number i - WIKI1-mar-aguera

M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
ARITHMETIC
UNIT1. CARDINAL AND ORDINAL NUMBERS.





Cardinals for counting.
How do you read and write cardinal numbers?
Ordinals for arranging.
How do you read and write ordinal numbers?
Some exceptions.
UNIT2. THE FOUR OPERATIONS ON NUMBERS.
 The four basic operations: Addition, subtraction,
multiplication and division.
 How do you read arithmetic expressions?
 Review some different methods of multiplication and division.
UNIT3. BIG NUMBERS.
 The place value of numbers.
 Rounding big numbers.
UNIT4. POWERS.




Power, index and base.
The squares and the cubes of the first natural numbers.
Numbers in expanded form and in index form.
General rules of powers.
1
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT5. MULTIPLES AND FACTORS.






Multiples.
Factors.
Prime numbers.
Test of divisibility.
The lowest common multiple.
The highest common divisor.
UNIT6. INTEGERS.
 What is an integer?
 Operating with integers.
UNIT7. DECIMAL NUMBERS.




How do you read decimal numbers?
The place value of a number.
Writing in order.
Operating with decimals.
UNIT8. FRACTIONS, DECIMALS AND PERCENTAGES.
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





What is a fraction?
How do you read fractions?
Different types of fractions.
Fractions and decimals.
A fraction like an operator.
Equivalent fractions.
Comparing fractions.
2
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme







Fractions, decimals and percentages.
Ratio and proportion.
Adding and subtracting fractions.
Multiplying fractions.
Dividing fractions.
Combined operations with fractions.
Fractions with the calculator.
UNIT9. SPECIAL NUMBER SEQUENCES.
 Even and odd numbers.
 Square and cube numbers.
 Triangle numbers.
UNIT10. THE REAL NUMBER.


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

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Rational numbers.
Irrational numbers.
Square root.
Cube root.
The n-th root.
The rules of
Operating with roots.
3
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 1.CARDINAL AND ORDINAL NUMBERS.
Los números desde el punto de vista gramatical son adjetivos numerales y pueden aparecer de
dos formas, como números cardinales y como números ordinales.
A) CARDINAL NUMBERS (números para contar / counting numbers /numbers used for counting /
to count)
NUMBERS
CARDINALS
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
30
40
50
60
70
80
90
100
101
1000
1001
1000000
Nought, zero
one
two
three
four
five
six
seven
eight
nine
ten
eleven
twelve
thirteen
fourteen
fifteen
sixteen
seventeen
eighteen
nineteen
twenty
twenty-one
thirty
forty
fifty
sixty
seventy
eighty
ninety
one hundred
one hundred and one
one thousand
one thousand and one
one million
4
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
OBSERVACIONES sobre números cardinales:

Las unidades se separan de las decenas por un guión.
25
67
31

En un número de tres cifras one hundred, two hundred, van siempre seguidos de
and.
123
237
986

three thousand and eighty-one
Los números de cuatro cifras, especialmente las fechas pueden leerse por
parejas
1975
3786
2007

one thousand three hundred and fifty
Si el lugar de las centenas lo ocupa un cero, los millares van seguidos de and.
3081
4098
1072

one hundred and twenty-three
Los millares no se unen a las centenas ni con guión, ni con and, ni con coma.
1350
4567
5281

twenty-five
nineteen seventy-five
Los números de cuatro cifras terminados en dos ceros, pueden leerse con las dos
primeras cifras seguidas de hundred
2500
7800
twenty-five hundred
5
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme

En lenguaje coloquial, hundred, thousand and million suelen ir precedidos de a
en lugar de one
a hundred
a thousand
a million

Hundred, thousand and million, cuando van precedidos de otro número o
determinante numeral, o sea, cuando funcionan como adjetivos, no llevan s final
en el plural
En cambio si funcionan como sustantivos si termina en s el plural.
Doscientos dolares
Miles de pajaros
Varios cientos de libras
Tres millones de euros

Two hundred dollars
Thousands of birds
En Inglés, se utiliza una coma ó un espacio para marcar los millares (nunca un
punto como nosotros)
25,670
25 670

El cero: 0
nought
zero
o
nothing
nil
love
mathematical digits
point zero, zero degrees
telephone numbers
coloquial
In football
In tennis
6
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
B) ORDINAL NUMBERS (números para ordenar/ arranging numbers / numbers used for
arranging / to arrange)
NUMBERS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
30
40
50
60
70
80
90
100
CARDINAL NUMBERS
ORDINAL NUMBERS
First
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
Tenth
Eleventh
Twelfth
Thirteenth
Fourteenth
Fifteenth
Sixteenth
Seventeenth
Eighteenth
Nineteenth
Twentieth
Twenty-first
Thirtieth
Fourtieth
Fiftieth
Sixtieth
Seventieth
Eightieth
Ninetieth
One hundredth
101
One hundred and first
1000
One thousandth
1001
One thousand and first
1000000
One millionth
7
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
OBSERVACIONES sobre números ordinales:

1576
Los números ordinales se forman a partir del correspondiente cardinal, con sólo
añadir la terminación th a las unidades.
One thousand five hundred
and seventy-six
One thousand five hundred
and seventy-sixth
29
341
1002308

Excepto los tres primeros ordinales, que son totalmente irregulares.
1
2
3

5º
8º
9º
12º
20º,30º,40º,…

One
Two
Three
first
No obstante observa las siguientes modificaciones gráficas.
Cambio de ve por f
Se añade sólo h por terminar en t
Pérdida de la e final
Cambio de ve por f + th
Cambio de y por ie + th
fifth
Los números ordinales se abrevian añadiendo al número cardinal
correspondiente las dos últimas letras de dicho ordinal.
1º
2º
3º
4º
18º
1st
2nd
3rd
4th
18th
8
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme

Situaciones en las que se utilizan números ordinales
FECHAS
TITULOS
PREMIOS
SIGLOS
CLASIFICACIONES

