Operaciones con números racionales

We find numbers everywhere. People have used numbers throughout the history in different ways: for
counting, ordering, comparing, measuring, etc.
Set of Natural Numbers:
ℕ={ 0, 1, 2, 3, 4 ...} . They are useful for counting and ordering.
We can add or multiply two any natural numbers and the result is also a natural number; but the
subtraction of two natural numbers is not always a natural number.
Set of Integer Numbers:
ℤ={... , −3, −2, −1, 0, 1, 2, 3, ...} . Here the addition,
multiplication and subtraction of two integer numbers is always an integer number; but the quotient
of two integers is not always an integer.
Set of Rational Numbers:
ℚ={
a
/ a , b∈ℤ , b≠0 } .
b
3º ESO – Rational Numbers - 1
So the rational numbers are all those numbers that can be expressed as a fraction of two integer
numbers.
Examples of rational numbers:
2
; 2.13 ;
3
−175
24
0. 3̂
;
; 5 ; 0
; - 7 ….
As you can see, all natural numbers are integer numbers and, at the same time, all integer numbers are
rational numbers as well.
ℚ
ℕ
ℤ
Rational numbers can be expressed in different ways: fraction, decimal, percentage, graphically and
using the scientific notation.
Fraction
Decimal
Quotient of two integers:
numerator and denominator
We can write 52.678 in the following expanded forms
52.678 = 5 · 10 + 2 · 1 + 0.07 + 0.008
4
−8
−20
2
=
=
= .... =
6 −12 −30
3
= 5·10 + 2·1+ 6·0.1 + 7·0.01 + 8·0.001
−8
8
−16
4
=
=
= .... = −
6
−6
12
3
= 5·10 + 2·1 + 6·
1
1
1
+7·
+8·
10
100
1000
= 5· 101 +2· 100 +6·
1
1
1
1 +7·
2 +8·
10
10
10 3
= 5· 101 +2· 100 +6· 10−1 +7· 10−2 +8· 10−3
Percentage
A percent is a fraction with a denominator of 100. Ratios are often expressed as percentages.
1
= 1 % = 0.01
100
25 % =
25
100
= 0.25
2
40
=
= 40 % = 0.40
5 100
3º ESO – Rational Numbers - 2
200 % =
200
=2
100
Graphically
Scientific Notation
Large numbers
Small numbers
The distance to the sun is aproximately
The diameter of an atom is about
147 000 000 km
0.000 000 015 cm
= 1.47 x 108
= 1.5 x 10−8
RELATIONSHIP BETWEEN FRACTIONS AND DECIMALS
A fraction can be :
a) An integer number. Examples:
14
7
;
−8
2
;
15
3
b) A decimal number:
b.1) A terminating decimal. Example:
3
= 0.1875
16
b.2) A repeating or periodic decimal. Examples:
1
= 0.33333...= 0. 3
3
1
= 0.166666...= 0.1 6
6
There are also no-terminating and no-repeating decimals (irrational numbers ) such as
the number π = 3.14159926535897932384626433832795...... But these numbers can´t
be written as a fraction of integer numbers. Why? Because a fraction has always a
terminating decimal expression or a repeating decimal expression.
3º ESO – Rational Numbers - 3
Find the decimal expression of the following fractions. You can use the calculator, but may be,
in some cases, you will need to divide by yourself !
13
5
;
35
4
;
3
20
;
18
75
;
7
3
;
25
6
;
1
7
;
1
13
;
Investigate what fractions are terminating decimals and what are periodic decimals.
3º ESO – Rational Numbers - 4
1
17
Working the other way. Express as a fraction (in its lowest terms) the following decimal
numbers :
1.52 =
12.203 =
7.048 =
0.3502 =
Find out, with the calculator, the decimal expressions for the following fractions:
1
=
9
2
=
9
4
=
9
3
=
9
5
=
9
6
=
9
7
=
9
8
=
9
Do you see any pattern? All of them are ________________ or _______________ decimals. The
digit that repeats (period) is the ____________________ and it starts to repeat after the decimal point.
Would you know how to finish the following process to get the fraction that represents each
decimal number?
a) 1.5555..... = 1  0.5555.... = 1  0. 5 = 1 

