Fraction in lowest phrases is mentioned right here.
If numerator and denominator of a fraction don’t have any frequent issue apart from 1(one), then the fraction is alleged to be in its easy type or in lowest time period.
In different phrases, a fraction is in its lowest phrases or in lowest type, if the HCF of its numerator and denominator is 1.
Contemplate the equal fractions:
(frac{2}{3}), (frac{4}{6}), (frac{6}{9}), (frac{8}{12}), (frac{10}{15}) ……..
That’s , (frac{10 ÷ 5}{15 ÷ 5}) = (frac{2}{3}); (frac{10}{15}) = (frac{2}{3})
(frac{8 ÷ 4}{12 ÷ 4}) = (frac{2}{3}); (frac{8}{12}) = (frac{2}{3})
(frac{6 ÷ 3}{9 ÷ 3}) = (frac{2}{3}); (frac{6}{9}) = (frac{2}{3})
(frac{2}{3}) is the only type of the fraction (frac{10}{15}) or (frac{8}{12}) or (frac{6}{9})
A fraction is within the lowest phrases if the one frequent issue of the numerator and denominator is 1.
Observe the fractions represented by the coloured portion in
the next figures.
In determine A coloured half is represented by fraction (frac{8}{16}).
The coloured half in determine B is represented by fraction (frac{4}{8}).
In determine C the coloured half represents the fraction (frac{2}{4}) and
In determine D coloured half represents (frac{1}{2}).
When numerator and denominator of fraction (frac{8}{16}) are divided by 2. We get (frac{4}{8}) and in the identical means (frac{4}{8}) offers (frac{2}{4}) after which (frac{1}{2}).
So, we discover that (frac{8}{16}), (frac{4}{8}), (frac{2}{4}) are equal to fraction for (frac{1}{2}). Thus, (frac{1}{2}) is the only or lowest type of all its equal fractions like (frac{2}{4}), (frac{4}{8}), (frac{8}{16}), (frac{16}{32}), (frac{32}{64}), …… and many others.
Now, if we take all of the components of the numerator 8 and denominator 16 of the fraction (frac{8}{16}), we get the next:
All components of 8 are 1, 2, 4, 8.
All components of 16 are 1, 2, 4, 8, 16.
We discover that highest frequent issue (HCF) of 8 and 16 is 8.
On dividing each numerator and denominator by highest frequent issue we get (frac{1}{2}).
Since, each numerator and denominator of fraction (frac{1}{2}) don’t have any frequent issue apart from 1, we are saying that the fraction (frac{1}{2}) is in its lowest phrases or easiest type.
There are two strategies to cut back a given fraction to its easiest type, viz., H.C.F. Methodology and Prime Factorization Methodology.
H.C.F. Methodology
Discover the H.C.F. of the numerator and denominator of the given fraction.
So as to cut back a fraction to its lowest phrases, we divide its numerator and denominator by their HCF.
Instance to cut back a fraction in lowest time period, utilizing H.C.F. Methodology:
1. Cut back the fraction ²¹/₅₆ to its easiest type.
Resolution:
Due to this fact H.C.F. of 21 and 56 is 7.
We now divide the numerator and denominator of the given fraction by 7.
²¹/₅₆ = (frac{21 ÷ 7}{56 ÷ 7}) = ³/₈.
2. Cut back ⁴⁸/₆₄ to its lowest type.
Resolution:
First we discover the HCF of 48 and 64 by factorization methodology.
The components of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The components of 64: 1, 2, 4, 8, 16, 32, and 64.
The frequent components of 48 and 64 are: 1, 2, 4, 8, 12 and 16.
Due to this fact, HCF of 48 and 64 is 16.
Now ⁴⁸/₆₄ = (frac{48 ÷ 16}{64 ÷ 16})
[Dividing numerator and denominator by the HCF of 48 and 64 i.e., 16]
⇒ ⁴⁸/₆₄ = ³/₄
3. Cut back ⁴⁴/₇₂ to its lowest type.
Resolution:
First we discover the HCF of 44 and 72 by factorization methodology.
The components of 44: 1, 2, 4, 11, 22 and 44.
The components of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36.
The frequent components of 44 and 72 are: 1, 2 and 4.
Due to this fact, HCF of 44 and 72 is 4.
Now ⁴⁴/₇₂ = (frac{44 ÷ 4}{72 ÷ 4})
[Dividing numerator and denominator by the HCF of 44 and 72 i.e., 4]
⇒ 44/72 = 11/18
To alter a fraction to lowest phrases:
4. Cut back (frac{10}{15}) to its lowest phrases:
Resolution:
Step I:
Discover the most important frequent issue of 10 and 15.
Elements of 10: 1, 2, 5, 10
Elements of 15: 1, 3, 5, 15
Widespread components: 1, 5
H.C.F of 10 and 15 = 5
Step II:
Divide each the numerator and denominator by the H.C.F.
(frac{10 ÷ 5}{15 ÷ 5}) = (frac{2}{3})
