MATHS SYMBOLS - PROPORTIONS and FIRST EQUATIONS - DEFINITION - 4 PROPERTIES - CROSS PRODUCTS - EXTREMES SWITCHING - MEANS SWITCHING - UPSIDE-DOWN - DENOMINATOR ADDITION - DENOMINATOR SUBTRACTION - PROPORTIONS and EQUATIONS - SOLUTIONS by 3 METHODS - PROOFS - EXAMPLES and CALCULATIONS STEP by STEP

1. ENZO EXPOSYTO
MATHS
SYMBOLS
PROPORTIONS and FIRST EQUATIONS
Enzo Exposyto 1

2. a : b = c : d a x d = b x c d : b = c : a a : c = b : d b : a = d : c
PROPORTIONS
and FIRST
EQUATIONS
x : b = c : d a : x = c : d a : b = x : d a : b = c : x
Enzo Exposyto 2

3.
Enzo Exposyto 3

4. 1 - Proportions - Definition 5
2 - Proportions - First 4 Properties 9
3 - Proportions and First Equations - Method (1) 16
4 - Proportions and First Equations - Method (2) 29
5 - Proportions and First Equations - Method (3) 42
6 - SitoGraphy 63
Enzo Exposyto 4

5. PROPORTIONS
-
Definition
Enzo Exposyto 5

6. 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 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 6

7. 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
Because 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 7

8. 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 8

9. PROPORTIONS
-
First 4 Properties
Enzo Exposyto 9

10. Proportions - First 4 Properties - (1)
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
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 10

11. Proportions - First 4 Properties - (2a)
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
1) From the proportion 2:4 = 3:6
if we switch 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 11

12. Proportions - First 4 Properties - (2b)
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
1) From the proportion 2:4 = 3:6
if we switch 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 switch 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 12

13. Proportions - First 4 Properties - (3)
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
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 13

14. Proportions - First 4 Properties - (4a)
a:b=c:d proportion - Property (4a) - Denominator Addition Property
a + b = c + d with b ≠ 0 and d ≠ 0
b d
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 14

15. Proportions - First 4 Properties - (4b)
a:b=c:d proportion - Property (4b) - Denominator Subtraction Property
a - b = c - d with b ≠ 0 and d ≠ 0
b d
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 15

16. PROPORTIONS
and FIRST
EQUATIONS
Method (1)
Enzo Exposyto 16

17. Proportions and First Equations - Method (1) - 1a
x:b=c:d equation - Find x
By the Cross-Products Property, we can write:
x x d = b x c
If we divide both sides by d, we have:
x x d = b x c
d d
and, then:
x = b x c
d
x (1st extreme) is equal to the product of the means
divided
by the 2nd extreme
Enzo Exposyto 17

18. Proportions and First Equations - Method (1) - 1b
x:b=c:d equation - Find x
Example 1
x : 4 = 3 : 6
By the Cross-Products Property, we can write:
x x 6 = 4 x 3
If we divide both sides by 6, we have:
x x 6 = 4 x 3
6 6
and, then:
x = 4 x 3 = 12 = 2
6 6
Enzo Exposyto 18

19. Proportions and First Equations - Method (1) - 1c
x:b=c:d equation - Find x
Example 2
x : 5 = 9 : 3
By the Cross-Products Property, we can write:
x x 3 = 5 x 9
If we divide both sides by 3, we have:
x x 3 = 5 x 9
3 3
and, then:
x = 5 x 9 = 45 = 15
3 3
Enzo Exposyto 19

20. Proportions and First Equations - Method (1) - 2a
a:x=c:d equation - Find x
By the Cross-Products Property, we can write:
a x d = x x c
If we divide both sides by c, we have:
a x d = x x c
c c
and, then:
x = a x d
c
x (1st mean) is equal to the product of the extremes
divided
by the 2nd mean
Enzo Exposyto 20

21. Proportions and First Equations - Method (1) - 2b
a:x=c:d equation - Find x
Example 1
2 : x = 3 : 6
By the Cross-Products Property, we can write:
2 x 6 = x x 3
If we divide both sides by 3, we have:
2 x 6 = x x 3
3 3
and, then:
x = 2 x 6 = 12 = 4
3 3
Enzo Exposyto 21

