Fraction in lowest terms is discussed here.
If numerator and denominator of a fraction have no common factor other than 1(one), then the fraction is said to be in its simple form or in lowest term.
In other words, a fraction is in its lowest terms or in lowest form, if the HCF of its numerator and denominator is 1.
There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method.
H.C.F. Method
Find the H.C.F. of the numerator and denominator of the given fraction.
In order to reduce a fraction to its lowest terms, we divide its numerator and denominator by their HCF.
Example to reduce a fraction in lowest term, using H.C.F. Method:
1. Reduce the fraction ²¹/₅₆ to its simplest form.
Solution:
Therefore 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. Reduce ⁴⁸/₆₄ to its lowest form.
Solution:
First we find the HCF of 48 and 64 by factorization method.
The factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The factors of 64: 1, 2, 4, 8, 16, 32, and 64.
The common factors of 48 and 64 are: 1, 2, 4, 8, 12 and 16.
Therefore, 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. Reduce ⁴⁴/₇₂ to its lowest form.
Solution:
First we find the HCF of 44 and 72 by factorization method.
The factors of 44: 1, 2, 4, 11, 22 and 44.
The factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36.
The common factors of 44 and 72 are: 1, 2 and 4.
Therefore, 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
Prime Factorization Method
Express both numerator and denominator of the given fraction as the product of prime factors and then cancel the common factors from them.
Example to reduce a fraction in lowest term, using Prime Factorization Method:
Reduce \(\frac{120}{360}\) to the lowest term.
Solution:
120 = 2 × 2 × 2 × 3 × 5 = 1
360 2 × 2 × 2 × 3 × 3 × 5 3
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