Due to this fact, (frac{10}{15}) = (frac{2}{3}) (in its lowest phrases)
2. Cut back (frac{18}{45}) to its lowest phrases.
Resolution: H.C.F. of 18 and 45 is 3 × 3 = 9 (frac{18 ÷ 9}{45 ÷ 9}) = (frac{2}{5}) Due to this fact, (frac{18}{45}) = (frac{2}{5}) (in its lowest phrases) 
Prime Factorization Methodology
Categorical each numerator and denominator of the given fraction because the product of prime components after which cancel the frequent components from them.
Instance to cut back a fraction in lowest time period, utilizing Prime Factorization Methodology:
Cut back (frac{120}{360}) to the bottom time period.
Resolution:
120 = 2 × 2 × 2 × 3 × 5 = 1
360 2 × 2 × 2 × 3 × 3 × 5 3
Clear up Examples on Decreasing Fractions to Lowest Phrases:
1. Categorical (frac{28}{140}) within the easiest type.
Resolution:
Allow us to discover all of the components of each numerator and denominator.
Elements of 28 are 1, 2, 4, 7, 14, 28
Elements of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140
The best frequent issue is 28. Now dividing each numerator and denominator by 28, we get (frac{1}{5}). The numerator 1 and denominator 5 don’t have any frequent components apart from 1. So, (frac{1}{5}) is the only type of (frac{28}{140}).
2. Is (frac{48}{168}) in its easiest type?
Resolution:
Allow us to discover HCF of numerator and denominator after which divide each by the very best frequent issue.
The best frequent issue is 2 × 2 × 2 × 3 = 24
Allow us to divide each numerator and denominator by 24. We get (frac{2}{7}).
So, the fraction (frac{48}{168}) is just not in its easiest type.
Simplifying a Fraction:
3. Simplify (frac{42}{84})
Methodology I:
Steps I: Divide numerator and denominator by 2. 42 ÷ 2 = 21; 84 ÷ 2 = 42; we get (frac{21}{42}) Steps II: Divide 21 and 42 by 3. 21 ÷ 3 = 7; 42 ÷ 3 = 14; we get (frac{7}{14}) Steps III: Divide 7 and 14 by 7. 7 ÷ 7 = 1; 14 ÷ 7 = 2; we get (frac{1}{2}) Due to this fact, (frac{42}{84}) = (frac{1}{2}) 
Now divide the numerator and denominator by the H.C.F. i.e., 42
(frac{42 ÷ 42}{84 ÷ 42}) = (frac{1}{2}) in its lowest phrases.
Questions and Solutions on Cut back a Fraction to its Easiest Kind:
1. Convert the given fractions in lowest type:
(i) (frac{2}{4})
(ii) (frac{3}{9})
(iii) (frac{4}{16})
(iv) (frac{12}{15})
(v) (frac{7}{28})
(vi) (frac{6}{10})
(vii) (frac{9}{72})
(viii) (frac{24}{36})
Solutions:
1. (i) (frac{1}{2})
(ii) (frac{1}{3})
(iii) (frac{1}{4})
(iv) (frac{4}{5})
(v) (frac{1}{4})
(vi) (frac{3}{5})
(vii) (frac{1}{8})
(viii) (frac{2}{3})
2. Cut back the next fractions to their lowest phrases.
(i) (frac{12}{60})
(ii) (frac{13}{169})
(iii) (frac{7}{35})
(iv) (frac{12}{28})
(v) (frac{3}{27})
(vi) (frac{80}{100})
(vii) (frac{14}{18})
(viii) (frac{29}{58})
(ix) (frac{9}{63})
(x) (frac{90}{128})
Reply:
2. (i) (frac{1}{5})
(ii) (frac{1}{13})
(iii) (frac{1}{5})
(iv) (frac{3}{7})
(v) (frac{1}{9})
(vi) (frac{4}{5})
(vii) (frac{7}{9})
(viii) (frac{1}{2})
(ix) (frac{1}{7})
(x) (frac{45}{64})
3. Write the fraction which is within the lowest phrases in every set of equal fractions.
(i) [(frac{15}{65}), (frac{3}{13}), (frac{30}{130})]
(ii) [(frac{1}{9}), (frac{8}{72}), (frac{5}{45})]
(iii) [(frac{50}{70}), (frac{5}{7}), (frac{25}{35})]
(iv) [(frac{3}{11}), (frac{33}{121}), (frac{15}{55})]
Reply:
3. (i) (frac{3}{13})
(ii) (frac{1}{9})
(iii) (frac{5}{7})
(iv) (frac{3}{11})
4. State true or false:
(i) (frac{5}{8}) = (frac{55}{8})
(ii) (frac{6}{48}) = (frac{1}{8})
(iii) (frac{6}{9}) = (frac{48}{75})
(iv) (frac{7}{8}) = (frac{9}{10})
(v) (frac{8}{6}) = (frac{28}{21})
Reply:
4. (i) False
(ii) True
(iii) False
(iv) False
(v) False
5. Match the given fractions:
(i) (frac{12}{15}) (ii) (frac{6}{9}) (iii) (frac{8}{36}) (iv) (frac{24}{32}) (v) (frac{15}{25}) 
(a) (frac{3}{4}) (b) (frac{2}{9}) (c) (frac{3}{5}) (d) (frac{4}{5}) (e) (frac{2}{3}) 
Solutions:
5.
(i) (frac{12}{15}) (ii) (frac{6}{9}) (iii) (frac{8}{36}) (iv) (frac{24}{32}) (v) (frac{15}{25}) 
(d) (frac{4}{5}) (e) (frac{2}{3}) (b) (frac{2}{9}) (a) (frac{3}{4}) (c) (frac{3}{5}) 
6. Write the fraction for given statements and convert them
to the bottom type.
Assertion 
Fraction 
Lowest Kind 
(i) Ten minutes to an hour 