22. Proportions and First Equations - Method (1) - 2c
a:x=c:d equation - Find x
Example 2
15 : x = 9 : 3
By the Cross-Products Property, we can write:
15 x 3 = x x 9
If we divide both sides by 9, we have:
15 x 3 = x x 9
9 9
and, then:
x = 15 x 3 = 45 = 5
9 9
Enzo Exposyto 22

23. Proportions and First Equations - Method (1) - 3a
a:b=x:d equation - Find x
By the Cross-Products Property, we can write:
a x d = b x x
If we divide both sides by b, we have:
a x d = b x x
b b
and, then:
x = a x d
b
x (2nd mean) is equal to the product of the extremes
divided
by the 1st mean
Enzo Exposyto 23

24. Proportions and First Equations - Method (1) - 3b
a:b=x:d equation - Find x
Example 1
2 : 4 = x : 6
By the Cross-Products Property, we can write:
2 x 6 = 4 x x
If we divide both sides by 4, we have:
2 x 6 = 4 x x
4 4
and, then:
x = 2 x 6 = 12 = 3
4 4
Enzo Exposyto 24

25. Proportions and First Equations - Method (1) - 3c
a:b=x:d equation - Find x
Example 2
15 : 5 = x : 3
By the Cross-Products Property, we can write:
15 x 3 = 5 x x
If we divide both sides by 5, we have:
15 x 3 = 5 x x
5 5
and, then:
x = 15 x 3 = 45 = 9
5 5
Enzo Exposyto 25

26. Proportions and First Equations - Method (1) - 4a
a:b=c:x equation - Find x
By the Cross-Products Property, we can write:
a x x = b x c
If we divide both sides by a, we have:
a x x = b x c
a a
and, then:
x = b x c
a
x (2nd extreme) is equal to the product of the means
divided
by the 1st extreme
Enzo Exposyto 26

27. Proportions and First Equations - Method (1) - 4b
a:b=c:x equation - Find x
Example 1
2 : 4 = 3 : x
By the Cross-Products Property, we can write:
2 x x = 4 x 3
If we divide both sides by 4, we have:
2 x x = 4 x 3
2 2
and, then:
x = 4 x 3 = 12 = 6
2 2
Enzo Exposyto 27

28. Proportions and First Equations - Method (1) - 4c
a:b=c:x equation - Find x
Example 2
15 : 5 = 9 : x
By the Cross-Products Property, we can write:
15 x x = 5 x 9
If we divide both sides by 15, we have:
15 x x = 5 x 9
15 15
and, then:
x = 5 x 9 = 45 = 3
15 15
Enzo Exposyto 28

29. PROPORTIONS
and FIRST
EQUATIONS
Method (2)
Enzo Exposyto 29

30. Proportions and First Equations - Method (2) - 1a
x:b=c:d equation - Find x
We can write:
x = c
b d
If we multiply both sides by b, we have:
x x b = c x b
b d
and, then:
x = c x b
d
The number b has been moved from the left to the right side
and from the denominator to the numerator
Enzo Exposyto 30

31. Proportions and First Equations - Method (2) - 1b
x:b=c:d equation - Find x
Example 1
x : 4 = 3 : 6
We can write:
x = 3
4 6
Now, moving the number 4 from the left to right side
and from the denominator to the numerator, we obtain:
x = 3 x 4 = 12 = 2
6 6
Enzo Exposyto 31

32. Proportions and First Equations - Method (2) - 1c
x:b=c:d equation - Find x
Example 2
x : 5 = 9 : 3
We can write:
x = 9
5 3
Now, moving the number 5 from the left to right side
and from the denominator to the numerator, we obtain:
x = 9 x 5 = 45 = 15
3 3
Enzo Exposyto 32

33. Proportions and First Equations - Method (2) - 2a
a:x=c:d equation - Find x
We can write:
a = c
x d
By the Upside-Down Property, the equation becomes
x = d
a c
If we multiply both sides by a, we get:
x x a = d x a
a c
and, finally:
x = d x a
c
Enzo Exposyto 33

34. Proportions and First Equations - Method (2) - 2b
a:x=c:d equation - Find x
Example 1
2 : x = 3 : 6
We can write:
2 = 3
x 6
By the Upside-Down Property, the equation becomes
x = 6
2 3
Now, moving the number 2 from the left to right side
and from the denominator to the numerator, we obtain:
x = 6 x 2 = 12 = 4
3 3
Enzo Exposyto 34

