have
3 - 5 = 9 - 15 If we simplify, it means
5 15
- 2 = - 6 which is equivalent to - 2 = - 2 and is true
5 15 5 5
Enzo Exposyto 40

41. Proportions - Properties From (2a) - d : b = c : a - (4d + 3)
d:b=c:a proportion - Property (4d + 3)
b = a with d-b ≠ 0 and c-a ≠ 0
d - b c - a
1) From the proportion 6:4 = 3:2 we get
4 = 2 If we simplify, it means
6 - 4 3 - 2
4 = 2 that's equivalent to 2 = 2 and is true
2 1
2) From the proportion 3:5 = 9:15 we have
5 = 15 If we simplify, it means
3 - 5 9 - 15
- 5 = - 15 which is equivalent to - 5 = - 5 and is true
2 6 2 2
Enzo Exposyto 41

42. Proportions - Properties From (2a) - d : b = c : a - (5a)
d:b=c:a proportion - Property (5a)
d + b = c + a with d-b ≠ 0 and c-a ≠ 0
d - b c - a
1) From the proportion 6:4 = 3:2 we get
6 + 4 = 3 + 2 If we simplify, it means
6 - 4 3 - 2
10 = 5 that's equivalent to 5 = 5 and is true
2 1
2) From the proportion 3:5 = 9:15 we have
3 + 5 = 9 + 15 If we simplify, it means
3 - 5 9 - 15
- 8 = - 24 which is equivalent to -4 = -4 and is true
2 6
Enzo Exposyto 42

43. Proportions - Properties From (2a) - d : b = c : a - (5a + 3)
d:b=c:a proportion - Property (5a + 3)
d - b = c - a with d+b ≠ 0 and c+a ≠ 0
d + b c + a
1) From the proportion 6:4 = 3:2 we get
6 - 4 = 3 - 2 If we simplify, it means
6 + 4 3 + 2
2 = 1 that's equivalent to 1 = 1 and is true
10 5 5 5
2) From the proportion 3:5 = 9:15 we have
3 - 5 = 9 - 15 If we simplify, it means
3 + 5 9 + 15
- 2 = - 6 which is equivalent to - 1 = - 1 and is true
8 24 4 4
Enzo Exposyto 43

44. TABLE - V
TABLE - VI
PROPERTIES from (2a) d : b = c : a OR d = c
b a
4c 4c + 3 4d 4d + 3 5a 5a + 3
d + b = c + a
b a
b = a
d + b c + a
d - b = c - a
b a
b = a
d - b c - a
d + b = c + a
d - b c - a
d - b = c - a
d + b c + a
PROPERTIES from (2a) 6 : 4 = 3 : 2 OR 6 = 3
4 2
4c 4c + 3 4d 4d + 3 5a 5a + 3
6 + 4 = 3 + 2
4 2
4 = 2
6 + 4 3 + 2
6 - 4 = 3 - 2
4 2
4 = 2
6 - 4 3 - 2
6 + 4 = 3 + 2
6 - 4 3 - 2
6 - 4 = 3 - 2
6 + 4 3 + 2
Enzo Exposyto 44

45. PROPORTIONS
-
Properties
From (2b)
a : c = b : d
Enzo Exposyto 45

46. Proportions - Properties From (2b) - a : c = b : d
a:b=c:d proportion - Property (2b) - Means Switching Property
a : c = b : d
FROM THIS PROPORTION
WE CAN GET
OTHERS PROPERTIES …
Enzo Exposyto 46

47. Proportions - Properties From (2b) - a : c = b : d - (4e)
a:c=b:d proportion - Property (4e) - Denominator Addition Property
a + c = b + d with c ≠ 0 and d ≠ 0
c d
1) From the proportion 2:3 = 4:6 we get
2 + 3 = 4 + 6 If we simplify, it means
3 6
5 = 10 that's equivalent to 5 = 5 and is true
3 6 3 3
2) From the proportion 15:9 = 5:3 we have
15 + 9 = 5 + 3 If we simplify, it means
9 3
24 = 8 which is equivalent to 8 = 8 and is true
9 3 3 3
Enzo Exposyto 47