1st April, 1st of April, April the 1st, April 1st
Isabel II Elisabeth II Elisabeth the second
The first prize, the 2nd prize, ……
21st century, 18th century, ….
The 4th position
El primero y el último
EL PRIMERO
THE FIRST
EL ÚLTIMO
THE LAST
EL SEGUNDO
THE SECOND
EL PENÚLTIMO
THE SECOND LAST
EL TERCERO
THE THIRD
EL ANTEPENÚLTIMO
THE THIRD LAST
9
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 2.
THE FOUR BASIC OPERATIONS ON
NUMBER
 Do you know the four basic operations on numbers?
OPERATION
Addition
Subtraction
Multiplication
Division
MEANING
SYMBOL
To add,
To collect,
To combine,
To put toguether
+
To subtract,
To take away.
It’s the inverse of addition
To multiply,
To add some copies of the
same number.
It’s a repeated addition.
To divide,
To share in equal portions.
It’s the inverse of
multiplication.
-
×
÷
EXAMPLE
Plus
Minus
Multiplied by
or
Times
Divided by
 How do you read?
ARITHMETIC
SYMBOLS
Equals,
=
Is equal to,
Is
Plus
+
-


Minus
IN FIGURES
IN WORDS
6  9  15
…………………………………….....
17  35  52
.............................................................
10  5  5
…………………………………….....
23  8  15
.............................................................
Times
or
Multiplied by
5  6  30
…………………………………….....
12  5  60
.............................................................
Divided by
40  5  8
……………………………………….
72  9  8
.............................................................
10
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme

You need to be able to add and subtract numbers (one digit
numbers, two digit numbers, three digit numbers, ...):
In a addition, don’t forget:
To line up the unit digits.
To write first the number with more digits.
To start from the right.
EXERCISE:
456  379  15 
1. Calculate the result using the
columnar addition.
2. Circle in green colour the
addends.
3. Circle in red colour the
sum or total.
In a subtraction, don’t forget:
To line up the units.
To write first the greatest number.
To start from the right.
1. Work out the solution, using the
columnar
subtraction:
1. Tick in blue colour the
minuend
.
2. Tick in red colour the
subtrahend.
3. Tick in green colour the
difference.
11
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme

You need to know the multiplication tables up 10  1 0 , they are
essential for both multiplication and division:

You need to do perfectly short multiplication and short
division. (The multiplier and the divisor are one digit number).
EXERCISE:

There are some different methods for long
multiplication and long
division.
EXAMPLES:
1.
Small tins of fried tomato weigh 157 grams. How much do 35 tins weigh?
The question is 157  35  .
We can solve it using different methods:
12
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
1. STANDARD COLUMN METHOD.
STEP 1
Write down the multiplication
lining up the units. (Write first the
number with more digits).
STEP 2
Split the second number in their
units and tens.
STEP 3
Multiply 157 by the units (5).
STEP 4
Multiply 157 by the tens (30).
STEP 5
Finally add both answers to get
the total.
The answer is 5,495
2. BOX METHOD.
STEP 1
Draw a box 3 squares by 2 squares (it is a
multiplication of 3 digits by 2 digits).
Split the numbers into their hundreds,
tens and units along the top and down the
left-hand side.
STEP 2
Multiply the numbers at the top and the
numbers at the side and put each answer
in each box.
STEP 3
Take the number from the boxes and
write them down, lining up the units.
Finally add them together.
The answer is 5,495
13
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
3.
CHINESE MULTIPLICATION. (Observa el ejemplo e intenta
describir en español los tres pasos)
STEP 1
STEP 2
STEP 3
The answer is 5,495
4. RUSSIAN METHOD.
Look at the video and take notes.
14
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
5. EGYPCIAN METHOD.
Look at the video and take notes.
6. SIMPLE GRAPHICAL METHOD.
Pay attention carefully to the video and take notes.
15
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
2. We are 806 students in my school distributed in 31 class-rooms. How
many pupils are in each group?
The question is 806  31 .
Do this by the METHOD
SIMILAR TO SHORT DIVISION.
STEP 1
31 into 8 does not go.
So carry the next digit (the
0).
STEP 2
31 into 80 goes 2 times but
with remainder 18 (Resto
18).
STEP 3
Finally carry the 6
and divide again.
31 into 186 goes 6 times
exactly (remainder 0).
The answer is 26
16
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 3.
BIG NUMBERS
 You need to understand the place value of numbers.
Each digit in a number has a value which depends on its place in the number. (Decimal
System).
The 4 represents 4 hundreds  400 units.
The 5 represents 5 tens  50 units
The 3 represents 3 units  3 units
So 453 = 400 + 50 + 3
In a number such as 453
(four hundred and fifty-three)
TRY WITH THE NUMBER 859:
 Learn the name of numbers with more than nine digits:
-----
- -
billions
billones
----Thousand
millions
Miles de
millones/
millardos
- -
-----
- -
-----
-----
-----
-----
millions
thousands
hundreds
tens
units
millones
millares
centenas
decenas
unidades
 ROUNDED BIG NUMBERS.
In real life is not always necessary to use exact numbers.
Rounded numbers may be used.
Numbers can be rounded to different degrees of accuracy, for example, to the nearest
thousand, hundred, ten,..
REMEMBER
1. Look at the digit you are rounding.
.
2. Now look at the digit on its right.
3. If it is 4 or less the digit being rounded stays the
same.
4. If it is 5 or more the digit you are rounding goes up
by 1.
17
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
 EXAMPLE:
 EXERCISES:
1. Round to the nearest ten the following numbers:
23,456
245,205
1,672
2. Round to the nearest hundred the following numbers:
24,012
122,655
1,345,008
3. Round to the nearest thousand the following numbers:
1,506
34,907
12,345,001
4. Round to nearest million the following number:
13,405,543
18
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 4.
POWERS
REMEMBER:
In a power, the index tells us how many times the base has to be multiplied by itself.
EXAMPLE:
10 2  10  10
(10 squared has two of the same factors)
73  7  7  7
( 7 cubed has three of the same factors)
EXERCISES:
1. Calculate the following: 5 3 =
34 
Write down what the base and index is in each case.
2. Calculate the following powers:
a) 0 2 ,0 7 ,010
b) 18 ,13 ,12
c) 31 ,51 ,91
d) 2 0 ,50 ,12 0
e) 101 ,10 2 ,10 3 ,10 4 ,105 ,10 6
f) 210 ,35 ,5 4 ,7 3
3. Calculate the squares of the first 15 natural numbers:
Number
Square
1
2
3
4
5
6
7
8
9
10
11
12
13
14
4. Calculate the cubes of the first 10 natural numbers:
Number
Cube
1
2
3
4
5
6
7
5. Check:
7 3  35  7  7  7  3  3  3  3  3 
 343  243  100
6. Calculate:
6 2  14  43 
19
8
9
10
15
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
7. Read and write in words:
“5 squared” or
5
2
7
3
“5 to the power of 2”
“7 cubed” or
“7 to the power of 3”
34
25