35. Proportions and First Equations - Method (2) - 2c
a:x=c:d equation - Find x
Example 2
15 : x = 9 : 3
We can write:
15 = 9
x 3
By the Upside-Down Property, the equation becomes
x = 3
15 9
Now, moving the number 15 from the left to right side
and from the denominator to the numerator, we get:
x = 3 x 15 = 45 = 5
9 9
Enzo Exposyto 35

36. Proportions and First Equations - Method (2) - 3a
a:b=x:d equation - Find x
We can write:
a = x or x = a
b d d b
If we multiply both sides by d, we have:
x x d = a x d
d b
and, then:
x = a x d
b
The number d has been moved from the left to the right side
and from the denominator to the numerator
Enzo Exposyto 36

37. Proportions and First Equations - Method (2) - 3b
a:b=x:d equation - Find x
Example 1
2 : 4 = x : 6
We can write:
2 = x or x = 2
4 6 6 4
Now, moving the number 6 from the left to right side
and from the denominator to the numerator, we obtain:
x = 2 x 6 = 12 = 3
4 4
Enzo Exposyto 37

38. Proportions and First Equations - Method (2) - 3c
a:b=x:d equation - Find x
Example 2
15 : 5 = x : 3
We can write:
15 = x or x = 15
5 3 3 5
Now, moving the number 3 from the left to right side
and from the denominator to the numerator, we obtain:
x = 15 x 3 = 45 = 9
5 5
Enzo Exposyto 38

39. Proportions and First Equations - Method (2) - 4a
a:b=c:x equation - Find x
We can write:
a = c
b x
By the upside-down property, the equation becomes
b = x or x = b
a c c a
If we multiply both sides by c, we get:
x x c = b x c
c a
and, finally:
x = b x c
a
Enzo Exposyto 39

40. Proportions and First Equations - Method (2) - 4b
a:b=c:x equation - Find x
Example 1
2 : 4 = 3 : x
We can write:
2 = 3
4 x
By the upside-down property, the equation becomes
4 = x or x = 4
2 3 3 2
Now, moving the number 3 from the left to right side
and from the denominator to the numerator, we obtain:
x = 4 x 3 = 12 = 6
2 2
Enzo Exposyto 40

41. Proportions and First Equations - Method (2) - 4c
a:b=c:x equation - Find x
Example 2
15 : 5 = 9 : x
We can write:
15 = 9
5 x
By the upside-down property, the equation becomes
5 = x or x = 5
15 9 9 15
Now, moving the number 9 from the left to right side
and from the denominator to the numerator, we obtain:
x = 5 x 9 = 45 = 3
15 15
Enzo Exposyto 41

42. PROPORTIONS
and FIRST
EQUATIONS
Method (3)
Enzo Exposyto 42

43. Proportions and First Equations - Method (3) - 1a
x:b=c:d equation - Find x
We can write:
x = c
b d
The target is to have the 2nd side denominator equal to
the 1st side denominator …
Enzo Exposyto 43

44. Proportions and First Equations - Method (3) - 1b
x:b=c:d equation - Find x
Example 1
x : 4 = 3 : 6
We can write:
x = 3 (let’s reduce the 2nd fraction to lowest terms) x = 1
4 6 4 2
The target is to have the 2nd side denominator equal to
the 1st side denominator : let's multiply numerator
and denominator of the 2nd side by the same number …
x = 1 . 2 and, then, x = 2
4 2 2 4 4
Now, it’s a simple equation and we can see that:
x = 2
Enzo Exposyto 44

45. Proportions and First Equations - Method (3) - 1c
x:b=c:d equation - Find x
Example 2
x : 5 = 9 : 3
We can write:
x = 9 (let’s reduce the 2nd fraction to lowest terms) x = 3
5 3 5 1
The target is to have the 2nd side denominator equal to
the 1st side denominator : let's multiply numerator
and denominator of the 2nd side by the same number …
x = 3 . 5 and, then, x = 15
5 1 5 5 5
Now, it’s a simple equation and we can see that:
x = 15
Enzo Exposyto 45

46. Proportions and First Equations - Method (3) - 1d
x:b=c:d equation - Find x
Example 3
x : 4 = 6 : 12
The target is to have the 2nd side denominator equal to
the 1st side denominator; then
1) we can write:
x = 6 (let’s reduce the 2nd fraction to lowest terms) x = 1
4 12 4 2
and so on … (see previous examples)
2) we can divide numerator and denominator of the 2nd side
by the same number and achieve the goal …
x = 6 : 3 and, then, x = 2
4 12 : 3 4 4
Now, it’s a simple equation and we can see that:
x = 2
Enzo Exposyto 46