48. Proportions - Properties From (2b) - a : c = b : d - (4e + 3)
a:c=b:d proportion - Property (4e + 3)
c = d with a+c ≠ 0 and b+d ≠ 0
a + c b + d
1) From the proportion 2:3 = 4:6 we get
3 = 6 If we simplify, it means
2 + 3 4 + 6
3 = 6 that's equivalent to 3 = 3 and is true
5 10 5 5
2) From the proportion 15:9 = 5:3 we have
9 = 3 If we simplify, it means
15 + 9 5 + 3
9 = 3 which is equivalent to 3 = 3 and is true
24 8 8 8
Enzo Exposyto 48

49. Proportions - Properties From (2b) - a : c = b : d - (4f)
a:c=b:d proportion - Property (4f) - Denominator Subtraction Property
a - c = b - d with c ≠ 0 and d ≠ 0
c d
1) From the proportion 2:3 = 4:6 we get
2 - 3 = 4 - 6 If we simplify, it means
3 6
- 1 = - 2 that's equivalent to - 1 = - 1 and is true
3 6 3 3
2) From the proportion 15:9 = 5:3 we have
15 - 9 = 5 - 3 If we simplify, it means
9 3
6 = 2 which is equivalent to 2 = 2 and is true
9 3 3 3
Enzo Exposyto 49

50. Proportions - Properties From (2b) - a : c = b : d - (4f + 3)
a:c=b:d proportion - Property (4f + 3)
c = d with a-c ≠ 0 and b-d ≠ 0
a - c b - d
1) From the proportion 2:3 = 4:6 we get
3 = 6 If we simplify, it means
2 - 3 4 - 6
- 3 = - 6 that's equivalent to -3 = -3 and is true
1 2
2) From the proportion 15:9 = 5:3 we have
9 = 3 If we simplify, it means
15 - 9 5 - 3
9 = 3 which is equivalent to 3 = 3 and is true
6 2 2 2
Enzo Exposyto 50

51. Proportions - Properties From (2b) - a : c = b : d - (5b)
a:c=b:d proportion - Property (5b)
a + c = b + d with a-c ≠ 0 and b-d ≠ 0
a - c b - d
1) From the proportion 2:3 = 4:6 we get
2 + 3 = 4 + 6 If we simplify, it means
2 - 3 4 - 6
- 5 = - 10 that's equivalent to -5 = -5 and is true
1 2
2) From the proportion 15:9 = 5:3 we have
15 + 9 = 5 + 3 If we simplify, it means
15 - 9 5 - 3
24 = 8 which is equivalent to 4 = 4 and is true
6 2
Enzo Exposyto 51

52. Proportions - Properties From (2b) - a : c = b : d - (5b + 3)
a:c=b:d proportion - Property (5b + 3)
a - c = b - d with a+c ≠ 0 and b+d ≠ 0
a + c b + d
1) From the proportion 2:3 = 4:6 we get
2 - 3 = 4 - 6 If we simplify, it means
2 + 3 4 + 6
- 1 = - 2 that's equivalent to - 1 = - 1 and is true
5 10 5 5
2) From the proportion 15:9 = 5:3 we have
15 - 9 = 5 - 3 If we simplify, it means
15 + 9 5 + 3
6 = 2 which is equivalent to 1 = 1 and is true
24 8 4 4
Enzo Exposyto 52

53. TABLE - VII
TABLE - VIII
PROPERTIES from (2b) a : c = b : d OR a = b
c d
4e 4e + 3 4f 4f + 3 5b 5b + 3
a + c = b + d
c d
c = d
a + c b + d
a - c = b - d
c d
c = d
a - c b - d
a + c = b + d
a - c b - d
a - c = b - d
a + c b + d
PROPERTIES from (2b) 2 : 3 = 4 : 6 OR 2 = 4
3 6
4e 4e + 3 4f 4f + 3 5b 5b + 3
2 + 3 = 4 + 6
3 6
3 = 6
2 + 3 4 + 6
2 - 3 = 4 - 6
3 6
3 = 6
2 - 3 4 - 6
2 + 3 = 4 + 6
2 - 3 4 - 6
2 - 3 = 4 - 6
2 + 3 4 + 6
Enzo Exposyto 53

54. PROPORTIONS
-
Properties
From (3)
b : a = d : c
Enzo Exposyto 54

55. Proportions - Properties From (3) - b : a = d : c
a:b=c:d proportion - Property (3) - Upside-Down Property
b : a = d : c
FROM THIS PROPORTION
WE CAN GET
OTHERS PROPERTIES …
Enzo Exposyto 55

