Complex, Mixed & Compound Fractions

complex-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 + [latex]{\dfrac{3}{4}}[/latex] here the plus sign is implicitly written as  5 [latex]{\dfrac{3}{4}}[/latex]

Reducing the mixed fraction to an improper fraction:

Here  5 [latex]{\dfrac{3}{4}}[/latex]  can be reduced as  [latex]{\dfrac{23}{4}}[/latex]

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 [latex]{\dfrac{23}{4}}[/latex]

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 = [latex]{\dfrac{18}{1}}[/latex].

Reciprocal :

A fraction with it’s numerator and denominator inversed is the reciprocal of any given fraction for example reciprocal of 29 is [latex]{\dfrac{1}{29}}[/latex]. 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 [latex]\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}[/latex] and [latex]\dfrac{2 \dfrac{1}{2}}{5}[/latex]

Example i):

[latex]\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}[/latex]

=  [latex]{\dfrac{3}{4}}[/latex] ÷ [latex]{\dfrac{1}{2}}[/latex]

=  [latex]{\dfrac{3}{4}}[/latex] x [latex]{\dfrac{2}{1}}[/latex]

= [latex]{\dfrac{3}{2}}[/latex]

= [latex]1 \dfrac{1}{2}[/latex]

Example ii):

[latex]\dfrac{2 \dfrac{1}{2}}{5}[/latex]

= [latex]{\dfrac{5}{2}}[/latex] ÷ [latex]{\dfrac{1}{5}}[/latex]

= [latex]{\dfrac{5}{2}}[/latex] x [latex]{\dfrac{5}{1}}[/latex]

= [latex]{\dfrac{25}{2}}[/latex]

Example iii):

[latex]{\dfrac{3 \dfrac{1}{2}}{5 \dfrac{3}{4}}}[/latex]

= [latex]{\dfrac{7}{2}}[/latex] ÷ [latex]{\dfrac{23}{4}}[/latex]

= [latex]{\dfrac{7}{2}}[/latex] x [latex]{\dfrac{4}{23}}[/latex]

= [latex]{\dfrac{14}{23}}[/latex]

Compound Fraction:

A fraction of a fraction is known as compound fraction. The “of” word  is represented as multiplication of the fractions. For example [latex]{\dfrac{2}{3}}[/latex] of [latex]{\dfrac{3}{18}}[/latex] can be written as [latex]{\dfrac{2}{3}}[/latex] x [latex]{\dfrac{3}{18}}[/latex] = 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 [latex]{\dfrac{92}{100}}[/latex] = [latex]{\dfrac{92}{10^2}}[/latex] = 92 x [latex]10^{-2}[/latex]

0.000000000042 = [latex]{\dfrac{42}{10^{12}}}[/latex] = 42 x [latex]10^{-12}[/latex]

0.000000000000234 = [latex]{\dfrac{234}{10^{15}}}[/latex] = 234 x [latex]10^{-15}[/latex]

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,

[latex]0.{\overline{9564}}[/latex]=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:

[latex]0.{\overline{3}}[/latex]=3/9

[latex]0.{\overline{78}}[/latex]=78/99

[latex]0.{\overline{563}}[/latex]=563/999

[latex]0.{\overline{3821}}[/latex]=3821/9999

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

[latex]0.0{\overline{34}}[/latex] = 34/990
[latex]0.00{\overline{56}}[/latex] = 56/9900
[latex]0.000{\overline{78}}[/latex] =  78/99000
[latex]0.0000{\overline{562}}[/latex] =  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,

[latex]0.567{\overline{345}}[/latex] =  0.567 + 0.000345 =  567/1000 + 345/999000  =   566778/999000

[latex]2.5{\overline{6}}[/latex]

=  2 + [latex]0.5{\overline{6}}[/latex]

= 2 + 0.50 +[latex]0.0{\overline{6}}[/latex]

= 2 + [latex]{\dfrac{5}{100}}[/latex] + [latex]{\dfrac{6}{90}}[/latex]

=    2 + [latex]{\dfrac{105}{900}}[/latex]

=  2 [latex]{\dfrac{7}{60}}[/latex]

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

89.68 %  =  89[latex]{\dfrac{68}{100}}[/latex]

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

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

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

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

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 = [latex]{10^{-12}}[/latex] = 0.000000000001
Per trillion values can be expressed as mixed fractions. For example:
2.713256785457 ppt = 2[latex]{\dfrac{713256785457}{1000,000,000,000}}[/latex]

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 = [latex]{10^{-15}}[/latex] = 0.000000000000001
Per quadrillion values can be expressed as mixed fractions. For example:
6.672391548912 ppq = 6[latex]{\dfrac{672391548912}{1000,000,000,000,000}}[/latex]

Adding complex or mixed fractions:

example i):
[latex]\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}[/latex] + [latex]\dfrac{2 \dfrac{1}{2}}{5}[/latex]

