Complex, Mixed & Compound Fractions

Proper Fraction:

A fraction in which the numerator is lesser than the denominator, example: 1/2.

Improper Fraction:

A fraction where the numerator is greater than the denominator, example: 9/2.

Mixed Fraction:

The sum of non zero integer and a proper fraction is termed as a mixed fraction.

For example: 5 + ${\dfrac{3}{4}}$ here the plus sign is implicitly written as  5 ${\dfrac{3}{4}}$

Reducing the mixed fraction to an improper fraction:

Here  5 ${\dfrac{3}{4}}$  can be reduced as  ${\dfrac{23}{4}}$

Multiply  the denominator which is 4 with the whole integer to the left which is 5, that will be 4 x 5=20.  Now  to the product obtained,  in this case 20,  add the numerator which is  3,  we get 20 + 3 = 23. Write the denominator as it is, in our case 4 with the numerator  as 23, that is ${\dfrac{23}{4}}$

Ratios:

Ratio is relationship between two or more numbers which can also be converted to fractions.

8 red dots and 16 green dots expressed in ratio as 8:16 can be written in fraction form as 1/2

3:12 can be written as 1/4

4:32 can be written as 1/8

Invisible denominator :

Whenever any number is written without a denominator for example 18 it is implicit that it’s denominator is always 1,  that is = ${\dfrac{18}{1}}$.

Reciprocal :

A fraction with it’s numerator and denominator inversed is the reciprocal of any given fraction for example reciprocal of 29 is ${\dfrac{1}{29}}$. Every nonzero integer or fraction has a reciprocal.

Complex Fraction:

In a complex fraction either the numerator or the denominator, or both are  simple fraction or  mixed fraction.  For example $\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}$ and $\dfrac{2 \dfrac{1}{2}}{5}$

Example i):

$\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}$

=  ${\dfrac{3}{4}}$ ÷ ${\dfrac{1}{2}}$

=  ${\dfrac{3}{4}}$ x ${\dfrac{2}{1}}$

= ${\dfrac{3}{2}}$

= $1 \dfrac{1}{2}$

Example ii):

$\dfrac{2 \dfrac{1}{2}}{5}$

= ${\dfrac{5}{2}}$ ÷ ${\dfrac{1}{5}}$

= ${\dfrac{5}{2}}$ x ${\dfrac{5}{1}}$

= ${\dfrac{25}{2}}$

Example iii):

${\dfrac{3 \dfrac{1}{2}}{5 \dfrac{3}{4}}}$

= ${\dfrac{7}{2}}$ ÷ ${\dfrac{23}{4}}$

= ${\dfrac{7}{2}}$ x ${\dfrac{4}{23}}$

= ${\dfrac{14}{23}}$

Compound Fraction:

A fraction of a fraction is known as compound fraction. The “of” word  is represented as multiplication of the fractions. For example ${\dfrac{2}{3}}$ of ${\dfrac{3}{18}}$ can be written as ${\dfrac{2}{3}}$ x ${\dfrac{3}{18}}$ = 1/9

Decimal values as fractions:

Decimal values can be expressed as fractions by specifying 10 to the power of an integer in it’s denominator. For example:

0.92 can be expressed as ${\dfrac{92}{100}}$ = ${\dfrac{92}{10^2}}$ = 92 x $10^{-2}$

0.000000000042 = ${\dfrac{42}{10^{12}}}$ = 42 x $10^{-12}$

0.000000000000234 = ${\dfrac{234}{10^{15}}}$ = 234 x $10^{-15}$

Decimal values with infinite numbers to the right of decimal point represent infinite series which can also be expressed as fractions, for example , 0.22222….. as fraction can be written as  2/10 + 2/100 + 2/1000…….∞

Infinite Repeating Decimal:

Fractions can be more precisely accurate than decimal values in some cases. Infinite repeating decimal values can be conveniently expressed as fractions. Repeating decimal value are indicated with a bar placed over them. For example,

$0.{\overline{9564}}$=0.9564956495649564

Infinite Repeating Decimal as fraction:

To precisely express infinite repeating decimal values  as fractions divide it by as many number of nines as repetitive numbers after the decimal point. For example:

$0.{\overline{3}}$=3/9

$0.{\overline{78}}$=78/99

$0.{\overline{563}}$=563/999

$0.{\overline{3821}}$=3821/9999

In case of zeros preceding the repetetive decimal numbers, suffix the nine laden denominator with as many trailing zeros. For example,

$0.0{\overline{34}}$ = 34/990
$0.00{\overline{56}}$ = 56/9900
$0.000{\overline{78}}$ =  78/99000
$0.0000{\overline{562}}$ =  562/9990000

In case of non zero numbers preceding the repetitive decimal numbers split the number as sum of   decimal numbers. Then covert them to fractions, ultimately add them. For example,

$0.567{\overline{345}}$ =  0.567 + 0.000345 =  567/1000 + 345/999000  =   566778/999000