56. Proportions - Properties From (3) - b : a = d : c - (4g)
b:a=d:c proportion - Property (4g) - Denominator Addition Property
b + a = d + c with a ≠ 0 and c ≠ 0
a c
1) From the proportion 4:2 = 6:3 we get
4 + 2 = 6 + 3 If we simplify, it means
2 3
6 = 9 that's equivalent to 3 = 3 and is true
2 3
2) From the proportion 5:15 = 3:9 we have
5 + 15 = 3 + 9 If we simplify, it means
15 9
20 = 12 which is equivalent to 4 = 4 and is true
15 9 3 3
Enzo Exposyto 56

57. Proportions - Properties From (3) - b : a = d : c - (4g + 3)
b:a=d:c proportion - Property (4g + 3)
a = c with b+a ≠ 0 and d+c ≠ 0
b + a d + c
1) From the proportion 4:2 = 6:3 we get
2 = 3 If we simplify, it means
4 + 2 6 + 3
2 = 3 that's equivalent to 1 = 1 and is true
6 9 3 3
2) From the proportion 5:15 = 3:9 we have
15 = 9 If we simplify, it means
5 + 15 3 + 9
15 = 9 which is equivalent to 3 = 3 and is true
20 12 4 4
Enzo Exposyto 57

58. Proportions - Properties From (3) - b : a = d : c - (4h)
b:a=d:c proportion - Property (4h) - Denominator Subtraction Property
b - a = d - c with a ≠ 0 and c ≠ 0
a c
1) From the proportion 4:2 = 6:3 we get
4 - 2 = 6 - 3 If we simplify, it means
2 3
2 = 3 that's equivalent to 1 = 1 and is true
2 3
2) From the proportion 5:15 = 3:9 we have
5 - 15 = 3 - 9 If we simplify, it means
15 9
- 10 = - 6 which is equivalent to - 2 = - 2 and is true
15 9 3 3
Enzo Exposyto 58

59. Proportions - Properties From (3) - b : a = d : c - (4h + 3)
b:a=d:c proportion - Property (4h + 3)
a = c with b-a ≠ 0 and d-c ≠ 0
b - a d - c
1) From the proportion 4:2 = 6:3 we get
2 = 3 If we simplify, it means
4 - 2 6 - 3
2 = 3 that's equivalent to 1 = 1 and is true
2 3
2) From the proportion 5:15 = 3:9 we have
15 = 9 If we simplify, it means
5 - 15 3 - 9
- 15 = - 9 which is equivalent to - 3 = - 3 and is true
10 6 2 2
Enzo Exposyto 59

60. Proportions - Properties From (3) - b : a = d : c - (5c)
b:a=d:c proportion - Property (5c)
b + a = d + c with b-a ≠ 0 and d-c ≠ 0
b - a d - c
1) From the proportion 4:2 = 6:3 we get
4 + 2 = 6 + 3 If we simplify, it means
4 - 2 6 - 3
6 = 9 that's equivalent to 3 = 3 and is true
2 3
2) From the proportion 5:15 = 3:9 we have
5 + 15 = 3 + 9 If we simplify, it means
5 - 15 3 - 9
- 20 = - 12 which is equivalent to -2 = -2 and is true
10 6
Enzo Exposyto 60

61. Proportions - Properties From (3) - b : a = d : c - (5c + 3)
b:a=d:c proportion - Property (5c + 3)
b - a = d - c with b+a ≠ 0 and d+c ≠ 0
b + a d + c
1) From the proportion 4:2 = 6:3 we get
4 - 2 = 6 - 3 If we simplify, it means
4 + 2 6 + 3
2 = 3 that's equivalent to 1 = 1 and is true
6 9 3 3
2) From the proportion 5:15 = 3:9 we have
5 - 15 = 3 - 9 If we simplify, it means
5 + 15 3 + 9
- 10 = - 6 which is equivalent to - 1 = - 1 and is true
20 12 2 2
Enzo Exposyto 61