=

b) 8.3333..... = 8. 3 = 8  0. 3 =
c) 10.77777...... =
Find out, with the calculator, the decimal expressions for the following fractions:
2
=
99
14
=
99
35
=
99
21
=
99
67
=
99
3º ESO – Rational Numbers - 5
Find the fraction that represents each decimal number:
 = 3  0. 14
 = 3 
a) 3.14141414...... = 3. 14


b) 21.353535...... =
 =
c) 5. 78
How can we get the fraction that represents a repeating decimal when the group of digits that repeats
doesn´t start immediately after the decimal point? (“mixed periodic decimal”)
¡ Pay attention to next example!
2

0. 2
9
2
1
0.0 2 =
=
=
=
10
10
90
45
Now it´s your turn:
a)
0.0 3 =
b)
0.00 5 =
c)
1.2 4 =
d)
3.0 7 =
e)
 =
0.2 43
f)
g)
1.357575757.... =
0.1345454545.... =
3º ESO – Rational Numbers - 6
GRAPHICAL REPRESENTATION
We use fractions to show part of a whole. For example, 3/4 represents three out of the four parts in
which a whole (or unit) can be divided.
Examples:
Ejemplos:
Proper fraction: the numerator is less than Improper fraction: the numerator is greater than
the denominator. A proper fraction is less the denominator. An improper fraction is greater
than 1
than 1. They can be written as a mixed number.
3
7
2
2
=1+ =1
5
5
5
5
State fractions to describe the shaded
parts.
In how many different ways is it possible
to divide the following figures into 2 equal
parts? And in 3 equal parts? In 4 equal parts?
What fraction of the rectangle is the
square?
3º ESO – Rational Numbers - 7
What fraction of the square is the
triangle?
A ) State the fraction that indicates the part
out of the totalof children:
______are girls
______are boys
______wear glasses.
b) What fractios of girls wear glasses?
State the fraction
represented for A, B, C and
D.
State fractions to
describe the shaded parts.
I gave three eighths of my savings to my brother and now I still have 900 euros. How
much money had I saved? Show a graphical representation of the problem.
3º ESO – Rational Numbers - 8
GRAPHICAL REPRESENTATION OF DECIMALS
Represent on the straight line the following decimals :
2.3
4.56
-1.28
SCIENTIFIC NOTATION
Scientific notation refers to expressing a number as a product of any number between 1 and 10 and a
10th power. It is mostly used when dealing with very large or very small quantities or numbers
containing many digits since it shortens the notation.
Examples:
Standard form
Scientific Notation
76 300
7.63 x 104
2 560 000
2.56 x 106
0.000066
6.6 x 10−5
0.005
Scientific Notation with a calculator
Writing
−3
5 x 10
1.45 x 10−26
3º ESO – Rational Numbers - 9
Practica el paso de notación científica a decimal en:
http://www.genmagic.org/mates2/nc1c.swf
http://www.educaplus.org/play-179-Notaci%C3%B3n-cient%C3%ADfica.html
Complete the table.
Number
Scientific Notation
0.000 000 000 345
0.000 6789
3 456 000 000 000
2300 000 000
0.0205
0.12
8 670 340 000 000 000 000
356
0.000000000000000002
23098
1.02 x 10-2
1.054678 x 106
1.00034 x 10-3
1.5487056 x 107
6.03 x 10-7
8 x 108
6.023 x 105
5.6 x 10-1
2.45 x 10-5
9.206 x 10-3
8.134 x 106
The number 69.27 x10-5 is not written in scientific notation. Write it in scientific notation and in
decimal form.
3º ESO – Rational Numbers - 10
PERCENTAGES
1.
CALCULATING PERCENTAGES
30
⋅200 = 0.30 · 200 = 60
100
30% of 200
45% of 120
0.45 · 120 = 54
7% of 300
0.07 · 300 = 21
180 % of 200 =
2. CALCULATING A TOTAL KNOWING THE PERCENTAGE
25% of the total is 13
The total is
13
= 52
0.25
40% of the total is 290
The total is
290
= 725
0.40
3% of a number is 36
The number is …
