unit 8. fractions, decimals and percentages. - WIKI1-mar

Transcripción

M.Mar Agüera de Pablo-Blanco
IES Caura. Coria del Río
Bilingual programme
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
1
Bilingual programme
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
One third
One quarter
2
3
5
2
3
4
 It is also possible to read a fraction as:
Cardinal number
over
Cardinal number
EXAMPLES
EXERCISES
2
11
Two over eleven
31
76
9
25
Nine over twenty-five
7
18
2
Bilingual programme
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:
3
Bilingual programme
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.
EXAMPLES
=........
Convert the following fractions into decimals.
So,
1
 0.125
8
(It’s a terminating decimal)
........
So,
(It’s a recurring decimal)
EXERCISES

Convert the following fractions into decimals.
Now, turn terminating decimals into fractions. You have to look at where the last
digit after the decimal point is.
4
Bilingual programme
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.
So, it’s the same as
3
10
12
100
2547
1000
EXERCISES:
1. Convert the following terminating decimals into fractions:
1.7
0.85
3.547
2. Investigate with recurring decimals.
5
Bilingual programme
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
6
Bilingual programme
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:
1

2
150

2250
EXERCISES:
1.Amplify the following fractions: (Look for 3)
(Obtención de fracciones equivalent es por ampliación)
2
7
1
5
7
3
5
6
7
Bilingual programme
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.
8
Bilingual programme
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)
9
Bilingual programme
EXERCISES: Place in order starting with the greatest(Use the method3)
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
10
Bilingual programme
 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%
11
Bilingual programme
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€
12
Bilingual programme
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?
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)
13
Bilingual programme
 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
14
Bilingual programme
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
15
Bilingual programme
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
16
Bilingual programme
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
17
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

18
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
19
Bilingual programme
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
n)   
 3
3
 1
o)   
 3
2
5
5
2
3
1 1
p)     
 2 5
2 2
 3  3
q)     
  1  2 
r)    
 2  


3
Remember:








=

20
Bilingual programme
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.
21
Bilingual programme

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
22

unit 8. fractions, decimals and percentages. - WIKI1-mar

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