62. TABLE - IX
TABLE - X
PROPERTIES from (3) b : a = d : c OR b = d
a c
4g 4g + 3 4h 4h + 3 5c 5c + 3
b + a = d + c
a c
a = c
b + a d + c
b - a = d - c
a c
a = c
b - a d - c
b + a = d + c
b - a d - c
b - a = d - c
b + a d + c
PROPERTIES from (3) 4 : 2 = 6 : 3 OR 4 = 6
2 3
4g 4g + 3 4h 4h + 3 5c 5c + 3
4 + 2 = 6 + 3
2 3
2 = 3
4 + 2 6 + 3
4 - 2 = 6 - 3
2 3
2 = 3
4 - 2 6 - 3
4 + 2 = 6 + 3
4 - 2 6 - 3
4 - 2 = 6 - 3
4 + 2 6 + 3
Enzo Exposyto 62

63. Properties
of the
Proportion
a : b = c : d
TABLE - XI
Enzo Exposyto 63

64. TABLE - XI
PROPERTIES of the PROPORTION a : b = c : d OR a = c
b d
denominators ≠ 0
1 2a 2b 3 4a 4a + 3 4b 4b + 3 5 5 + 3
a x d
=
b x c
d = c
b a
a = b
c d
b = d
a c
a + b = c + d
b d
b = d
a + b c + d
a - b = c - d
b d
b = d
a - b c - d
a + b = c + d
a - b c - d
a - b = c - d
a + b c + d
From 2a 4c 4c + 3 4d 4d + 3 5a 5a + 3
d + b = c + a
b a
b = a
d + b c + a
d - b = c - a
b a
b = a
d - b c - a
d + b = c + a
d - b c - a
d - b = c - a
d + b c + a
From 2b 4e 4e + 3 4f 4f + 3 5b 5b + 3
a + c = b + d
c d
c = d
a + c b + d
a - c = b - d
c d
c = d
a - c b - d
a + c = b + d
a - c b - d
a - c = b - d
a + c b + d
From 3 4g 4g + 3 4h 4h + 3 5c 5c + 3
b + a = d + c
a c
a = c
b + a d + c
b - a = d - c
a c
a = c
b - a d - c
b + a = d + c
b - a d - c
b - a = d - c
b + a d + c
Enzo Exposyto 64

65. Properties
of the
Proportion
2 : 4 = 3 : 6
TABLE - XII
Enzo Exposyto 65

66. TABLE - XII
PROPERTIES of the PROPORTION 2 : 4 = 3 : 6 OR 2 = 3
4 6
1 2a 2b 3 4a 4a + 3 4b 4b + 3 5 5 + 3
2 x 6
=
4 x 3
6 = 3
4 2
2 = 4
3 6
4 = 6
2 3
2 + 4 = 3 + 6
4 6
4 = 6
2 + 4 3 + 6
2 - 4 = 3 - 6
4 6
4 = 6
2 - 4 3 - 6
2 + 4 = 3 + 6
2 - 4 3 - 6
2 - 4 = 3 - 6
2 + 4 3 + 6
From 2a 4c 4c + 3 4d 4d + 3 5a 5a + 3
6 + 4 = 3 + 2
4 2
4 = 2
6 + 4 3 + 2
6 - 4 = 3 - 2
4 2
4 = 2
6 - 4 3 - 2
6+4 = 3+2
6 - 4 3 - 2
6 - 4 = 3 - 2
6 + 4 3 + 2
From 2b 4e 4e + 3 4f 4f + 3 5b 5b + 3
2 + 3 = 4 + 6
3 6
3 = 6
2 + 3 4 + 6
2 - 3 = 4 - 6
3 6
3 = 6
2 - 3 4 - 6
2+3 = 4+6
2 - 3 4 - 6
2 - 3 = 4 - 6
2 + 3 4 + 6
From 3 4g 4g + 3 4h 4h + 3 5c 5c + 3
4 + 2 = 6 + 3
2 3
2 = 3
4 + 2 6 + 3
4 - 2 = 6 - 3
2 3
2 = 3
4 - 2 6 - 3
4+2 = 6+3
4 - 2 6 - 3
4 - 2 = 6 - 3
4 + 2 6 + 3
Enzo Exposyto 66

