UNIT 1 (Act 2+Sol) Fractions and Decimals

Unit 1: Fractions and Decimals. ACTIVITIES 2 Mathematics - 3º ESO

UNIT 1: Fractions and Decimals

Rational Numbers

The numbers that fall between the integers on the number line are either rational or irrational. In this section, we discuss the rational numbers.

Rational numbers are used to express a part of a whole, a part of a quantity.

58→(Five eighths)

The number below the fraction line is called denominator, and expresses the number of parts into which the whole is divided. The number above the fraction line is called the numerator, and expresses the number of parts taken.

A more formal definition of rational numbers could be:

So any number that can be expressed as a quotient of two integers (denominator not zero) is a rational number:

56=0.833 ... 10

5=2−25

4=−6.25 3

7=0.42857142…

When the numerator and denominator have a common divisor, we can reduce the fraction to its lowest terms (or simplest form):

618

=39=13=0.3333…

A fraction is said to be in its lowest terms (or reduced, or simplified) when the numerator and the denominator are relatively prime.

IES Albayzín (Granada) Page 1

The set of rational numbers, denoted by Q, is the set of all numbers

of the form pq, where p and q are integers, and q≠0.


Now we can introduce the definition of equivalent fractions:

Proper and Improper Fractions. Mixed Numbers.

Rational numbers less than 1 or greater than -1 are represented by proper fractions. A proper fraction is a fraction whose numerator is less than its denominator:

47− 511

Consider the number 234. It is an example of a mixed number. It is called a

mixed number because it consists of an integer, 2, and a fraction 34, and it is

equal to 2+34. The mixed number −4

14 means −(4+ 14 ).

Rational numbers greater than 1 or less than -1 that are not integers may be represented as mixed numbers, or as improper fractions. An improper fraction is a fraction whose numerator is greater than its denominator.

114

=2+ 34


Two fractions are said to be equivalent when simplifying both of them produces the same fraction, which cannot be further reduced.

Equivalent fractions look different but represent the same portion of the whole.

Equivalent fractions have the same numerical value. They are represented by the same rational number.

Equivalent fractions are represented by the same point on the number line.

We can test if two fractions are equivalent by cross-multiplying (or cross-product) their numerators and denominators.


Exercises

1) How do you read fractions? Complete the following table using the rules to read and write fractions.

Fraction Write down how the fraction is read

12 One half

32 Three halves

53 Five thirds

34 Three quarters

25 Two fifths

18 One eighths

59 Five ninths

2) Simplify or cancel each of these fractions down to their simplest form.

a)1421

=23

b) 2575

=13

c) 2346

=12

d) 52130

=2665

=25

3) Fill in the required number:

a)3❑=12

20 b)

68=❑12

c) ❑21

= 428

d) 1018

=25❑

a) 3 ∙20=12 x→ x=3 ∙2012

=6012

=5



b) 6 ∙12=8 x→ x=6 ∙128

=728

=9

c) 28 x=4 ∙21→x=4 ∙2128

=8428

=3

d) 10 x=18 ∙25→x=18∙2510

=2∙32 ∙52

2 ∙5=32 ∙5=45

Finding Fractions of quantities: Problems Involving Fractions

1) Calculate in your head and then answer:

a) How many minutes are there in 1/5 of an hourb) How many minutes are there in 5/6 of an hourc) What fraction of an hour is 20 minutes?d) What fraction of an hour is 40 minutes?

a)15∙60=60

5=12minutes

b)56∙60=60 ∙5

6=50minutes

c)2060

=13

d)4060

=23

2) A shelf in a supermarket holds 80 one-quarter litre bottles and 44 one and a half litre bottles. How many litres of water are there on the self?

14∙80=20 litres (from the one-quarter litre bottles)

112∙ 44=(1+ 12 ) ∙44=32 ∙44=3 ∙442 =66 litres (from the

one and a half litre bottles)

There are 20+66=86 litres of water on the self.



3) When David Conway finished his first year of high school, his height was

6978

inches. When he returned to school after the summer, David’s

height was 7158

inches. How much did David’s height increase over the

summer? (1 inch = 2.54 cm)

7158−69 7

8=71+ 5

8−69−7

8=2−2

8=16−2

8=148

=74=¿1.75 inches

1.75 inches×2.54cminch

=4.445cm

David’s height increased 4.445 cm over the summer

4) A lorry’s tank contains 225 litres of diesel, and the gauge says the tank is ¾ full. How many litres can the tank hold?

34∙225=3∙225

4=168.75 litres

5) In a bicycle race, cyclist A has covered 4/5 of the total route and has 21 km left before the finish line. How many kilometres are left before cyclist B reaches the finish line, if he has covered 6/7 of the route?

Cyclist A

21 km

The route has 21 ∙5=105 km

Cyclist B

15 km

Cyclist B has to covert 17∙105=15 Km to reach the finish line.



6) A bag contains twenty beads numbered from 1 to 20.

a) Which of the bead numbers are multiples of 5?b) Express as a fraction the probability of obtaining a multiple of 5 by

drawing one bead at random from the bag

Don’t Forget

P [event ]=Favourable outcomesPossible outcomes

a) The multiples of 5 are:

5 10 15 20

b) P [event ]=Favourable outcomesPossible outcomes

= 420

=15=0.20

The probability of obtaining a multiple of 5 is one fifth or 20%.

7) A vineyard sells two-fifths of its harvest to a wholesaler, and one-third of the rest to a supermarket. If the vineyard still has 40 hectolitres left, how much did it produce?

Algebraic method

If the vineyard sells 2/5 of the harvest to a wholesaler, then the rest of his

harvest is 1−25=35.

If the vineyard sells 1/3 of the rest to a supermarket, then he still has 2/3 of the rest.

23∙35∙ x=40→ 2

5∙ x=40→x=40 ∙5

2=100

The vineyard produced 100 hectolitres



Graphic method

Wholesaler Wholesaler Supermarket 20 20

40 hectolitres

20hectolitres×5=100 hectolitres produced by the vineyard


Documents

UNIT 1 (Act 2+Sol) Fractions and Decimals