47. Proportions and First Equations - Method (3) - 1e
x:b=c:d equation - Find x
Example 4
x : 2 = 12 : 4
The target is to have the 2nd side denominator equal to
the 1st side denominator; then
1) we can write:
x = 12 (let’s reduce the 2nd fraction to lowest terms) x = 3
2 4 2 1
and so on … (see previous examples)
2) we can divide numerator and denominator of the 2nd side
by the same number and achieve the goal …
x = 12 : 2 and, then, x = 6
2 4 : 2 2 2
Now, it’s a simple equation and we can see that:
x = 6
Enzo Exposyto 47

48. Proportions and First Equations - Method (3) - 2a
a:x=c:d equation - Find x
We can write:
a = c
x d
The target is to have the 2nd side numerator equal to
the 1st side numerator …
Note that, by the upside-down property,
the equation becomes
x = d
a c
and we could use the algorithm of page 43,
with x as first extreme
Enzo Exposyto 48

49. Proportions and First Equations - Method (3) - 2b
a:x=c:d equation - Find x
Example 1
2 : x = 3 : 6
We can write:
2 = 3 (let’s reduce the 2nd fraction to lowest terms) 2 = 1
x 6 x 2
The target is to have the 2nd side numerator equal to
the 1st side numerator : let's multiply numerator
and denominator of the 2nd side by the same number …
2 = 1 . 2 and, then, 2 = 2
x 2 2 x 4
Now, it’s a simple equation and we can see that:
x = 4
Enzo Exposyto 49

50. Proportions and First Equations - Method (3) - 2c
a:x=c:d equation - Find x
Example 2
15 : x = 9 : 3
We can write:
15 = 9 (let’s reduce the 2nd fraction to lowest terms) 15 = 3
x 3 x 1
The target is to have the 2nd side numerator equal to
the 1st side numerator : let's multiply numerator
and denominator of the 2nd side by the same number …
15 = 3 . 5 and, then, 15 = 15
x 1 5 x 5
Now, it’s a simple equation and we can see that:
x = 5
Enzo Exposyto 50

51. Proportions and First Equations - Method (3) - 2d
a:x=c:d equation - Find x
Example 3
2 : x = 6 : 12
The target is to have the 2nd side numerator equal to
the 1st side numerator; then
1) we can write:
2 = 6 (let’s reduce the 2nd fraction to lowest terms) 2 = 1
x 12 x 2
and so on … (see previous examples)
2) we can divide numerator and denominator of the 2nd side
by the same number and achieve the goal …
2 = 6 : 3 and, then, 2 = 2
x 12 : 3 x 4
Now, it’s a simple equation and we can see that:
x = 4
Enzo Exposyto 51

52. Proportions and First Equations - Method (3) - 2e
a:x=c:d equation - Find x
Example 4
6 : x = 12 : 4
The target is to have the 2nd side numerator equal to
the 1st side numerator; then
1) we can write:
6 = 12 (let’s reduce the 2nd fraction to lowest terms) 6 = 3
x 4 x 1
and so on … (see previous examples)
2) we can divide numerator and denominator of the 2nd side
by the same number and achieve the goal …
6 = 12 : 2 and, then, 6 = 6
x 4 : 2 x 2
Now, it’s a simple equation and we can see that:
x = 2
Enzo Exposyto 52

53. Proportions and First Equations - Method (3) - 3a
a:b=x:d equation - Find x
We can write:
a = x
b d
The target is to have the 1st side denominator equal to
the 2nd side denominator …
Note that, if we exchange the sides
a = x or x = a
b d d b
we could use the same algorithm seen at page 43,
with x as first extreme
Enzo Exposyto 53

54. Proportions and First Equations - Method (3) - 3b
a:b=x:d equation - Find x
Example 1
2 : 4 = x : 6
We can write:
2 = x (let’s reduce the 1st fraction to lowest terms) 1 = x
4 6 2 6
The target is to have the 1st side denominator equal to
the 2nd side denominator : let's multiply numerator
and denominator of the 1st side by the same number …
1 . 3 = x and, then, 3 = x
2 3 6 6 6
Now, it’s a simple equation and we can see that:
x = 3
Enzo Exposyto 54