67. Properties
From
(4a)
a + b = c + d
b d
Enzo Exposyto 67

68. TABLE - XIII
PROPERTIES from (4a) - (a+b) : b = (c+d) : d OR a + b = c + d
b d
denominators ≠ 0
1a 2c 2d 3a
(a + b) x d
=
b x (c + d)
d = c + d
b a + b
a + b = b
c + d d
b = d
a + b c + d
4i 4i + 3 4j 4j + 3 5d 5d + 3
a + 2b = c + 2d
b d
b = d
a + 2b c + 2d
a = c
b d
b = d
a c
a + 2b = c + 2d
a c
a = c
a + 2b c + 2d
Enzo Exposyto 68

69. TABLE - XIV
PROPERTIES from (4a) - (2+4) : 4 = (3+6) : 6 OR 2 + 4 = 3 + 6
4 6
1a 2c 2d 3a
(2 + 4) x 6
=
4 x (3 + 6)
6 = 3 + 6
4 2 + 4
2 + 4 = 4
3 + 6 6
4 = 6
2 + 4 3 + 6
4i 4i + 3 4j 4j + 3 5d 5d + 3
2 + 2.4 = 3 + 2.6
4 6
4 = 6
2 + 2.4 3 + 2.6
2 = 3
4 6
4 = 6
2 3
2 + 2.4 = 3 + 2.6
2 3
2 = 3
2 + 2.4 3 + 2.6
Enzo Exposyto 69

70. Properties
From
(4a + 3)
b = d
a + b c + d
Enzo Exposyto 70

71. TABLE - XV
PROPERTIES from (4a + 3) - b : (a+b) = d : (c+d) OR __b__= __d__
a + b c + d
denominators ≠ 0
1b 2e 2f 3b
b x (c + d)
=
(a + b) x d
c + d = d
a + b b
b = a + b
d c + d
a + b = c + d
b d
4k 4k + 3 4l 4l + 3 5e 5e + 3
a + 2b = c + 2d
a + b c + d
a + b = c + d
a + 2b c + 2d
-a = _-c
a + b c + d
a + b = c + d
-a -c
a + 2b = c + 2d
-a -c
-a = -c
a + 2b c + 2d
Enzo Exposyto 71

72. TABLE - XVI
PROPERTIES from (4a + 3) - 4 : (2+4) = 6 : (3+6) OR __4__= __6__
2 + 4 3 + 6
1b 2e 2f 3b
4 x (3 + 6)
=
(2 + 4) x 6
3 + 6 = 6
2 + 4 4
4 = 2 + 4
6 3 + 6
2 + 4 = 3 + 6
4 6
4k 4k + 3 4l 4l + 3 5e 5e + 3
2 + 2.4 = 3 + 2.6
2 + 4 3 + 6
2 + 4 = 3 + 6
2 + 2.4 3 + 2.6
-2 = _-3
2 + 4 3 + 6
2 + 4 = 3 + 6
-2 -3
2 + 2.4 = 3 + 2.6
-2 -3
-2 = -3
2 + 2.4 3 + 2.6
Enzo Exposyto 72

73. Properties
From
(4b)
a - b = c - d
b d
Enzo Exposyto 73

74. TABLE - XVII
PROPERTIES from (4b) - (a-b) : b = (c-d) : d OR a - b = c - d
b d
denominators ≠ 0
1c 2g 2h 3c
(a - b) x d
=
b x (c - d)
d = c - d
b a - b
a - b = b
c - d d
b = d
a - b c - d
4m 4m + 3 4n 4n + 3 5f 5f + 3
a = c
b d
b = d
a c
a - 2b = c - 2d
b d
b = d
a - 2b c - 2d
a = c
a - 2b c - 2d
a - 2b = c - 2d
a c
Enzo Exposyto 74

75. TABLE - XVIII
PROPERTIES from (4b) - (2-4) : 4 = (3-6) : 6 OR 2 - 4 = 3 - 6
4 6
1c 2g 2h 3c
(2 - 4) x 6
=
4 x (3 - 6)
6 = 3 - 6
4 2 - 4
2 - 4 = 4
3 - 6 6
4 = 6
2 - 4 3 - 6
4m 4m + 3 4n 4n + 3 5f 5f + 3
2 = 3
4 6
4 = 6
2 3
2 - 2.4 = 3 - 2.6
4 6
4 = 6
2 - 2.4 3 - 2.6
2 = 3
2 - 2.4 3 - 2.6
2 - 2.4 = 3 - 2.6
2 3
Enzo Exposyto 75