If a number is written as a product of factors, then we say it’s written in
expanded form, and if it is written in the short form, then we say it’s written in
index form.
EXAMPLES:

Write 11111111 in index form.
Solution: 11111111  114

Write 3 6 in expanded form.
Solution: 36  3  3  3  3  3  3
EXERCISES:
1. Write in expanded form:
75 
10 4 
2. Write in index form:
3 3 3 
5 55 5 
10  10  10  10  10 
20
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
GENERAL RULES:
1. Any number to the power of zero is equal to one.
2. Any number to the power of one is equal to the same number.
3. The product of two powers with the same base number is another
power with the same base number whose index is the sum of the
other two indices.
a n  a m  a nm
EXAMPLES:
43  45  (4  4  4)  (4  4  4  4  4)  48  435
23  25  2  (2  2  2)  (2  2  2  2  2)  2  29  2351
EXERCISES:
1. Calculate the final answer, writing first the answers in index form (use this general rule):
23  27 
3 2  33 
53  5 
2. Write the answer in index form:
7 4  78  7 
13 2  133 
4 6  4 20 
4. The division of two powers with the same base number is another
power with the same base number and its index is the difference
between the other two indices.
a n  a m  a nm
EXAMPLES:
85  83  (8  8  8  8  8)  (8  8  8)  82  853
7 6  7 2  (7  7  7  7  7  7)  (7  7)  7 4  7 62
21
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
EXERCISE:
3. Calculate the following and write the answers in index form (use this general rule):
27  22 
3 7  33 
10 8  10 5 
5. A product raised to a power is equal to multiply these numbers
raised to the same power.
(a  b) n  a n  b n
EXAMPLE:
(2  3) 3  (2  3)  (2  3)  (2  3)  (2  2  2)  (3  3  3)  23  33
So, to calculate the result, we can multiply and cube the product, or, we can cube each of the
factors and multiply the results.
(2  3) 3  63  216
or
23  33  8  27  216
EXERCISE:
3. Calculate the following in the two ways:
(2  5) 6 
(3  4) 2 
6. A division raised to a power is equal to the quotient of both bases
raised to the same power.
( a / b) n  a n / b n
( a  b) n  a n  b n
EXAMPLE:
(10 / 5) 3  To calculate the result we have to divide and cube the division: (10 / 5) 3  23  8
Or
We can cube each of the factors and divide the results:
EXERCISE:
5. Calculate the following in the two ways:
(12 / 3) 3 
(8 / 4) 4 
22
103 / 53  1000 / 125  8
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
7. A power raised to another power is another power with the same
base number raised to the product of these two powers.
(a n ) m  a nm
EXAMPLE:
(45 ) 3  45  45  45  4555  415
EXERCISE:
6. Simplify the following, using this general rule:
(2 3 ) 6 
(5 4 ) 2 
NOTES:
The same base.
The same index.
No property.
No property.
23
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 5. MULTIPLES AND FACTORS.
PRIME NUMBERS. LOWEST COMMON
MULTIPLE AND HIGHEST COMMON
FACTOR.
 You must know what a multiple and a factor are.

A multiple is a number in the times table.
You can find them multiplying the number by 1, 2, 3, 4, 5,......
EXAMPLE: The multiples of 4 are:
4,8,12,16,20,24,…….
EXERCISE:
The
multiples
of 7 are:

A factor is a number that divides into another number exactly. I mean, when
the division is an exact division.
Remember, in a division: D
r
d
q
These terms are called:
D= dividend
d=divisor
q=quotient
r=remainder
Remember the rule in a division:
D  d q  r
When the remainder is 0 (r=0), the division is called exact division.
And now
EXAMPLE:
The factors of 20 are: 1,2,4,5,10,20.
(All these numbers divide into 20 exactly).
EXERCISE:
The
factors of
80 are:
24
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
 You must know what a prime number is.

A prime number is a number with only two factors, itself
and 1.

A composite number is a number with more than two factors.

To find prime numbers, you can use the following tests of divisibility:
A number is divisible by:




EXERCISE:

2 if the last digit is 0,2,4,6 or 8.
3 if the sum of the digits is a number on the 3 times table.
5 if the last digit is 0 or 5.
11 if the sum of every digit which is in odd place minus the sum
of the digits in even place is equal to 0 or 11.
Find the prime numbers up to 50 are:
FACTORISATION: To factorise a composite number is to write it as a
product of prime factors.
EXAMPLE:
20
10
5
Divide the number by
the first prime numbers
2,3,5,7,11,13,………,
and look for factors.
1
EXERCISE:
Write as a product
of prime factors
the number 50



50
25
2
2
5
 20  2  2  5 1
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
 You must know what the lowest common multiple and the
highest common factor are.
 Abbreviations:
ENGLISH
SPANISH
lowest common multiple LCM
mínimo común múltiplo
mcm
highest common factor
máximo común divisor
mcd
EXAMPLE:
STEP 1
STEP 2
STEP 3
HCF
Find the HCF of 640 and 192:
Write the
numbers as
the products
of their
prime
factors.
Ring the
factors in
common
84
42
21
7
1




2
2
3
7
360
180
90
45
15
5
1






2
2
2
3
3
5
360  2  2  2  3  3  5 1
84  2  2  3  7  1
360  2  2  2  3  3  5 1
84  2  2  3  7  1
2  2  3 1  12
Multiply all
these factors
in common
The HCF is 12
26
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
EXERCISE:
Find the HCF of 250 and 115:
STEP 1
STEP 2
STEP 3
The HCF is 5
EXAMPLE:
Find the LCM of 640 and 192:
STEP 1
Divide and look for the
prime factors
STEP 2
Write both numbers as a
product of their prime
factors.
STEP 3
Write in index form and
ring all the factors with
the highest index.
640
320
160
80
40
20
10
5
1