= {[latex]\dfrac{3}{4}[/latex] ÷ [latex]\dfrac{1}{2}[/latex]} + {[latex]\dfrac{5}{2}[/latex] ÷ [latex]\dfrac{5}{1}[/latex]}

= {[latex]\dfrac{3}{4}[/latex] x [latex]\dfrac{2}{1}[/latex]} + {[latex]\dfrac{5}{2}[/latex] x [latex]\dfrac{1}{5}[/latex]}

= {[latex]\dfrac{3}{2}[/latex] + [latex]\dfrac{1}{2}[/latex]}

= [latex]\dfrac{4}{2}[/latex]

= 2

example ii):

[latex]\cfrac{3 \dfrac{1}{2}}{5 \dfrac{3}{4}}[/latex] + [latex]4 \dfrac{1}{3}[/latex]

= {[latex]\dfrac{7}{2}[/latex] ÷ [latex]\dfrac{23}{4}[/latex]} + [latex]\dfrac{13}{3}[/latex]

= {[latex]\dfrac{7}{2}[/latex] x [latex]\dfrac{4}{23}[/latex]} + [latex]\dfrac{13}{3}[/latex]

= [latex]\dfrac{14}{23}[/latex] + [latex]\dfrac{13}{3}[/latex]

= [latex]\dfrac{42+299}{69}[/latex] = [latex]\dfrac{341}{69}[/latex]

example iii):

[latex]\cfrac{\dfrac{3}{4}}{\dfrac{1}{2}}[/latex] + [latex]\cfrac{\dfrac{5}{2}}{\dfrac{3}{4}}[/latex]

= {[latex]\dfrac{3}{4}[/latex] ÷ [latex]\dfrac{1}{2}[/latex]} + {[latex]\dfrac{5}{2}[/latex] ÷ [latex]\dfrac{3}{4}[/latex]}

= {[latex]\dfrac{3}{4}[/latex] x [latex]\dfrac{2}{1}[/latex]} + {[latex]\dfrac{5}{2}[/latex] x [latex]\dfrac{4}{3}[/latex]}

= [latex]\dfrac{3}{2}[/latex] + [latex]\dfrac{10}{3}[/latex]

= [latex]\dfrac{9+20}{6}[/latex]

= [latex]\dfrac{29}{6}[/latex]

Substracting complex or mixed fractions:

example i):

[latex]\cfrac{\dfrac{1}{4}}{\dfrac{2}{3}}[/latex] ─ [latex]\dfrac{5 \dfrac{1}{2}}{5}[/latex]

= {[latex]\dfrac{1}{4}[/latex] ÷ [latex]\dfrac{2}{3}[/latex]} ─ {[latex]\dfrac{11}{2}[/latex] ÷ [latex]\dfrac{5}{1}[/latex]}

= {[latex]\dfrac{1}{4}[/latex] x [latex]\dfrac{3}{2}[/latex]} ─ {[latex]\dfrac{11}{2}[/latex] x [latex]\dfrac{1}{5}[/latex]}

= {[latex]\dfrac{3}{8}[/latex] ─ [latex]\dfrac{11}{10}[/latex]}

= [latex]\dfrac{30 – 88}{80}[/latex]

= ─ [latex]\dfrac{58}{80}[/latex]

= ─ [latex]\dfrac{29}{40}[/latex]

example ii):

[latex]\cfrac{8 \dfrac{1}{2}}{\dfrac{3}{4}}[/latex] – [latex]1 \dfrac{1}{3}[/latex]

= {[latex]\dfrac{17}{2}[/latex] ÷ [latex]\dfrac{3}{4}[/latex]} ─ [latex]\dfrac{4}{3}[/latex]

= {[latex]\dfrac{17}{2}[/latex] x [latex]\dfrac{4}{3}[/latex]} ─ [latex]\dfrac{4}{3}[/latex]

= [latex]\dfrac{34}{3}[/latex] ─ [latex]\dfrac{4}{3}[/latex]

= [latex]\dfrac{30}{3}[/latex] = 10

example iii):

[latex]\cfrac{\dfrac{5}{4}}{\dfrac{1}{2}}[/latex] ─ [latex]\cfrac{\dfrac{1}{2}}{\dfrac{1}{4}}[/latex]

= {[latex]\dfrac{5}{4}[/latex] ÷ [latex]\dfrac{1}{2}[/latex]} ─ {[latex]\dfrac{1}{2}[/latex] ÷ [latex]\dfrac{1}{4}[/latex]}

= {[latex]\dfrac{5}{4}[/latex] x [latex]\dfrac{2}{1}[/latex]} ─ {[latex]\dfrac{1}{2}[/latex] x [latex]\dfrac{4}{1}[/latex]}

= [latex]\dfrac{5}{2}[/latex] ─ [latex]\dfrac{2}{1}[/latex]

= [latex]\dfrac{5 – 4}{2}[/latex]

= [latex]\dfrac{1}{2}[/latex]

Dividing complex or mixed fractions:

example i):

[latex]\cfrac{\dfrac{5}{4}}{\dfrac{1}{2}}[/latex] ─ [latex]\dfrac{3 \dfrac{1}{2}}{5}[/latex]

