1. Applying Operation in Real Numbers l

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Real Numbers Scheme
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3. The Definition of Rational Numbers
Rational Number is number that can be denoted by ,
with, a and b, are the members of integer numbers and b
Example:
6, ½ etc.
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4. The Definition of Irrational Numbers
Irrational number is numbers which cannot be denoted by fraction and the decimal number is unlimited.
Example:
Roots, ,decimal
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5. The Definition of Prime Numbers
Prime Numbers is number which only have two factors; 1 (one) and the number itself.
Example:
2, 3, 5, 7, ...etc
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6. The Definition of Compose Numbers
Compose number is number which have more than one factors.
Example:
4, 6, 8, 9…
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7. The Real Number Operation
A. Addition Operation
1. Integer Number
The Properties
Commutative: a +b = b + a
Example: = 3 + 2
Associative: a +(b + c)= (a + b)+ c
Example: 1 + (3 + 5) = (1 + 3) + 5
Have addition identity element, that are: a + 0 = 0 + a
Example : = 0 + 1
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8. Real Numbers Operation
Subtraction
Has addition inverse,
Example; inverse a = - a,
Then : a + (-a) = -a + a
Example : (-2) =
= 0
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Real Number Operation
A. Addition and Subtraction Operation
2. Fraction Number
Properties c b a + = or
where a, b, c B and c ≠ 0 ,
Where a, b, c, d B and c ≠ 0 bd bc ad d c b a - = bd bc ad d c b a - =
atau
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10. Multiplication and Division Operation
The properties:
1. Commutative: a x b = b x a
Example: a. 4 x 3 = 3 x 4
½ x ¾ = ¾ x ½
½ : ¾ = ½ x 4/3
2. Asassociative: (a x b) x c = a x ( b x c)
Example: { 5 x (-7)} x 2 = 5 x { (-7) x 2}
3. Have identity item 1, so: a . 1 = 1 . a = a
Example: = = 2
4. Have multiplication inverse for aR; a ≠ 0 ; so a x 1/a = 1, then inverse 1/a is multiplication inverse of a.
In multiplication and division of real numbers, we have:
a. a . ( -b) = - (ab) d. ( -a) : b = -a : ( -b)
b. ( -a) . b = - (ab) e. ( -a) . b = - (ab) c. ( -a) :(-b) = f. -a : (-b) = -
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11. Conversing percentage, or decimal fraction
Fraction converse is usually in percentage.
Change fraction into percentage by changing the denominator into 100.
Example:
a = = 40%
b = = 44%
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12. Conversing percentage, or decimal fraction
2. Conversing fraction into decimal
Changing the denominator into 10 or another exponent 10
Example:
a = x = = 0,4
b = X = = 3, 40
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13. Conversing percentage, or decimal fraction
3. Conversing the percentage into fraction or into decimal.
Example :
a. 20% = = 0,2 = 20%
b. 75% =
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Ratio Equivalent
Complete the table !
Quantity
Price 1
200 2
Perbandingan senilai
400 3 … 4 … …
1000 6 … 7 … X …
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Ratio Unequivalent
Learning Experience
A project of sewing clothes with 24 tailor is planned to finish in 48 days.
After working for 12 days, the job was delayed for 9 days.
How many tailor should be added so that the job will be finished on time?
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Ratio Unequivalent
Problem Solving:
Its ratio is unequivalent, so:
The rest of the job for 48–12 = 36 days, which should be finished by 24 tailors.
But the remain time is only 48–12–9 = 27 days.
so:
24 tailor  days
x tailor  days
Then: 32 27
864 36 . 24 = Û x
Then the additional tailor are 8 tailors
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Ratio Unequivalent
The ratio is not equivalent if the values of two comparatives are unequivalent.
Formula = or a . c = b . d
Example:
A farmer has food supply for 80 cattle for a month. If the farmer adds 20 cattle more, so how many days the food supply will be run out?
Answer:
Then: = ↔ 80 x 30 = 100 x d ↔ 2400
= 100d↔ d = 24
Cattle Quantity
Days
80 = a
30 = c
= 100= b d
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18. Perbandingan Senilai dan Berbalik Nilai
In another way :
If variable X from x1 becomes x2
and variable y from y1 becomes y2
then : 2 y 1 x =
Senilai ,if : 1 y 2 x =
Berbalik nilai if :
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Ratio Equivalent and unequivalent
Exercise
In constant speed, a car needs 5 liter fuel for 60 km. how many liter of fuel does it need for 150 km?
2. The distance between two towns can be reached by the vehicle with 72km/hour speed for 5 hours. Determine the rate speed of that vehicle to reach that town in 8 hours?
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Ratio Equivalent and unequivalent
Problem Solving:
Because the ratio is equivalent, then: x 5
150 60 =
The ratio is unequivalent, then: 5 8 72 = x
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21. Ratio Equivalent and unequivalent
Exercise
The mixture of liquid cake ingredient are palm oil and water with the scale 1 : 18.
How many liters of palm oil needed to get 9.5 the mixture of liquid cake ingredient?
A map in rectangle shape is drawn in scale of : 1 : and has length and width 4:3. while the map circumference is 112 cm.
Determine the real area of the map?
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Scale
Scale is ratio between the size in the drawing and the real size.
Scale 1 : n means, every 1 cm in the map represents n cm in the real distance.
Distance in Map
Distance
Real
(PICTURE)
Scale=
map
Scale=
Distance in Map (Picture)
Scale Distance
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Scale
Example:
In a map which has scale 1: , the distance between Surabaya and Malang is 2 cm.
How many kilometer is the real distance?
Answer:
Scale 1:
The distance in map =2 cm
The real distance = 2 x 4, =
= 85 km
Hal.: 23
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