2
2
2
2
2
2
2
5
192
96
48
24
12
6
3
1
640  2  2  2  2  2  2  2  5 1
192  2  2  2  2  2  2  3 1
640  2 7  5  1
192  2 6  3  1
2 7  3  5  1  1920
STEP 4
Multiply all these factors.
The answer is 1920
27







2
2
2
2
2
2
3
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
EXERCISE:
Find the LCM of 49 and 77:
STEP 1
STEP 2
STEP 3
STEP 4
The answer is 539
28
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
UNIT 6. INTEGERS.
A) WHAT IS AN INTEGER?
 DEFINITION:
Z  ......,4,3,2,1,0,1,2,3,4,5,.........

 Integers are whole numbers.
 They can be positive or negative.
 Positive integers are above zero and negative integers
are bellow zero.
 The integer zero is neutral. It is neither positive nor
negative.
 The sign of an integer is: Positive integers (+)
Negative integers (-)
Zero
(no sign)
 Two integers are opposites if the distance to 0 is equal,
always in opposite directions. One of them will have
positive sign and the other negative sign. For instance:  3
and  3 are opposites.
 Look at the number line:
 Absolute value: It’s the distance to the 0.
You have to take the number without the
sign.
EXAMPLES:
29
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
B) WHY DO INTEGERS APPEAR?
C) ORDERING INTEGERS
EXERCISE:
Write in order from least to greatest:
1. Place the numbers on the
number line.
2. The number which is on the left
is the least and the number on
the right is the greatest.
EXERCISES:
1. Order the following numbers from the least to the greatest:
2. Put in order from the greatest to the least:
3. Write the following integers in order starting from the least:
4. Compare the following whole numbers, circle the greatest and
underline the lowest:
30
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
D) ADDING AND SUBTRACTING INTEGERS
EXAMPLE:
 When the number to be added or subtracted is negative, the
normal direction of movement is reversed.
 When two signs are together, these rules are used:
 ()  
 ( )  
 ()  
 ( )  
same signs give a positive.
different signs
EXAMPLES:
 2  (3)  2  3  5
 3  (5)  3  5  8
5  (2)  5  2  3
6  (4)  6  4  10
31
give a negative.
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
EXERCISES:
1. Adding integers with same sign (you have to add).
a) 5  8  13
b)  2  3  5
c)
d)
e)
f)
g)
h)
i)
j)
k)
6452 
3 1 0  5 
64 
52 4 
 4  3  2 1 
17 5  4 
34 573 
2 574 
 1  3  7  4  50 
2. Adding two integers with different sign (you have to subtract).
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
8 5 
50  73 
7  20 
50  52 
 48  50 
 20  27 
35 
 2 1 
 30  43 
 502  505 
107 110 
 30  27 
32
M. Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
3. Adding several positives and negatives (you have to collect them).
a. 3 4 5 3 2 
3  5  3 4  2  11  6  5
b.
c.
d.
e.
f.
g.
h.
i.
j.
3  4  6  2 1 5  7 
 5  4  3  2 1 6 
4  3 6  5 1 2 7 8 
5  3 1 4 8  7 
 8  0  2  50  15  12 
7  3  5  8  7  4  20  7 
 20  3  1  7  2  50  3  7 
5  20  3  1  2  7  5  6 
 50  20  3  20  1  3  0 
4. Work out the answer (eliminate brackets)
  3  3
  3  3
  3  3
  3  3
e)
j)
a)   5 
  23 
  7 
f)
 5 
k)
 3 
b)  3 
g)
l)
  5 
  2 
h)
 3 
+ 2  
d)   2  
i)
  4 
c)
5. Work out the answer (remove brackets)
 5  3  2  2
 3  5   2  2
u)  32  50 
m)  4  7 
v)  32  50 
n)  4  7 
o)
p)
q)
r)
s)
t)
w)   5  2 
4  7 
4  7 
 2  5 
  2  5 
 2  5 
  50  64 
x)   4  7 
y)   4  7 
z)
 4  3 
aa)  7  10 
bb)  3  8 
33
6. Adding and subtracting more than two integers.
 3   5   4   3  5  4  5  7  2
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
8   4   5 
  7   5  4  
 5  7   5 
 8  5   4  
  6  8   7 
 3   5   3 
 8   8   8 
 5   3   3 
 3   5   22 
  4    7  2 
 7  6   3 
7 Work out the answer:
 10   4  12  30  10   34  12  10   22  10  22  12
a) 12  8   5  9 
b)  2  4  13   21 
c)  10   4  12  30 
d) 15   11  13  43 
e)  0   7   4   1 
f)
 2   2   5  1 
g)  24  58  79  48  31 
h) 3  11  15  7  2 
i)
 2  3  4    5  3 
j)
  7   5  2  7  3 
k)   7   5  2  7  3 
l)
m)
 3  15  11   2  7 
4   3  5   2  8 
34
E) MULTIPLYING AND DIVIDING INTEGERS

Multiply and divide the numbers as normal and then find the sign for
the answer using these rules:
()  ()  
(  )  ( )  
( )  (  )  
( )  (  )  
two of the same signs give positive.

two different signs give negative.
The same when dividing
EXAMPLES:
3  (5)  15
 20  (2)  10
 15  (3)  5
40  (8)  5
EXERCISES:
1. Complete the next tables:
– 2 +1 – 9
+3 – 6
–5
+7
–1
0
–5
+4
-12
+15
-18
+21
+8
+24
2. Fill in the gaps to make true equalities:
a)
b)
c)
d)
 5   35
  9  0
 6   18
 4   4
e)
f)
  7  7
 10 
0
40  (8) 
h) (21)  (3) 
g)
35
F) ADDING, SUBTRACTING, MULTIPLYING AND
DIVIDING INTEGERS. COMBINED
OPERATIONS.
1) Brackets
2) Indices