160% of a number is 832
The number is …
3. CALCULATING A PERCENT AS PART OF A WHOLE
What percentage is 308 out of 1400 ?
308
22
= 0.22 =
= 22 %
1400
100
What percentage is 32 out of 400 ?
32
8
= 0.08 =
=8%
400
100
What percentage is 12 out of 28 ?
12
= 0.42857 ≃ 42.86 %
28
What percentage is 45 out of 900 ?
What percentage is 5880 out of 35000?
3º ESO – Rational Numbers - 11
4.
PERCENT INCREASES
In a bar a soda drink costs 1.50 pounds. The owner wants to increase the price by 10 %. What will be
the new price?
First method:
Calculate 10% of 1.50 which is 0.15 and then add this to the original price.
1.50 + 0.15 = 1.65
So the new price will be 1.65 pounds.
Second and shorter method:
Multiply the original price by the number 1.1 because the final price will be 110% of the original
price. So we multiply 1.50 · 1.1 = 1.65
and the final price will be 1.65 pounds.
“The number 1.1 is called the VARIATION INDEX“
The population of a village is 2500. During the summer it increases by 35%. How many people
does the village have in summer?
5. PERCENT DECREASES
A sweatshirt that costs 60 euros is on sale for 25% off. How much does it cost now?
In this case he have to subtract 25% from the original price , so the final price will be the 75%
of the original one.
So the final price will be 60 · 0.75 = 45 euros.
“The number 0.75 is called the VARIATION INDEX“
We observe that:
.- In a percent increase, the variation index is bigger than 1
.- In a percent decrease, the variation index is less than 1
.- If an original quantity is transformed by a percent increase or a percent decrease, the final quantity
is obtained from this formula:
Final Quantity = Original Quantity · Variation Index
3º ESO – Rational Numbers - 12
For what number do you multiply a quantity to
a) calculate the 10% of the quantity
b) increase by 10% the quantity
c) decrease by 10% the quantity
d) increase by 1% the quantity
d) decrease by 1% the quantity
A swamp had 200 hl of water before summer. The amount of water has decreased by 45%.
What percentage of water remains? What amount of water is left?
In a class there are 15 girls and 5 boys. What part out of the total are the girls? Express the
number as a fraction in lowest terms and as a percentage.
Two out of five children of our country are obese. What percentage of obese children is it? In
another country the ratio of obese children is seven to fifteen. In which country the children obesity is
higher?
My savings have been reduced a 30 % off after giving some money to my brother. If I have
350 euros left, how much money have I saved?
3º ESO – Rational Numbers - 13
¿Es lo mismo aplicar un descuento a un producto y luego calcular el IVA que hacerlo al revés,
primero calcular el IVA y luego hacer el descuento?
Un producto sufre un aumento del 10% en su precio, ¿qué porcentaje debemos descontar para
que vuelva al precio inicial?
Complete the following table writing the different expressions of a number.
Fraction
Decimal
Percentage
Graphic
Scientific notation
6/7
6,54
5%
1,3.10-5
3º ESO – Rational Numbers - 14
OPERATIONS WITH RATIONAL NUMBERS
 ORDER OF OPERATIONS
ￇ

EXPONENT LAWS
 Multiplying powers with the same base
 Dividing powers with the same base
The power of a multiplication
 The power of a division
 The power of a power
3º ESO – Rational Numbers - 15
Calculate
a) 34=
b) (-3)4=
d) (-3)-4=
c) 3-4=
e) 30=
Express as a power
Express as a power
Powers in a calculator . Example: 23
3º ESO – Rational Numbers - 16

