Unit: 2FRACTIONS, DECIMALS

AND PERCENTAGES2.2 Types Of Decimals.

2.3 Negative Fractions and decimals

By the end of this presentation, you will be able to:•identify different types of decimal numbers and use the correct notation for writing them.

•calculate with positive and negative fractions and decimals

•This is a Grade 8 Power Point Presentation prepared for the Unit on Fractions, Decimals and Percentages.

•The sub-topics are: 2.2 Types of Decimals.

2.3 Negative Fractions and Decimals.

•You as a student, are required to:1. read through and understand this sub-topic by making your own notes,

2. make notes and understand the examples given in the presentations,

3. attempt similar activities given in the presentations and

There are two types of decimals:

1.Terminating decimals and

2.Non-terminating decimals

1.Terminating decimals

Are answers to a division that will result in another whole number and a decimal number with a clearly defined number of digits (terminating decimal).

Examples:

a.12

3= 12

b.18

5= 18 ÷ 5 = 3.6

c.9

8= 9 ÷ 8 = 1.125

d.88

200= 88 ÷ 200 = 0.44

Non- terminating decimalsThe result of two numbers divided together has an infinite number of digits(the digits

continue forever).

Called non-terminating decimals.

If some, or all, of the decimal digits follow a repeating pattern, it is recurring decimal.Example.

1.1

3= 1 ÷ 5 = 0.333 333 3…

2.4

15= 4 ÷ 15 = 0.266 666…

3. 11

7= 11 ÷ 7 = 1.5771428 571…

Cont...

Some decimals are non-terminating and non-recuring i.e. They don’t have an infinite number of digits and no pattern. These are called irrational numbers.

- Irrational numbers are numbers that:Cannot be written as a fractionInclude special numbers such as 𝜋 and the square roots of numbers.

Example 2, 3 𝑎𝑛𝑑 √5Surds

The only way to write the exact value of a surd is to write it in surd form; that is

2, 3 𝑎𝑛𝑑 5For example: 2 ≈ 1.4 (1. 𝑑. 𝑝. )

≈ 1.41 (2 𝑑. 𝑝. )≈ 1.414214 6 𝑑. 𝑝. (′ ≈ ′𝑚𝑒𝑎𝑛𝑠 𝑖𝑠 ′𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜′)

Structure of decimal numbers

Example

Calculate and classify them as either terminating, recurring, or irrational.

a. 12 ÷ 7 = 1.714285714…Recuring

b.5

16= 0.3125

Terminating

c. 50 = 7.071067812Irrational

d. 156.25 = 12.5Terminating

Activity 1

Calculate and classify them as either terminating, recurring, or irrational.a. 9 ÷ 8 e. 15 ÷ 6

b.11

12f.

5

9

c. 449 g. 216

d. 96.4 h. 342.25

Writing re-curing decimals.

To write a recuring decimal:Write any recuring digits, then the digits that form the repeating pattern.

Place a dot on top of tje first snd the last digits of the repeating pattern, or draw a dash accross the whole of the repeating pattern. For example, 0.357 357... Can be written as

Using a dot or dash notation is called writting the number in exact decimal form.

Example of recuring decimals written in exact decimal form is:1

3= 1 ÷ 3 = 0.333 333… and is written as 0. 3 or 0. 3

Example

Calculate each of the following fractions in exact decimal form.

a.1

7= 1 ÷ 7 = 0.142857142

the repeating pattern is 142 857.1

7=

b. 1

13= 1 ÷ 13 = 0.076923 076

the repeating pattern is 076 923.1

13=

c. 1

12= 1 ÷ 12 = 0.083333333

the repeating pattern is 3.1

12=

Activity 2

1. Write each of the following fractions in exact decimal form.

a.1

6f.

2

11

b.1

11g.

11

18

c.1

15h.

7

15

d.1

18i.

2

13

e.5

6j.

4

15

Writing recuring decimals as fractions

We have converted decimls to fractions using the table of values. Given recuring decimals does not have the last digit, we need a method to change the recuring decimals to fractions by:

1. Creating two expressions with the same decimal part.

2. Eliminate, when the two expressions are subtrated.

Example:2.4444 − 0.4444 = 2

Example

Write the followingrecuring decimals as fractions.

a. 0.15

steps:I. Let x = the recuring decimal

𝑥 = 0.151515… (1)

II. Create a second expression by multiplying by 100.100𝑥 = 15.151515… 2

III. Subtract the expressions2 − 1 ;100𝑥 = 15.151515…

− 𝑥 = 0.151515…99𝑥 = 15

IV. Isolate 𝑥 and simplify if necessary

𝑥 =15

99

𝑥 =5

33

Example

b. 1.24

steps:I. Let x = the recuring decimal

𝑥 = 1.244 44… (1)

II. Create a second expression by multiplying by the power of 10 that has the same numer of zeroes as the number of digits in the repeating pattern..10𝑥 = 12.4444… 2

III. Subtract the expressions2 − 1 ;10𝑥 = 12.4444…

− 𝑥 = 1.2444…9𝑥 = 11.2

IV. Isolate 𝑥 and simplify if necessary

𝑥 =11.2

9

𝑥 =11.2 × 10

9 × 10

𝑥 =112

90=

56

45𝑜𝑟 1

11

45

Activity 3

Example Place the following sets of numbers on seperate number lines.

a) 21

2,3

4,1

4, −2

1

3

Step 1Construct a number line

Locate each of the numbers

Number line Symmetry

Like integers, fractions and decimals can be located on a number line in such a way that we locate positive fractions and decimals.

Adding and subtracting negative decimals

•The symmetry of the number line can also help us add and subtract negative decimal numbers.

•Calculations on one side of the number line as an equivalent mirror image calculations on the opposite side.

Example

Calculate −3.7 + (−2.8)

Step 1

• Simplify the addition by writing a single sign between the two numbers

• −3.7 + −2.8 = 3.7 − 2.8

Step 2

• Visualise the calculation on the number line

Step 3

• Complete the mirror image calculation

• 3.7+2.5 = 6.5• ∴-3.7-2.8=-6.5

Activity

1. Use the number line provided and place the following sets of numbers on separate number line.

a. −31

2, −2

3

4, −

1

2, 1

1

4

b. −1.4, −1.9, 0.4, 2.2

c. −3.6, −2.6, −6.1, 1.6

d. −6

5,−5

6, −2

1

6,1

5

Number line work sheet

Use the number

lines on this

worksheet to

complete the

questions on the

previous slide

Adding and subtracting negative fractions

To add or subtract negative fractions, the same method you used ealier will be used again.

Example

Calculate the following and give answers to the simplest form.

a. −4

5+

1

2

Step 1

Find the LCD that is 10.

−4

5+1

2= −

8

10+

5

10

Step 2

Add the numerators and simplify the answer if

possible.

= −3

10

b. −13

4+ (−

5

6)

Step 1

Simplify the addition by witting a single sign between

the two fractions.

−13

4−5

6Step 2

Write any mix number into improper fractions.

= −7

4−

5

6Step 3

Write the fractions as equivalent with a common

denominator

= −21

12−

10

12

Step 4

Subtract the numerators and simplify the answer if

possible.

= −31

12𝑜𝑟 − 2

7

12

Multiplying and dividing negative fractions and decimals

Activity 1

Activity 2