The right order in your operations
3) M ultiplications and Divisions
4) Additions and Subtractions
To remember the right order memorize the word
BIDMAS
EXERCISES:
1. Work out.
5  3  2  5  6  11
a)  3  2  4 
b) 2  3  7 
c) 2   3  7 
d)  7  5  5 
e)  7  5   5 
f)  3  4  7 
g) 5   3  20 
h) 5   3  4 
i)
5   3  4 
5  2  3 2 
5  3 4  7 
8  23 4 
63  8  3 
5  3  64 
1 25  4 
p) 7  3  4   2 
j)
k)
l)
m)
n)
o)
q) 2  3   3  6 
r)
1 73  5 
36
2. Work out.
a)
b)
c)
d)
52  7 
10 : 2  3 
2  50 : 5  1 
  7  8 : 4 
e)  3  2  4  2 
f)  5   3 
 5  3 
h)  5  3  7 
g)
i)
j)
2  20 : 10  3 
3   7  20 
k)  5  3   16 
l)
  3  5  4  2 
3. Work out:
2  3  4   2  7  14
a. 2  5  3 
b. 2  5  20 
20  25  4 
d. 5  7  2 
e. 25  30  5 
c.
f. 3  7  7  4 
g. 3  50  55 
h. 2  3 5  2 
i.
2  3 5  7 
j.
1  4 3  2 
37
4. Work out.




10  8 : 2 
23 4 
7  24 
 35 1 


 22  3  5 
 3 23 
2   32 

  42   7 

  6  7 



6  4 8 
7  3   8 



 25  5 
 4  4 8 
3   2  5 

52  8 

4  24 
5. Work out.
i. 2  3  4 
ii. 2  3  4  
iii. 3  5  7  4 
iv. 3  5  7  4 
v. 3  5  7  4  
vi. 3  5  7  4 
vii.  2  2  5  7  4 
 2  2  5  7  4 
viii.
ix.  2  2  5  7  4 
 2  2  5  7  4 
xi.  2  2  5  7  4 
xii.  2  2  5  7  4  
x.
xiii.
 3  3  3  3 
xiv.
5  3  2  4  2  5 
2 3

38
6. Work out (more than one bracket)
a)
 3  5  2   8 
b) 3  4  2  6  5  7 
c) 3  4  2  6  5  7 
d) 4  2  3  4  3  5  7 
e) 8  5  2  3  4  5  6  4 
f)
8  5  2  3  5  5  6  7 
g) 16  (9  5  3  12)
h) ( 21  11)  13  (4  15  9)
i)
 5  3  (4  6)  7  ( 8  3)
j)
(17  5)  3  (5  3  2)
k)
l)
5  (8  2  3)  (4)  6  (2  7)
(7)  4  (3  8)  5  (8  5)
m) 4  (5  7  3)  ( 3)  (8  3  4)
n) 10  5  (8  3)  5  4  3 2
39
G) TEST ABOUT INTEGERS.
1.
Write an integer to represent each situation:
a)
b)
c)
d)
10 degrees above zero.
A loss of 25 €.
In the second basement.
13.2 m of height.
2.
Write the four opposite situations.
3.
What is the opposite of each integer?
-13
+7
-38
+49
0
4.
Which is the absolute value of each number in the exercise number 3.
5.
Name in English six real life situations in which integers can be used.
Spending and earning money
Floors in a hotel
Temperature
6.
How do you read?
(4)  (5)  1
Positive four minus positive five equals negative one
(5)  (8)  13
Negative five plus negative eight is equal to negative thirteen
(3)  (7)  21
Negative three times negative seven equals positive twenty-one
(8)  (2)  4
Positive eight divided by negative two is negative four
(7)  (1) 
(7)  (2) 
(5)  (1000) 
(400)  (10) 
40
UNIT 7.
DECIMAL NUMBERS.

A decimal number has a whole number and a fraction.

The decimal point in a number separates the whole number from the fraction.

The decimal point always comes after the unit digit.
 Remember the place value of a number:
ESPAÑOL
----
----
----
----
,
----
----
----
Se llama
Millar
Centena
Decena
Unidad
Coma
decimal
Décimas
Centésimas
Milésimas
Abreviatura
Valor
M
1000
C
100
D
10
U
1
d
0,1
c
0,01
m
0,001
----
----
----
----
----
----
Tenth
Hundredth
Thousandth
t
0.1
h
0.01
th
0.001
ENGLISH
----
,
It’s called
Thousand
Hundred
Ten
Unit
Abbrevation
Place value
Th
1000
H
100
T
10
U
1
EXAMPLE:
452.945
Decimal
point
Hundreds
Tens
Units
Tenths
Centenas
Decenas
Unidades
Décimas
Centésimas
Milésimas
400=
4 x 100
50=
5 x 10
2=
2x1
0.9=
9 x 0.1
0.04=
4 x 0.01
0.005=
5 x 0.001
41
Hundredths Thousandths
 EXERCISES:
1.Look at the worked examples and do the same with the other two
numbers:
One unit and
five hundredths
1.05
2.25
1.05  1  0.05
0.5
3.2
Three units and
two tenths
3.2  3  0.2
2.How do you read a decimal number?
452.945
Four hundred and fifty-two point nine four five.
Zero point eight.
0.8
3.68
507.913
 THE NUMBER LINE
We can place decimal numbers on the number line just dividing the units in fractions.
1. First, you have to divide the unit in 10 equal parts, so, you have
tenths.
2. Second, if it’s necessary you divide each tenth in another 10 equal
parts. Now, you have hundredths.
3. And so on.
42
Observe with attention the figures bellow:
|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Every decimal number can be represented on a number line. Look at the units, then divide the unit
in tenths or hundredths.
EXAMPLE:
2.35
|______|______|___|___|______|______|______|______|______|______|__
2
2.1
2.2 2.35 2.3
2.9
 COMPARING DECIMAL NUMBERS
To compare two or more decimal numbers first compare the units, second the first decimal place
value, then the second and so on.
EXAMPLE:
Place the next numbers in order, starting with the largest:
3.25
0.6
0.56
0.616
0.66
0.62
There is only one with 3 units, the rest of them have no units.
So the first is 3.25
Now arrange by comparing the first decimal place. Four of them have the same
tenths 0.6 0.616
0.66
0.62
The last one has only 5 tenths
0.56
Now compare the second decimal place (The hundredths):
0.60 0.61
0.66
0.62
So 3.25 > 0.66 > 0.62 > 0.616 > 0.6 > 0.56
(is greater than)
EXERCISE:
43
Place in order the next numbers, starting with the smallest:
1.7
1.77
1.78
1.786
1.5
(Use the sign “< , less than”)
7
0.35
 ADDING DECIMALS
You must know to add decimals.

