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

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

8th Grade Math Practice 

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