Chapter 2.1 (8th Grade Mathemathics)

Fractions and decimals

  • Fractions
  • Decimals
  • Converting a fraction to a decimal 
  • Converting a decimal to a fraction
  • Infinite decimal numbers
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Review what you already know

Fraction

Fraction

A fraction is less than one.

The fraction \frac{3}{4}\ means that the whole is divided into four equal parts, from which 3 parts are selected.

Improper fraction

An improper fraction is greater than one (or equal to one).

The improper fraction \frac{4}{3} means that the whole is divided into three equal parts and 4 parts are selected.

Mixed number

A mixed number consists of a whole number and a fraction.

The whole is divided into eight equal parts.

2\frac{3}{8} means two and three eighths.

Decimal number

A decimal number consists of an integer, a comma and the following decimal places.

Examples

  • 1.24
  • 67.912
  • 3000.5
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Finite decimal number

If the fraction can be expanded so that the denominator is

10, 100, 1 000, 10 000 etc, 

then the decimal number is a finite decimal number.

Note

Any fraction can be converted to a decimal number. 

Conversion is easier if the denominator of the fraction is

 10, 100, 1000 ...

\frac{1}{10}=0.1

\frac{1}{100}=0.01

\frac{1}{1000}=0.001

\frac{1}{10\ 000}=0.0001

Example 1

Expansion

Convert \frac{2}{5} to a decimal number.

  1. Expand the fraction by 2 to get a fraction in tenths.

\frac{2}{5}=\frac{4}{10}

  1. Divide the numerator by the denominator.

4:10=0.4

Answer

\frac{2}{5}=0.4

0.4 is a finite decimal number.

Mixed number expansion

Convert 3\frac{1}{20} to a decimal number.

  1. Expand the fraction by 5 to get fractions in hundredths.

3\frac{1}{20}=3+\frac{5}{100}

  1. Divide the numerator by the denominator.

3+5:100=3+0.05

Answer

3\frac{1}{20}=3.05

3.05 is a finite decimal number.

Dividing the numerator by the denominator

Convert \frac{3}{4} to a decimal number.

Divide the numerator by the denominator.

Answer

\frac{3}{4}=0.75

0.75 is a finite decimal number.

  1. \frac{7}{10}= 
  2. \frac{13}{100}= 
  3. \frac{81}{1000}= 
  1. 12=10=
  2. 325=100= 
  3. 2950=100= 
  1. 5\frac{9}{10}= 
  2. 13\frac{21}{100}= 
  3. 10\frac{1}{4}= 
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Infinite decimal number

  • If the numerator of a fraction is divided by the denominator, and the same number starts repeating, it is an infinite decimal number.
  • Such a repetition of a number or series of numbers is called a period.

Example

3.123123... 

Period: 123

Infinite decimal numbers are indicated by three dots at the end of the decimal part.

Example 2

Convert \frac{1}{6}  into a decimal number.

  1. Divide the numerator by the denominator.
  1. Note that the decimal part repeats the number 6. 
    Thus, the decimal 0.166... has a period of 6.

Answer

\frac{1}{6}=0.166...

Convert 9\frac{5}{11} into a decimal number.

  1. Divide the numerator by the denominator.
  1. See that the decimal 0.4545... has a period of 45.

Answer

9\frac{5}{11}=9.4545...

  1. \frac{4}{3}= 
  2. \frac{7}{9}= 
  3. \frac{8}{11}= 
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Converting a decimal to a fraction

Any decimal number can be converted to a fraction, except for non-terminating non-sequential decimals (e.g. π).

Note

When converting a decimal number to a fraction, the fraction should be reduced if possible.

Example 3

Let's convert the decimal numbers to fractions.

  1. 0.3=\frac{3}{10}
  2. 0.19=\frac{19}{100}
  3. 0.123=\frac{123}{1000}
  1. 5.7=5\frac{7}{10}
  2. 28.39=28\frac{39}{100}
  3. 10.193=10\frac{193}{1000}
  1. 2.8=2\frac{8}{10}=2\frac{4}{5}
  2. 9.15=9\frac{15}{100}=9\frac{3}{20}
  3. 0.125=\frac{125}{1000}=\frac{1}{8}
  1. 0.31 = 
  2. 33.7 = 
  3. 1.111 = 
  1. 0.55 = 
  2. 5.8 = 
  3. 10.12 = 
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Practise and solve

  • 13
  • 14
  • 48
  • 22
  • 64
  • 0.25
  • 0.5
  • 1.0
  • 1.25
  • 1.5

Fraction

Decimal number

  • 23
  • 68
  • 53
  • 32
  • 83
  • 0.75
  • 0.66...
  • 1.33...
  • 1.5
  • 1.66...
  • 2.33...
  • 2.66...

Fraction

Decimal number

      • 18
      • 49
      • 10815
      • 1336
      • 61124
      • 2150
      • 1716
      • 5922
      • 3925
      • 7332
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      Good to know

      Tangram

      The picture shows a tangram consisting of seven patterns: five triangles and two squares.

      1st pattern:  1 = 

      6th pattern:  1  = 

      3rd pattern:  1 = 

      7th pattern:  1 = 

      Read and think

      1. A hike to the top of Halti
      2. A hike to a tenth of the height of Mt. Everest
        \frac{8849}{10}=  m

      Answer

      The higher hiking trail is 

      1. One quarter of the hike is\frac{1157}{4} or  km
      2. 289.33 ... km

      Answer

      The longer hike is 

      1. \frac{314600}{1000} is  km
      2. \frac{1295}{4} is  km

      Answer

      The longer distance is 

      1. One fiftieth of 121 metres is
          m
      2. One hundredth of 953 decimetres is
          dm

      Answer

      Device  was used to see deeper.

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      Extra

      Note

      A decimal number corresponding to a fraction is always finite if the denominator of the fraction has no prime factors other than 2 and 5.

      For example:

      \frac{7}{20}=0.35

      is a finite decimal number, because the denominator

      20=2\cdot2\cdot5.

      Example
      Fraction: 112=0.0833...
      Denominator: 12=22·3 or 12=2·2·3

      Fraction

      Denominator

      \frac{4}{15}=  

      15= 

      6\frac{9}{32}= 

      32= 

      \frac{11}{24}= 

      24= 

      2\frac{12}{125}= 

      125= 

      \frac{27}{40}= 

      40= 

      1\frac{13}{60}= 

      60= 

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