76. Properties
From
(4b + 3)
b = d
a - b c - d
Enzo Exposyto 76

77. TABLE - XIX
PROPERTIES from (4b + 3) - b : (a-b) = d : (c-d) OR __b__ = __d__
a - b c - d
denominators ≠ 0
1d 2i 2j 3d
b x (c - d)
=
(a - b) x d
c - d = d
a - b b
b = a - b
d c - d
a - b = c - d
b d
4o 4o + 3 4p 4p + 3 5g 5g + 3
a = c
a - b c - d
a - b = c - d
a c
-a + 2b = -c + 2d
a - b c - d
a - b = c - d
-a + 2b -c + 2d
a = c
-a + 2b -c + 2d
-a + 2b = -c + 2d
a c
Enzo Exposyto 77

78. TABLE - XX
PROPERTIES from (4b + 3) - 4 : (2-4) = 6 : (3-6) OR __4__ = __6__
2 - 4 3 - 6
1d 2i 2j 3d
4 x (3 - 6)
=
(2 - 4) x 6
3 - 6 = 6
2 - 4 4
4 = 2 - 4
6 3 - 6
2 - 4 = 3 - 6
4 6
4o 4o + 3 4p 4p + 3 5g 5g + 3
2 = 3
2 - 4 3 - 6
2 - 4 = 3 - 6
2 3
-2 + 2.4 = -3 + 2.6
2 - 4 3 - 6
2 - 4 = 3 - 6
-2 + 2.4 -3 + 2.6
2 = 3
-2 + 2.4 -3 + 2.6
-2 + 2.4 = -3 + 2.6
2 3
Enzo Exposyto 78

79. Properties
From
(5)
a + b = c + d
a - b c - d
Enzo Exposyto 79

80. TABLE - XXI
PROPERTIES from (5) - (a+b) : (a-b) = (c+d) : (c-d) OR a + b = c + d
a - b c - d
denominators ≠ 0
1e 2k 2l 3e
(a + b) x (c - d)
=
(a - b) x (c + d)
c - d = c + d
a - b a + b
a + b = a - b
c + d c - d
a - b = c - d
a + b c + d
4q 4q + 3 4r 4r + 3 5h 5h + 3
2a = 2c
a - b c - d
a - b = c - d
2a 2c
2b = 2d
a - b c - d
a - b = c - d
2b 2d
a = c
b d
b = d
a c
Enzo Exposyto 80

81. TABLE - XXII
PROPERTIES from (5) - (2+4) : (2-4) = (3+6) : (3-6) OR 2 + 4 = 3 + 6
2 - 4 3 - 6
1e 2k 2l 3e
(2 + 4) x (3 - 6)
=
(2 - 4) x (3 + 6)
3 - 6 = 3 + 6
2 - 4 2 + 4
2 + 4 = 2 - 4
3 + 6 3 - 6
2 - 4 = 3 - 6
2 + 4 3 + 6
4q 4q + 3 4r 4r + 3 5h 5h + 3
2.2 = 2.3
2 - 4 3 - 6
2 - 4 = 3 - 6
2.2 2.3
2.4 = 2.6
2 - 4 3 - 6
2 - 4 = 3 - 6
2.4 2.6
2 = 3
4 6
4 = 6
2 3
Enzo Exposyto 81

82. Properties
From
(5 + 3)
a - b = c - d
a + b c + d
Enzo Exposyto 82

83. TABLE - XXIII
PROPERTIES from (5+3) - (a-b) : (a+b) = (c-d) : (c+d) OR a - b = c - d
a + b c + d
denominators ≠ 0
1f 2m 2n 3f
(a - b) x (c + d)
=
(a + b) x (c - d)
c + d = c - d
a + b a - b
a - b = a + b
c - d c + d
a + b = c + d
a - b c - d
4s 4s + 3 4t 4t + 3 5i 5i + 3
2a = 2c
a + b c + d
a + b = c + d
2a 2c
-2b = -2d
a + b c + d
a + b = c + d
-2b -2d
- a = - c
b d
- b = - d
a c
Enzo Exposyto 83