When adding decimals it is important to line up the decimal point.

Start from the right.
The terms in the addition are called addends and sum or total.
Work out the sum
18€ + 4.65€ + 3.99€ =.
44
 SUBTRACTING DECIMALS
You must know to subtract decimals.

When subtracting decimals it is important to line up the decimal point.

Don’t forget the minuend has to be greater than the subtrahend.

And, start from the right.
The terms in the subtraction are called minuend, subtrahend and difference.
Work out the difference:
337.495m – 91.65m =
 MULTIPLYING DECIMALS.
You must know to multiply decimals.

When multiplying decimals it is important to place the decimal number with more digits
above.

Multiply the numbers using the same method as you use with natural numbers, I mean, as
normal.

When you have the product, count how many decimal places the factors have and move the
decimal point to the left by this number of places.
The terms in the multiplication are called multiplying, multiplier end product.
Work out the product:
34.197
----------------
 DIVIDING DECIMALS
45
You must know to divide decimals.

When dividing decimals, it is important to know that the divisor has to be an integer.

So, if the divisor has no decimal point divide as normal.

But, if the divisor has decimal point, you must first to remove it by multiplying by a power
of 10. Now, multiply as normal again.
23.7  5
15.2  2.4
250  6.5
985.42  7.3
46
 MULTIPLYING AND DIVIDING DECIMALS AND
WHOLE NUMBERS BY 10, 100 AND 1000.
1. This is what happens when you multiply decimals and whole numbers by a power of
10 (add 0s or move the decimal point to the right so many times as the exponent
indicates).
12.5  10  125
17.68  10  176.8
43  10  430
26.97  100  2697
2.457  100  245.7
3.9  100  390
84  1000  84000
8.456  1000  8456
76.31  1000  76310
2. This what happens when you divide decimals and whole numbers by a power of 10
(remove 0s or move the decimal point to the left so many times as the index indicates).
4500  10  450
369  10  36.9
18.5  10  1.85
7900  100  79
1234  100  12.34
68.7  100  0.687
98760  1000  98.76
6543  1000  6.543
12  1000  0.012
3. Now you try:
123  10 
45.7  10 
23.89  10 
4.478  10 
479  100 
3.98  100 
2345  100 
4.98  100 
3.4  1000 
34.5  1000 
47
UNIT 8. FRACTIONS, DECIMALS AND
PERCENTAGES.
A) WHAT IS A FRACTION?
 Do you remember decimals? Well, fractions are another way to show numbers that
are between the whole numbers.
 When you divide the whole in equal parts, the fraction appears.
 A fraction is an indicated quotient between two whole numbers “ ” and “ ”.

a
has two terms. The top number "a" is called numerator and the
b
bottom number "b" is called denominator.
A fraction
B) WHEN FRACTIONS ARE USED?
EXAMPLE:
Gemma has done an exam with 8 questions. She has got 3 questions wrong. What
fraction of the exam did she fail?
What fraction of the exam did she pass?
ANSWER:

3
8
5
She past 5 of the 8 questions. So, it`s
8
She felt 3 of the 8 questions. So, it´s
EXERCISE: Explain the meaning of the following fractions with fraction bars
(diagrams):
1
means 1 part out of 6.
6
7
means 7 parts out of 4.
4
means 2 parts out of 5
48
C) READING FRACTIONS.
 You always read the numerator as a cardinal number and the
denominator as an ordinal number.
EXAMPLES
Two fifths
2
5
8
6
1
10

EXERCISES
4
7
6
9
3
8
Eight sixths
One tenth
There are three special cases:
1
2
1
3
1
4
One half
2
3
5
2
3
4
One third
One quarter
 It is also possible to read a fraction as:
Cardinal number
over
Cardinal number
EXAMPLES
2
11
9
25
EXERCISES
Two over eleven
31
76
Nine over twenty-five
7
18
49
D) DIFFERENT TYPES OF FRACTIONS:

A fraction like
2 1 3
, , ,.... is called a proper fraction.
5 6 4

A fraction like
6 7 4
, , ,.... is called an improper fraction.
5 2 3

A fraction like
6 8 12
, , ,..... is called a fraction equal to 1.
6 8 12

A proper fraction with its numerator equals one, for example:
1 1 1 1
, , , ,.... is
2 3 4 5
called a unitary fraction.

An improper fraction written in this way:
6 5 1
1
is called a mixed fraction.
   1
5 5 5
5

Positive fractions are:

Negative fractions are:

Any fraction like:

Any expression like:
, ......
, ......
..... is equal to 0.
it’s not a fraction.
EXERCISES:
1.
Explain the difference between proper and improper fractions:
……………………………………………………………………………………………………………………
………………..........................................................................................................................................................
................................................................................................................................................................................. .
.........................................................................................................................................................................
2.
Explain what a unitary fraction is:
..................................................................................................................................................................................
..................................................................................................................................................................................
.............................................................................................................
3.
Write down 2 fractions equals 1 and 2 fractions equals 0:
4.
Write like a mixed fraction the following improper fractions:
50
5

3
7

5
11

6
E) FRACTIONS AND DECIMALS.


It’s very easy to turn a fraction into a decimal. Just divide the top by the bottom.
You can obtain a terminating decimal or a recurring decimal.
Convert the following fractions into decimals.
EXAMPLES
=........
So,
1
 0.125
8
(It’s a terminating decimal)
........
So,
(It’s a recurring decimal)
Convert the following fractions into decimals.
EXERCISES

Now, turn terminating decimals into fractions. You have to look at where the last
digit after the decimal point is.
51
EXAMPLES:
0.3
The last digit is in the tenths
column.
It’s 3 tenths.
0.12
The last digit is in the
hundredths column.
It’s 12 hundredths.
2.547
The last digit is in the
thousandths column.
It’s 2 547
thousandths.
EXERCISES:
1. Convert the following terminating decimals into fractions:
1.7
0.85
3.547
2. Investigate with recurring decimals.
52
So, it’s the same as
3
10
So, it’s the same as
12
100
So, it’s the same as
2547
1000
F)
A FRACTION LIKE AN OPERATOR.
 When they’re talking about fractions, people say “of” when they mean
“times or multiplied by”.
EXAMPLE:
What is
1
of 40?
4
ANSWER:
1
of 40 is just
4
1
1  40
 40 
 10
4
4
EXERCISES:
Find one-third of each amount:
3, 150, 111 and 96
Find three-quarters of each amount:
120, 300, 8000 and 448
Find three tenths of 60, 210 and 75
53
G)
EQUIVALENT FRACTIONS.

