Lowest Form of a Rational Number
What is the lowest form of a rational number?
A rational number a/b is said to be in the lowest form or simplest form if a and b have no common factor other than 1.
In other words, a rational number \(\frac{a}{b}\) is said to be in the simplest form, if the HCF of a and b is 1, i.e., a and b are relatively prime.
The rational number \(\frac{3}{5}\) is in the lowest form, because 3 and 5 have no common factor other than 1. However, the rational number \(\frac{18}{60}\) is not in the lowest form, because 6 is a common factor to both numerator and denominator.
How to convert a rational number into lowest form or simplest form?
Every rational number can be put in the lowest form using the following steps:
Step I: Let us obtain the rational number \(\frac{a}{b}\).
Step II: Find the HCF of a and b.
Step III: If k = 1, then \(\frac{a}{b}\) is in lowest form.
Step IV: If k ≠ 1, then \(\frac{a ÷ k}{b ÷ k}\) is the lowest form of a/b.
The following examples will illustrate the
above procedure
to convert a rational number into lowest form.
1. Determine whether the following rational numbers are in the lowest form or not.
(i) \(\frac{13}{81}\)
Solution:
We observe that 13 and 81 have no common factor, i.e., their HCF is 1.
Therefore, \(\frac{13}{81}\) is the lowest form of a rational number.
(ii) \(\frac{72}{960}\)
Solution:
We have, 24 = 2 × 2 × 2 × 3 × 3 and 320 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5
Thus, HCF of 72 and 960 is 2 × 2 × 2 × 3 = 24.
Therefore, \(\frac{72}{960}\) is not in the lowest form.
2. Express each of the following rational numbers to the lowest form.
(i) \(\frac{18}{30}\)
Solution:
We have,
18 = 2 × 3 × 3 and 30 = 2 × 3 × 5
Therefore, HCF of 18 and 30 is 2 × 3 = 6.
So, \(\frac{18}{30}\) is not in lowest form.
Now, dividing numerator and denominator of \(\frac{18}{30}\) by 6, we get
\(\frac{18}{30}\) = \(\frac{18 ÷ 6}{30 ÷ 6}\) = \(\frac{3}{5}\)
Therefore, \(\frac{3}{5}\) is the lowest form of a rational number \(\frac{18}{30}\).
(ii) \(\frac{-60}{72}\)
Solution:
We have
60 = 2 × 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3
Therefore, HCF of 60 and 72 is 2 × 2 × 3 = 12
So, \(\frac{-60}{72}\) is not in lowest form.
Dividing numerator and denominator of \(\frac{-60}{72}\) by 12, we get
\(\frac{-60}{72}\) = \(\frac{(-60) ÷ 12}{72 ÷ 12}\) = \(\frac{-5}{6}\)
Therefore, \(\frac{-5}{6}\) is the lowest form of \(\frac{-60}{72}\).
More examples on simplest form or lowest form of a rational number:
3. Express each of the following rational numbers to the simplest form.
(i) \(\frac{-24}{-84}\)
Solution:
We have, 24 = 2 × 2 × 2 × 3 and 84 = 2 × 2 × 3 × 7
Therefore, HCF of 24 and 84 is 2 × 2 × 3 = 12
Dividing numerator and denominator of \(\frac{-24}{-84}\) by 12, we get
\(\frac{-24}{-84}\) = \(\frac{(-24) ÷ 12}{(-84) ÷ 12}\) = \(\frac{-2}{-7}\)
Therefore, \(\frac{-2}{-7}\) is the simplest form of rational number \(\frac{-24}{-84}\).
(ii) \(\frac{91}{-364}\)
Solution:
We have, 91 = 7 × 13 and 364 = 2 × 2 × 7 × 13
Therefore, HCF of 91 and 364 is 13 × 7 = 91.
Dividing numerator and denominator by 91, we get
\(\frac{91}{-364}\) = \(\frac{91 ÷ 91}{(-364) ÷ 91}\) = \(\frac{1}{-4}\)
Therefore, \(\frac{1}{-4}\) is the simplest form of \(\frac{91}{-364}\).
4. Fill in the blanks:
\(\frac{90}{165}\) = \(\frac{-6}{.....}\) = \(\frac{.....}{-55}\)
Solution:
Here, 90 = 2 × 3 × 3 × 5 and 165 = 3 x 5 x 11
Therefore, HCF of 90 and 165 is 15.
So, \(\frac{90}{165}\) is not in lowest form of rational number.
Dividing numerator and denominator by 15, we get
\(\frac{90}{165}\) = \(\frac{90 ÷ 15}{165 ÷ 15}\) = \(\frac{6}{11}\)
Thus, the rational number \(\frac{90}{165}\) in the lowest form equals \(\frac{6}{11}\)
Now, (-6) ÷ 6 = -1
Therefore, \(\frac{6}{11}\) = \(\frac{6 × (-1)}{11 × (-1)}\) = \(\frac{-6}{-11}\)
Similarly, we have (-55) ÷ 11 = -5
Therefore, \(\frac{6}{11}\) = \(\frac{6 × (-5)}{11 × (-5)}\) = \(\frac{-30}{-55}\)
Hence, \(\frac{90}{165}\) = \(\frac{-6}{-11}\) = \(\frac{-30}{-55}\)
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
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Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
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Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
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Rational Expressions Involving Addition, Subtraction and Multiplication
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