84. TABLE - XXIV
PROPERTIES from (5+3) - (2-4) : (2+4) = (3-6) : (3+6) OR 2 - 4 = 3 - 6
2 + 4 3 + 6
1f 2m 2n 3f
(2 - 4) x (3 + 6)
=
(2 + 4) x (3 - 6)
3 + 6 = 3 - 6
2 + 4 2 - 4
2 - 4 = 2 + 4
3 - 6 3 + 6
2 + 4 = 3 + 6
2 - 4 3 - 6
4s 4s + 3 4t 4t + 3 5i 5i + 3
2.2 = 2.3
2 + 4 3 + 6
2 + 4 = 3 + 6
2.2 2.3
-2.4 = -2.6
2 + 4 3 + 6
2 + 4 = 3 + 6
-2.4 -2.6
- 2 = - 3
4 6
- 4 = - 6
2 3
Enzo Exposyto 84

85. Others
Properties
of the
Proportion
a : b = c : d
(Tables XXV-XXVI)
Enzo Exposyto 85

86. OTHER PROPERTIES of the PROPORTION a : b = c : d OR a/b = c/d [FROM 4a, 4a+3, 4b] (denominators ≠ 0)
1a 2c 2d 3a
(a + b) x d
=
b x (c + d)
d = c + d
b a + b
a + b = b
c + d d
b = d
a + b c + d
4i 4i + 3 4j 4j + 3 5d 5d + 3
a + 2b = c + 2d
b d
b = d
a + 2b c + 2d
a = c
b d
b = d
a c
a + 2b = c + 2d
a c
a = c
a + 2b c + 2d
1b 2e 2f 3b
b x (c + d)
=
(a + b) x d
c + d = d
a + b b
b = a + b
d c + d
a + b = c + d
b d
4k 4k + 3 4l 4l + 3 5e 5e + 3
a + 2b = c + 2d
a + b c + d
a + b = c + d
a + 2b c + 2d
-a = _-c
a + b c + d
a + b = c + d
-a -c
a + 2b = c + 2d
-a -c
-a = -c
a + 2b c + 2d
1c 2g 2h 3c
(a - b) x d
=
b x (c - d)
d = c - d
b a - b
a - b = b
c - d d
b = d
a - b c - d
4m 4m + 3 4n 4n + 3 5f 5f + 3
a = c
b d
b = d
a c
a - 2b = c - 2d
b d
b = d
a - 2b c - 2d
a = c
a - 2b c - 2d
a - 2b = c - 2d
a c
Enzo Exposyto 86

87. OTHER PROPERTIES of the PROPORTION a : b = c : d OR a/b = c/d [FROM 4b+3, 5, 5+3] (denominators ≠ 0)
1d 2i 2j 3d
b x (c - d)
=
(a - b) x d
c - d = d
a - b b
b = a - b
d c - d
a - b = c - d
b d
4o 4o + 3 4p 4p + 3 5g 5g + 3
a = c
a - b c - d
a - b = c - d
a c
-a + 2b = -c + 2d
a - b c - d
a - b = c - d
-a + 2b -c + 2d
a = c
-a + 2b -c + 2d
-a + 2b = -c + 2d
a c
1e 2k 2l 3e
(a + b) x (c - d)
=
(a - b) x (c + d)
c - d = c + d
a - b a + b
a + b = a - b
c + d c - d
a - b = c - d
a + b c + d
4q 4q + 3 4r 4r + 3 5h 5h + 3
2a = 2c
a - b c - d
a - b = c - d
2a 2c
2b = 2d
a - b c - d
a - b = c - d
2b 2d
a = c
b d
b = d
a c
1f 2m 2n 3f
(a - b) x (c + d)
=
(a + b) x (c - d)
c + d = c - d
a + b a - b
a - b = a + b
c - d c + d
a + b = c + d
a - b c - d
4s 4s + 3 4t 4t + 3 5i 5i + 3
2a = 2c
a + b c + d
a + b = c + d
2a 2c
-2b = -2d
a + b c + d
a + b = c + d
-2b -2d
- a = - c
b d
- b = - d
a c
Enzo Exposyto 87