Equivalent fractions are fractions which have the same value.

Fractions can be changed into their equivalent, multiplying or dividing the
numerator and denominator by the same number.

When you multiply the numerator and denominator by the same number, you are
amplifying fractions.

When you divide the numerator and denominator by the same number, you are
simplifying fractions.
EXAMPLES AND EXERCISES:
1 2

2 4
150
15

2250 225
1
2
and
are
2
4
equivalent
fractions.
You have
multiplied the
numerator and
denominator
by 2.
1. Find 3 different equivalent
fractions amplifying the first
one:
150
15
and
2250
225
are equivalent
fractions.
You have
divided the
numerator and
denominator
by 10.
2. Find 2 different equivalent
fractions simplifying the
first one:
EXERCISES:
1.Amplify the following fractions: (Look for 3)
(Obtención de fracciones equivalent es por ampliación)
2
7
1
5
7
3
5
6
54
1

2
150

2250
2.Simplify the next fractions (Look for 2)
(Obtención de fracciones equivalentes por simplificación.)
12
18
25
125
33
121
100
120
3.Write as simple as possible  Look for the irreducible fraction  Find the fraction in lowest terms.
(Obtención de fracciones irreducibles)
44
154
36
27
13
26
200
2550
H) COMPARING FRACTIONS
There are three different ways to order fractions:

METHOD 1: Convert fractions into decimals and put them in order.

METHOD 2: Use fraction bars and see which has the most shading.

METHOD 3: Find equivalent fractions with the same denominator and all you
have to do is compare the numerators.
55
EXERCISES:
1.Put
1 2 3
, , in order starting with the biggest. (Use the method1)
3 5 7
2.Put
5 1 2
, , in order starting with the smallest. (Use the method2)
6 2 3
EXAMPLE:
1.Place in order
7 4
3
, and , by converting fractions to a common denominator. (Write
8 5
2
the smallest first).
(Para ordenar fracciones has de buscar sus equivalentes con
denominador común. Recuerda el denominador común es el m.c.m. de los
denominadores)
L.C.M (8,5,2)  40 . So the common denominator is 40.
7 35

(5)
8 40
4 32

(8)
5 40
3 60

(20)
2 40
Now, you can order them, using the equivalent fractions:
32 35 60


40 40 40

4 7 3
 
5 8 2
(less than)
EXERCISES: Place in order starting with the greatest(Use the method3)
56
1 1
,
5 6
7 1
,
4 2
2 3
,
5 4
1 2 5
3
, , and
2 7 14
28
I)
FRACTIONS, DECIMALS AND PERCENTAGES.

A percentage is a part of a whole, expressed in hundredths.
EXAMPLES:
3%
25%
50%
3
100
25
100
50
100
57
 So, fractions, decimals and percentages are all
different ways of expressing parts of a whole.
 Any fraction can be expressed as a decimal and as a
percentage.
EXAMPLES:
1
 0.5  50%
2
1
 0.25  25%
4
3

4
1

3
EXERCISES:
1.Find 10% and 25% of each amount:
$100
$300
$50
2.Match these numbers into six sets that show the same number:
0.5
1
1
25% 0.3333.. 5% 0.25
10% 0.1
10
4
50% 0.8
Fraction
Decimal number
Percentage
4
5
1
1
33% 0.05
3
20
1
80%
2
1
2
0.5
50%
58
J) RATIO AND PROPORTION
 WHAT IS A RATIO?
A ratio is a comparison between two or more quantities.
EXAMPLE: A bag of carrots weighs 300g and a bag of potatoes 1.5kg. Calculate the
ratio of weight of carrots to weight of potatoes.
Both quantities must be in the same units. So:
1.5kg=1 500g
So, ratio is
300g : 1 500g
Or simplifying (  300 )
1 : 5
A ratio can be written as a fraction.
So, ratio is
1
5
You can say that the ratio is “ 1 to 5 “ ( 1 es a 5 )