(ii) Amy ate 3 out of the 9 slices of a pizza 

(iii) Eight months to a yr 

(iv) Kelly coloured 4 out of 12 elements of a drawing 

(v) Jack works for 8 hours in a day. 
Solutions:
6.
Assertion 
Fraction 
Lowest Kind 
(i) Ten minutes to an hour 
(frac{50}{60}) 
(frac{5}{6}) 
(ii) Amy ate 3 out of the 9 slices of a pizza 
(frac{3}{9}) 
(frac{1}{3}) 
(iii) Eight months to a yr 
(frac{8}{12}) 
(frac{2}{3}) 
(iv) Kelly coloured 4 out of 12 elements of a drawing 
(frac{4}{12}) 
(frac{1}{3}) 
(v) Jack works for 8 hours in a day. 
(frac{8}{24}) 
(frac{1}{3}) 
7. Give the fraction of the coloured determine and convert in
the bottom type.
Determine 
Fraction 
Lowest Kind 

(i) 

(ii) 

(iii) 

(iv) 
7.
Solutions:
Determine 
Fraction 
Lowest Kind 

(i) 
(frac{2}{8}) 
(frac{1}{4}) 

(ii) 
(frac{4}{8}) 
(frac{1}{2}) 

(iii) 
(frac{6}{12}) 
(frac{1}{2}) 

(iv) 
(frac{2}{6}) 
(frac{1}{3}) 
8. Simplify the next fractions:
(i) (frac{75}{80})
(ii) (frac{12}{20})
(iii) (frac{25}{45})
(iv) (frac{18}{24})
(v) (frac{125}{500})
Reply:
8. (i) (frac{15}{16})
(ii) (frac{3}{5})
(iii) (frac{5}{9})
(iv) (frac{3}{4})
(v) (frac{1}{4})
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