= {[latex]\dfrac{5}{4}[/latex] ÷ [latex]\dfrac{1}{2}[/latex]} ÷ {[latex]\dfrac{7}{2}[/latex] ÷ [latex]\dfrac{5}{1}[/latex]}

= {[latex]\dfrac{5}{4}[/latex] x [latex]\dfrac{2}{1}[/latex]} ÷ {[latex]\dfrac{7}{2}[/latex] x [latex]\dfrac{1}{5}[/latex]}

= [latex]\dfrac{5}{2}[/latex] ÷ [latex]\dfrac{7}{10}[/latex]

= [latex]\dfrac{5}{2}[/latex] x [latex]\dfrac{10}{7}[/latex]

= [latex]\dfrac{25}{7}[/latex]

example ii):

[latex]\cfrac{5 \dfrac{1}{2}}{3 \dfrac{3}{4}}[/latex] ÷ [latex]3 \dfrac{1}{3}[/latex]

= {[latex]\dfrac{11}{2}[/latex] ÷ [latex]\dfrac{15}{4}[/latex]} ÷ [latex]\dfrac{10}{3}[/latex]

= {[latex]\dfrac{11}{2}[/latex] x [latex]\dfrac{4}{15}[/latex]} ÷ [latex]\dfrac{10}{3}[/latex]

= [latex]\dfrac{22}{15}[/latex] x [latex]\dfrac{3}{10}[/latex]

= [latex]\dfrac{22}{50}[/latex]

= [latex]\dfrac{11}{25}[/latex]

example iii):

[latex]\cfrac{\dfrac{1}{4}}{\dfrac{3}{5}}[/latex] ÷ [latex]\cfrac{\dfrac{4}{3}}{\dfrac{2}{5}}[/latex]

= {[latex]\dfrac{1}{4}[/latex] ÷ [latex]\dfrac{3}{5}[/latex]} ÷ {[latex]\dfrac{4}{3}[/latex] ÷ [latex]\dfrac{2}{5}[/latex]}

= {[latex]\dfrac{1}{4}[/latex] x [latex]\dfrac{5}{3}[/latex]} ÷ {[latex]\dfrac{4}{3}[/latex] x [latex]\dfrac{5}{2}[/latex]}

= [latex]\dfrac{5}{12}[/latex] ÷ [latex]\dfrac{10}{3}[/latex]

= [latex]\dfrac{5}{12}[/latex] x [latex]\dfrac{3}{10}[/latex]

= [latex]\dfrac{1}{8}[/latex]

Multiplying complex or mixed fraction:

example i):

[latex]\cfrac{\dfrac{6}{5}}{\dfrac{1}{3}}[/latex] x [latex]\dfrac{5 \dfrac{1}{2}}{3}[/latex]

= {[latex]\dfrac{6}{5}[/latex] ÷ [latex]\dfrac{1}{3}[/latex]} x {[latex]\dfrac{11}{2}[/latex] ÷ [latex]\dfrac{3}{1}[/latex]}

= {[latex]\dfrac{6}{5}[/latex] x [latex]\dfrac{3}{1}[/latex]} x {[latex]\dfrac{11}{2}[/latex] x [latex]\dfrac{1}{3}[/latex]}

= [latex]\dfrac{18}{5}[/latex] x [latex]\dfrac{11}{6}[/latex]

= [latex]\dfrac{33}{5}[/latex]

example ii):

[latex]\cfrac{2 \dfrac{1}{5}}{6 \dfrac{3}{4}}[/latex] x [latex]1 \dfrac{1}{3}[/latex]

= {[latex]\dfrac{11}{5}[/latex] ÷ [latex]\dfrac{27}{4}[/latex]} x [latex]\dfrac{4}{3}[/latex]

= {[latex]\dfrac{11}{5}[/latex] x [latex]\dfrac{4}{27}[/latex]} x [latex]\dfrac{4}{3}[/latex]

= [latex]\dfrac{44}{135}[/latex] x [latex]\dfrac{4}{3}[/latex]

= [latex]\dfrac{176}{405}[/latex]

example iii):

[latex]\cfrac{\dfrac{3}{4}}{\dfrac{1}{5}}[/latex] x [latex]\cfrac{\dfrac{2}{3}}{\dfrac{7}{5}}[/latex]

= {[latex]\dfrac{3}{4}[/latex] ÷ [latex]\dfrac{1}{5}[/latex]} x {[latex]\dfrac{2}{3}[/latex] ÷ [latex]\dfrac{7}{5}[/latex]}

= {[latex]\dfrac{3}{4}[/latex] x [latex]\dfrac{5}{1}[/latex]} x {[latex]\dfrac{2}{3}[/latex] x [latex]\dfrac{5}{7}[/latex]}

= [latex]\dfrac{15}{4}[/latex] x [latex]\dfrac{10}{21}[/latex]

= [latex]\dfrac{25}{14}[/latex]

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