Dividing a quantity in a given ratio.
EXAMPLE: 60€ is to be divided between Jon and pat in the ratio
2 : 3. How much
money does each one receive?
We need to divide 60€ in the ratio 2 : 3.
The digits in the ratio represent parts.
Jon gets 2 parts.
Pat gets 3 parts.
And the total is 2+3=5 parts.
And the total amount is 60€.
So:
5 parts  60€
1 part  12€ (dividing by 5).
So, the final answer is:
Jon  two parts  2  12  24€
Pat  three parts  3  12  36€
Check: 24€+36€=60€
EXERCISE: Three brothers aged 6, 9 and 15 decide to share a tin of toffees in the ratio of their
ages (and not in equal parts ). If the tin contains 240 toffees. How many toffees does each brother get?
59
K)
OPERATIONS WITH FRACTIONS
 Addition of fractions.
Two or more fractions can be added very easily looking for their equivalents
with common denominator.
 Subtraction of fractions.
As for addition, two or more fractions can be subtracted by looking for their
equivalents with common denominator. (Be careful with signs)
60
 Multiplication of fractions.
To multiply fractions all you have to do is multiply together their numerators
and their denominators.
(Factorise)
(Cancel)
(Multiply)
 Division of fractions.
To divide two fractions turn the second fraction upside down, change the
division sign to a multiplication sign and now is the same as multiplying
fractions.
(Turn upside down) (Factorise)
J)
(Cancel)
(Operate)
TEST1. INTRODUCCIÓN FRACCIONES
1º Escribe la fracción que representa la parte coloreada de cada una de las siguientes
figuras:
2º Decir qué fracción de una hora representan:
a)
b)
c)
d)
e)
15 minutos
30 minutos
45 minutos
10 minutos
20 minutos
3º ¿A cuántos minutos equivalen los
7
de una hora? (Sol: 84 minutos)
5
61
4º Calcula:
2
a)
de 60
3
3
b)
de 100
4
3
c)
de 500
500
d) La mitad de
3
5
12
7
f) La mitad de la quinta
parte
de 6
e) La tercera parte de
5º Escribe cinco números naturales, cuatro números enteros negativos, tres números
fraccionarios positivos y tres números fraccionarios negativos. ¿Son todos
racionales?
6º Clasificar los números que figuran a continuación y escribir dos números racionales
4
2  108
equivalentes a cada uno de ellos: 2, ,  6, 3  ,
8
5 72
7º Escribir cuatro fracciones propias, y otras tantas impropias, cuyo denominador sea
7.
8º Escribe las siguientes fracciones impropias como suma de un número entero y una
fracción propia:
19
179
a)
c)
25
5
67
 1147
b)
d)
15
76
62
9º Expresa mediante fracción irreducible, los puntos señalados en los siguientes
segmentos de la recta numérica:
10º Transforma en fracciones los siguientes números mixtos:
3
9

a) 2 
c)   8  
4
 10 
5
7
b) 7 
d) 13 
9
11
11º Representa gráficamente los siguientes números racionales:
2 4
13 15  1 16
;
; 4; ;  ; ; ;  3
3 5
2
4 2 3
_________________________________________________________
12º Escribe cuatro números racionales que sean equivalentes a 6 y tengan por
denominador los números: 2, 5, 8 y 15.
13º Escribe una fracción equivalente a
3
cuyo denominador sea 6.
9
14º Añade el término desconocido en las siguientes igualdades:
a)
3
Sol: 39

13 169
c)
2
Sol: 8

9 36
b)
16 32

Sol: 18
9
d)
7
Sol: 56

5 40
63
15º Busca la fracción irreducible equivalente a las siguientes fracciones:
144
20


96
21
75
35


105
60
222
243


333
432
540
7200


40500
450
16º Reduce a común denominador las siguientes fracciones:
a)
5 4 7
, ,
12 9 18
c)
5 8 1 4
, , ,
9 21 63 7
b)
8 3 4
, ,
25 50 75
d)
11 1 5 7
, , ,
45 2 18 30
17º Ordena de menor a mayor las siguientes fracciones:
a)
1 3 1 3 5
, , , ,
2 4 3 2 6
b) 2,
 1 19
7
, ,  1, 0,
9 9
9
18º Escribe dos números racionales, comprendidos entre:
2 3
y
a)
3 4
b)
1 1
y
5
7
c)
7 8
y
5 5
64
19º Jorge ha comido los dos quintos de una tortilla mientras que su hermana Susana ha comido los
tres séptimos. ¿Quién ha comido más? Sol: Susana
9
5
de la edad de su padre, en tanto que su hermano Luis es los
.
20
12
¿Quién es mayor? Sol: José
20º La edad de José es los
L)TEST2. OPERACIONES CON FRACCIONES
1º Efectúa las siguientes sumas de fracciones, simplificando los resultados:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
9 5

13 13
5 9

8 16
7 5 1
 
12 12 12
2 7 6
 
11 11 11
7 5 3
 
12 6 4
7   3 3  2 

    
8  4  10  5 
1  3   7  11
      
8  4   12  24
5   3 3   7 

 

8  20  4  5 
1 1  1
1 2
  1   2   
2 3  2
3 5
1 2
3  
6 3
2 
7

 2    5  
3 
2

1

5    2
3

65
2º Efectúa las siguientes multiplicaciones, simplificando los resultados:
a)
8 5

15 12
b)
4 2  5
  
3 5  6
c)
4  7
 
7  3
d)
6  14 
 5
 4

7  23 
 6 
e)
1  3
      4 
2  7
f)
 3   7   11  13
   
 4   9   13  7
3º Efectúa las siguientes divisiones, simplificando los resultados:
a)
2
:4
7
b) 3 :
9
4
c)
4  10 
:  
9  3
d)
11  6 
:  
4  7
 9

 16 
e)  6 :  
f)
 5  5
  :  
 4  8
66
Bilingual Programme.
M. Mar Agüera de Pablo-Blanco.
IES Caura. Coria del Río.
M)
2
 3
TEST3. POTENCIAS DE FRACCIONES.
3
g)  
 1

 4 
2
h) 
i)
2


 5 
0
2
 3
j)  
4
k) 2 4
2
 4
l)   
 5
m)  2
3
 5
 3
3
 1
o)   
 3
2
n)   
5
5
2
3
1 1
p)     
 2 5
2 2
q)     
 3  3
  1  2 
r)    
 2  


3
Remember:








=

67
Bilingual Programme.
M. Mar Agüera de Pablo-Blanco.
IES Caura. Coria del Río.
M)
TEST4. ABOUT FRACTIONS
1.Translate into English:

Fracciones irreducibles.

Multiplicar las fracciones
del primer ejercicio.

Fracciones propias e
impropias.

Ordena las siguientes
fracciones de menor a
mayor.

Simplifica la fracción.

Para sumar y restar
fracciones, busca su
denominador común.
2.Exercises:
48
120

Simplify:

Write from greatest to least:
1 2 7
, ,
2 5 3

Convert the fraction
10
into a
30
decimal.

Convert the percentage 80% into
a fraction.
68
Bilingual Programme.
M. Mar Agüera de Pablo-Blanco.
IES Caura. Coria del Río.

Write the next improper fractions
as mixed numbers:
4 11
, .
3 2

Work out:
1 3
 
2 4

Work out:
1 1 6
   
 2 3 5

Work out:
1 4 5
  
2 25 7

Find
3
of 65 Kg.
5

Find
30% of 45 €
3.How do you read?

3 7 43
 
4 5 20

1 3 3


2 5
10
69