88. Others
Properties
of the
Proportion
2 : 4 = 3 : 6
(Tables XXVII-XXVIII)
Enzo Exposyto 88

89. OTHER PROPERTIES of the PROPORTION 2 : 4 = 3 : 6 OR 2/4 = 3/6 [FROM 4a, 4a+3, 4b]
1a 2c 2d 3a
(2 + 4) x 6
=
4 x (3 + 6)
6 = 3 + 6
4 2 + 4
2 + 4 = 4
3 + 6 6
4 = 6
2 + 4 3 + 6
4i 4i + 3 4j 4j + 3 5d 5d + 3
2 + 2.4 = 3 + 2.6
4 6
4 = 6
2 + 2.4 3 + 2.6
2 = 3
4 6
4 = 6
2 3
2 + 2.4 = 3 + 2.6
2 3
2 = 3
2 + 2.4 3 + 2.6
1b 2e 2f 3b
4 x (3 + 6)
=
(2 + 4) x 6
3 + 6 = 6
2 + 4 4
4 = 2 + 4
6 3 + 6
2 + 4 = 3 + 6
4 6
4k 4k + 3 4l 4l + 3 5e 5e + 3
2 + 2.4 = 3 + 2.6
2 + 4 3 + 6
2 + 4 = 3 + 6
2 + 2.4 3 + 2.6
-2 = _-3
2 + 4 3 + 6
2 + 4 = 3 + 6
-2 -3
2 + 2.4 = 3 + 2.6
-2 -3
-2 = -3
2 + 2.4 3 + 2.6
1c 2g 2h 3c
(2 - 4) x 6
=
4 x (3 - 6)
6 = 3 - 6
4 2 - 4
2 - 4 = 4
3 - 6 6
4 = 6
2 - 4 3 - 6
4m 4m + 3 4n 4n + 3 5f 5f + 3
2 = 3
4 6
4 = 6
2 3
2 - 2.4 = 3 - 2.6
4 6
4 = 6 _
2 - 2.4 3 - 2.6
2 = 3
2 - 2.4 3 - 2.6
2 - 2.4 = 3 - 2.6
2 3
Enzo Exposyto 89

90. OTHER PROPERTIES of the PROPORTION 2 : 4 = 3 : 6 OR 2/4 = 3/6 [FROM 4b+3, 5, 5+3]
1d 2i 2j 3d
4 x (3 - 6)
=
(2 - 4) x 6
3 - 6 = 6
2 - 4 4
4 = 2 - 4
6 3 - 6
2 - 4 = 3 - 6
4 6
4o 4o + 3 4p 4p + 3 5g 5g + 3
2 = 3
2 - 4 3 - 6
2 - 4 = 3 - 6
2 3
-2 + 2.4 = -3 + 2.6
2 - 4 3 - 6
2 - 4 = 3 - 6
-2 + 2.4 -3 + 2.6
2 = 3
-2 + 2.4 -3 + 2.6
-2 + 2.4 = -3 + 2.6
2 3
1e 2k 2l 3e
(2 + 4) x (3 - 6)
=
(2 - 4) x (3 + 6)
3 - 6 = 3 + 6
2 - 4 2 + 4
2 + 4 = 2 - 4
3 + 6 3 - 6
2 - 4 = 3 - 6
2 + 4 3 + 6
4q 4q + 3 4r 4r + 3 5h 5h + 3
2.2 = 2.3
2 - 4 3 - 6
2 - 4 = 3 - 6
2.2 2.3
2.4 = 2.6
2 - 4 3 - 6
2 - 4 = 3 - 6
2.4 2.6
2 = 3
4 6
4 = 6
2 3
1f 2m 2n 3f
(2 - 4) x (3 + 6)
=
(2 + 4) x (3 - 6)
3 + 6 = 3 - 6
2 + 4 2 - 4
2 - 4 = 2 + 4
3 - 6 3 + 6
2 + 4 = 3 + 6
2 - 4 3 - 6
4s 4s + 3 4t 4t + 3 5i 5i + 3
2.2 = 2.3
2 + 4 3 + 6
2 + 4 = 3 + 6
2.2 2.3
-2.4 = -2.6
2 + 4 3 + 6
2 + 4 = 3 + 6
-2.4 -2.6
- 2 = - 3
4 6
- 4 = - 6
2 3
Enzo Exposyto 90

91. … and so on …
Enzo Exposyto 91

92. SitoGraphy
Enzo Exposyto 92

93. https://en.m.wikipedia.org/wiki/List_of_mathematical_symbols
http://psychstat3.missouristate.edu/Documents/IntroBook3/sbk10.htm
https://www.cliffsnotes.com/study-guides/geometry/similarity/properties-of-proportions
Enzo Exposyto 93