OPERATIONS WITH FRACTIONS
Addition / Subtraction
Multiplication
Division
Power
Evaluate
1)
1 3 2
+ ⋅
4 2 3
5 4 3 20
3) ⋅ − ⋅
=
6 15 5 18
2)
12 −1 3
:( + ) =
18
2 8
4
5
3
8
4) (  2)  ( 
5
) =
6
3º ESO – Rational Numbers - 17
5)
3 18 5
: − =
8 24 6
3 1 −14
7) ( + ):
=
5 10 15
9)
6)
−7 1
3 3 2
: −[− +( − ) =
8 2
8 5 3
3
−7 3
8) (− +1):( ⋅ +1) =
8
3 4
−4 7 5
⋅( − ) =
5 3 4
10)
(
1 3 5
 ):
2 4 6


3º ESO – Rational Numbers - 18
 OPERATIONS WITH NUMBERS IN SCIENTIFIC NOTATION
Adition / Subtraction
a) 5·10 6 + 2·10 6 = 7·10 6
b) 9·10 3 + 9.6·10 2 = 9·10 3 + 0.96·10 3 = 9.96·10 3
(4·10 6 )·(2·10 6 ) = 8·10 1 2
Multiplication:
(4·10 1 2 ):(2·10 6 ) =2.10 6
Division:
Power:
(3·10 6 ) 2 = 9·10 1 2
Calculate and check with the calculator. Express the result in scientific notation.
1.-
1.54 · 108 - 4.34 ·107
2.-
( 6.2 · 10– 4 ) · ( 5 ·10– 7 )
3.-
( 3 · 105 ) : ( 8 ·10– 9 )
4.-
( 2 · 105 + 3 · 107 ) · ( 5 ·10– 8 )
5.-
( 2 · 105 )4 · ( 4 ·10 9 )-3
6.-
( 4.8 · 10– 12 ) : ( 1.9 · 10– 3 + 5 · 10– 4 )
Simplify
2
1)
3−
3)
5)
 
2
1
3
1−
 −2
3
4
8
 
1
3
− −1
2 4
3
1
4
 
 
1 3 2
3− ⋅ −
4 5 15
4 1 3
6 ⋅ −
25 2 4
2)
[    ]  
2 1
2
−  13
−1
3 9
3
 
 
4)
3 1
−
5 3
4 6
−2 −
3 5
6)
  
 
−3
2 5 3 5
− ⋅ −
3 9 4 6
7 5 4
− ⋅ 1
12 6 3
3º ESO – Rational Numbers - 19
2
: −
2
3
Evaluate
−3
−2
  
3
−1
2
1)
1
:
2
2)
−2
 
1
2
3
−2
3)
⋅3
−1 3
[  ]
1
1
2
Use the Laws of Exponents to simplify the following:
1.-
6 4⋅82
32⋅23⋅24
5.-
6 ⋅9
3
2 ⋅−32⋅42
2
8.-
152⋅42
12 2⋅10
3.-
2
−3

a
b
2.-
−2
⋅ a−1 
6.-
9.-
−3
2−5⋅43
16
2−5⋅8⋅9⋅3−2
2−4⋅42⋅6−1
7.-
−2

1
a
a
b
25⋅32⋅4−1
2 3⋅9−1
4.-
10.-
−4

a
b
−3 −1
[  ] 
b
a
a3
b2
a −1⋅b 
−2
Complete this magic square:
3/ 8
1/2
3/4
1
Martha has spent three fourths of her savings on a trip and two thirds of the rest on clothing.
She has 150 euros left so how much money had she saved? (Draw a picture to figure out the solution
but work out with fractions too !)
Calculate the following writing each decimal number as a fraction.
a) 0 . 
12 − 0.2
b) 1. 6 − 1.0 2
c) 2. 3  4. 6

d) 6. 
17  3. 82
3º ESO – Rational Numbers - 20