$2.5{\overline{6}}$

=  2 + $0.5{\overline{6}}$

= 2 + 0.50 +$0.0{\overline{6}}$

= 2 + ${\dfrac{5}{100}}$ + ${\dfrac{6}{90}}$

=    2 + ${\dfrac{105}{900}}$

=  2 ${\dfrac{7}{60}}$

Per cent(%) here’s how it can be expressed as fraction:
Percent means parts per 100, that is 1/100 =  ${10^{-2}}$ = 0.01
Percent values can also be expressed as mixed fractions. For example:
3.56 % can be expressed as 3  ${\dfrac{56}{100}}$

89.68 %  =  89${\dfrac{68}{100}}$

Permille(‰)  here’s how it can be expressed as fraction:
Permille(‰) means parts per 1000, that is 1/1000 =  ${10^{-3}}$ = 0.001
Per mille  values can also be expressed as mixed fractions. For example:
23.784 ‰  = 23${\dfrac{784}{1000}}$

Permyriad (‱) here’s how it can be expressed as mixed fraction:
Per myriad (‱) means parts per 10,000, that is 1/10000 = ${10^{-4}}$ = 0.0001
Per myriad  values can be also expressed as mixed fractions. For example:
45.63,478 = 45${\dfrac{63478 }{10,000}}$

Per million(ppm) here’s how it can be expressed as fraction:
Per million means parts per 1000000, that is 1/1000,000 = ${10^{-6}}$ = 0.000001
Per million values can be expressed as mixed fractions. For example:
5.352846 ppm = 5${\dfrac{352846 }{1000,000}}$

Per billion(ppb) here’s how it can be expressed as fraction:
Per billion means parts per 1000000000, that is 1/1000,000,000 = ${10^{-9}}$ = 0.000000001
Per billion values can be expressed as mixed fractions. For example:
62.713256785 ppb = 62${\dfrac{713256785}{1000,000,000}}$

Per trillion(ppt) here’s how it can be expressed as fraction:
Per trillion means parts per 1000000000000, that is 1/1000,000,000,000 = ${10^{-12}}$ = 0.000000000001
Per trillion values can be expressed as mixed fractions. For example:
2.713256785457 ppt = 2${\dfrac{713256785457}{1000,000,000,000}}$

Per quadrillion(ppq) here’s how it can be expressed as fraction:
Per quadrillion means parts per 1000000000000000, that is 1/1000,000,000,000,000 = ${10^{-15}}$ = 0.000000000000001
Per quadrillion values can be expressed as mixed fractions. For example:
6.672391548912 ppq = 6${\dfrac{672391548912}{1000,000,000,000,000}}$

example i):
$\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}$ + $\dfrac{2 \dfrac{1}{2}}{5}$

= {$\dfrac{3}{4}$ ÷ $\dfrac{1}{2}$} + {$\dfrac{5}{2}$ ÷ $\dfrac{5}{1}$}

= {$\dfrac{3}{4}$ x $\dfrac{2}{1}$} + {$\dfrac{5}{2}$ x $\dfrac{1}{5}$}

= {$\dfrac{3}{2}$ + $\dfrac{1}{2}$}

= $\dfrac{4}{2}$

= 2

example ii):

$\cfrac{3 \dfrac{1}{2}}{5 \dfrac{3}{4}}$ + $4 \dfrac{1}{3}$

= {$\dfrac{7}{2}$ ÷ $\dfrac{23}{4}$} + $\dfrac{13}{3}$

= {$\dfrac{7}{2}$ x $\dfrac{4}{23}$} + $\dfrac{13}{3}$

= $\dfrac{14}{23}$ + $\dfrac{13}{3}$

= $\dfrac{42+299}{69}$ = $\dfrac{341}{69}$

example iii):

$\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}$ + $\cfrac{\dfrac{5}{2}}{\dfrac{3}{4}}$

= {$\dfrac{3}{4}$ ÷ $\dfrac{1}{2}$} + {$\dfrac{5}{2}$ ÷ $\dfrac{3}{4}$}

= {$\dfrac{3}{4}$ x $\dfrac{2}{1}$} + {$\dfrac{5}{2}$ x $\dfrac{4}{3}$}

= $\dfrac{3}{2}$ + $\dfrac{10}{3}$

= $\dfrac{9+20}{6}$

= $\dfrac{29}{6}$

Substracting complex or mixed fractions:

example i):

$\cfrac{\dfrac{1}{4}}{\dfrac{2}{3}}$ ─ $\dfrac{5 \dfrac{1}{2}}{5}$

= {$\dfrac{1}{4}$ ÷ $\dfrac{2}{3}$} ─ {$\dfrac{11}{2}$ ÷ $\dfrac{5}{1}$}

= {$\dfrac{1}{4}$ x $\dfrac{3}{2}$} ─ {$\dfrac{11}{2}$ x $\dfrac{1}{5}$}

= {$\dfrac{3}{8}$ ─ $\dfrac{11}{10}$}

= $\dfrac{30 – 88}{80}$

= ─ $\dfrac{58}{80}$

= ─ $\dfrac{29}{40}$

example ii):