55. Proportions and First Equations - Method (3) - 3c
a:b=x:d equation - Find x
Example 2
15 : 5 = x : 3
We can write:
15 = x (let’s reduce the 1st fraction to lowest terms) 3 = x
5 3 1 3
The target is to have the 1st side denominator equal to
the 2nd side denominator : let's multiply numerator
and denominator of the 1st side by the same number …
3 . 3 = x and, then, 9 = x
1 3 3 3 3
Now, it’s a simple equation and we can see that:
x = 9
Enzo Exposyto 55

56. Proportions and First Equations - Method (3) - 3d
a:b=x:d equation - Find x
Example 3
4 : 12 = x : 6
The target is to have the 1st side denominator equal to
the 2nd side denominator; then
1) we can write:
4 = x (let’s reduce the 1st fraction to lowest terms) 1 = x
12 6 3 6
and so on … (see previous examples)
2) we can divide numerator and denominator of the 1st side
by the same number and achieve the goal …
4 : 2 = x and, then, 2 = x
12 : 2 6 6 6
Now, it’s a simple equation and we can see that:
x = 2
Enzo Exposyto 56

57. Proportions and First Equations - Method (3) - 3e
a:b=x:d equation - Find x
Example 4
8 : 24 = x : 6
The target is to have the 1st side denominator equal to
the 2nd side denominator; then
1) we can write:
8 = x (let’s reduce the 1st fraction to lowest terms) 1 = x
24 6 3 6
and so on … (see previous examples)
2) we can divide numerator and denominator of the 1st side
by the same number and achieve the goal …
8 : 4 = x and, then, 2 = x
24 : 4 6 6 6
Now, it’s a simple equation and we can see that:
x = 2
Enzo Exposyto 57

58. Proportions and First Equations - Method (3) - 4a
a:b=c:x equation - Find x
We can write:
a = c
b x
The target is to have the 1st side numerator equal to
the 2nd side numerator …
Note that, if we exchange the sides
a = c becomes c = a
b x x b
we could use the same algorithm seen at page 48
By the upside-down property, the equation becomes
b = x or x = b and we could use the algorithm of page 43
a c c a
Enzo Exposyto 58

59. Proportions and First Equations - Method (3) - 4b
a:b=c:x equation - Find x
Example 1
2 : 4 = 3 : x
We can write:
2 = 3 (let’s reduce the 1st fraction to lowest terms) 1 = 3
4 x 2 x
The target is to have the 1st side numerator equal to
the 2nd side numerator : let's multiply numerator
and denominator of the 1st side by the same number …
1 . 3 = 3 and, then, 3 = 3
2 3 x 6 x
Now, it’s a simple equation and we can see that:
x = 6
Enzo Exposyto 59

60. Proportions and First Equations - Method (3) - 4c
a:b=c:x equation - Find x
Example 2
15 : 5 = 9 : x
We can write:
15 = 9 (let’s reduce the 1st fraction to lowest terms) 3 = 9
5 x 1 x
The target is to have the 1st side numerator equal to
the 2nd side numerator : let's multiply numerator
and denominator of the 1st side by the same number …
3 . 3 = 9 and, then, 9 = 9
1 3 x 3 x
Now, it’s a simple equation and we can see that:
x = 3
Enzo Exposyto 60

61. Proportions and First Equations - Method (3) - 4d
a:b=c:x equation - Find x
Example 3
4 : 12 = 2 : x
The target is to have the 1st side numerator equal to
the 2nd side numerator; then
1) we can write:
4 = 2 (let’s reduce the 1st fraction to lowest terms) 1 = 2
12 x 3 x
and so on … (see previous examples)
2) we can divide numerator and denominator of the 1st side
by the same number and achieve the goal …
4 : 2 = 2 and, then, 2 = 2
12 : 2 x 6 x
Now, it’s a simple equation and we can see that:
x = 6
Enzo Exposyto 61

62. Proportions and First Equations - Method (3) - 4e
a:b=c:x equation - Find x
Example 4
8 : 24 = 2 : x
The target is to have the 1st side numerator equal to
the 2nd side numerator; then
1) we can write:
8 = 2 (let’s reduce the 1st fraction to lowest terms) 1 = 2
24 x 3 x
and so on … (see previous examples)
2) we can divide numerator and denominator of the 1st side
by the same number and achieve the goal …
8 : 4 = 2 and, then, 2 = 2
24 : 4 x 6 x
Now, it’s a simple equation and we can see that:
x = 6
Enzo Exposyto 62

63. SitoGraphy
Enzo Exposyto 63

64. 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 64