$\cfrac{8 \dfrac{1}{2}}{\dfrac{3}{4}}$ – $1 \dfrac{1}{3}$

= {$\dfrac{17}{2}$ ÷ $\dfrac{3}{4}$} ─ $\dfrac{4}{3}$

= {$\dfrac{17}{2}$ x $\dfrac{4}{3}$} ─ $\dfrac{4}{3}$

= $\dfrac{34}{3}$ ─ $\dfrac{4}{3}$

= $\dfrac{30}{3}$ = 10

example iii):

$\cfrac{\dfrac{5}{4}}{\dfrac{1}{2}}$ ─ $\cfrac{\dfrac{1}{2}}{\dfrac{1}{4}}$

= {$\dfrac{5}{4}$ ÷ $\dfrac{1}{2}$} ─ {$\dfrac{1}{2}$ ÷ $\dfrac{1}{4}$}

= {$\dfrac{5}{4}$ x $\dfrac{2}{1}$} ─ {$\dfrac{1}{2}$ x $\dfrac{4}{1}$}

= $\dfrac{5}{2}$ ─ $\dfrac{2}{1}$

= $\dfrac{5 – 4}{2}$

= $\dfrac{1}{2}$

Dividing complex or mixed fractions:

example i):

$\cfrac{\dfrac{5}{4}}{\dfrac{1}{2}}$ ─ $\dfrac{3 \dfrac{1}{2}}{5}$

= {$\dfrac{5}{4}$ ÷ $\dfrac{1}{2}$} ÷ {$\dfrac{7}{2}$ ÷ $\dfrac{5}{1}$}

= {$\dfrac{5}{4}$ x $\dfrac{2}{1}$} ÷ {$\dfrac{7}{2}$ x $\dfrac{1}{5}$}

= $\dfrac{5}{2}$ ÷ $\dfrac{7}{10}$

= $\dfrac{5}{2}$ x $\dfrac{10}{7}$

= $\dfrac{25}{7}$

example ii):

$\cfrac{5 \dfrac{1}{2}}{3 \dfrac{3}{4}}$ ÷ $3 \dfrac{1}{3}$

= {$\dfrac{11}{2}$ ÷ $\dfrac{15}{4}$} ÷ $\dfrac{10}{3}$

= {$\dfrac{11}{2}$ x $\dfrac{4}{15}$} ÷ $\dfrac{10}{3}$

= $\dfrac{22}{15}$ x $\dfrac{3}{10}$

= $\dfrac{22}{50}$

= $\dfrac{11}{25}$

example iii):

$\cfrac{\dfrac{1}{4}}{\dfrac{3}{5}}$ ÷ $\cfrac{\dfrac{4}{3}}{\dfrac{2}{5}}$

= {$\dfrac{1}{4}$ ÷ $\dfrac{3}{5}$} ÷ {$\dfrac{4}{3}$ ÷ $\dfrac{2}{5}$}

= {$\dfrac{1}{4}$ x $\dfrac{5}{3}$} ÷ {$\dfrac{4}{3}$ x $\dfrac{5}{2}$}

= $\dfrac{5}{12}$ ÷ $\dfrac{10}{3}$

= $\dfrac{5}{12}$ x $\dfrac{3}{10}$

= $\dfrac{1}{8}$

Multiplying complex or mixed fraction:

example i):

$\cfrac{\dfrac{6}{5}}{\dfrac{1}{3}}$ x $\dfrac{5 \dfrac{1}{2}}{3}$

= {$\dfrac{6}{5}$ ÷ $\dfrac{1}{3}$} x {$\dfrac{11}{2}$ ÷ $\dfrac{3}{1}$}

= {$\dfrac{6}{5}$ x $\dfrac{3}{1}$} x {$\dfrac{11}{2}$ x $\dfrac{1}{3}$}

= $\dfrac{18}{5}$ x $\dfrac{11}{6}$

= $\dfrac{33}{5}$

example ii):

$\cfrac{2 \dfrac{1}{5}}{6 \dfrac{3}{4}}$ x $1 \dfrac{1}{3}$

= {$\dfrac{11}{5}$ ÷ $\dfrac{27}{4}$} x $\dfrac{4}{3}$

= {$\dfrac{11}{5}$ x $\dfrac{4}{27}$} x $\dfrac{4}{3}$

= $\dfrac{44}{135}$ x $\dfrac{4}{3}$

= $\dfrac{176}{405}$

example iii):

$\cfrac{\dfrac{3}{4}}{\dfrac{1}{5}}$ x $\cfrac{\dfrac{2}{3}}{\dfrac{7}{5}}$

= {$\dfrac{3}{4}$ ÷ $\dfrac{1}{5}$} x {$\dfrac{2}{3}$ ÷ $\dfrac{7}{5}$}

= {$\dfrac{3}{4}$ x $\dfrac{5}{1}$} x {$\dfrac{2}{3}$ x $\dfrac{5}{7}$}

= $\dfrac{15}{4}$ x $\dfrac{10}{21}$

= $\dfrac{25}{14}$