Standard form of a Rational Number

What is the standard form of a rational number?

A rational number $\frac{a}{b}$ is said to be in the standard form if b is positive, and the integers a and b have no common divisor other than 1.

How to convert a rational number into standard form?

In order to express a given rational number in the standard form, we follow the following steps:

Step I: Obtain the rational number.

Step II: See whether the denominator of the rational number is positive or not. If it is negative, multiply or divide numerator and denominator both by -1 so that denominator becomes positive.

Step III: Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step IV: Divide the numerator and denominator of the given rational number by the GCD (HCF) obtained in step III. The rational number so obtained is the standard form of the given rational number.

The following examples will illustrate the above procedure to convert a rational number into standard form.

1. Express each of the following rational numbers in the standard form:

(i) $\frac{-9}{24}$ (ii) $\frac{-14}{-35}$ (iii) $\frac{27}{-72}$ (iv) $\frac{-55}{-99}$

Solution:

(i) $\frac{-9}{24}$

The denominator of the rational number $\frac{-9}{24}$ is positive. In order to express it in standard form, we divide its numerator and denominator by the greatest common divisor of 9 and 24 is 3.

Dividing the numerator and denominator of $\frac{-9}{24}$ by 3, we get

$\frac{-9}{24}$ = $\frac{(-9) ÷ 3}{24 ÷ 3}$ = $\frac{-3}{8}$

Thus, the standard form of $\frac{-9}{24}$ is $\frac{-3}{8}$.

(ii) $\frac{-14}{-35}$

The denominator of the rational number $\frac{-14}{-35}$ is negative. So, we first make it positive.

Multiplying the numerator and denominator of $\frac{-14}{-35}$ by -1 we get

$\frac{-14}{-35}$ = $\frac{(-14) × (-1)}{(-35) × (-1)}$ = $\frac{14}{35}$

The greatest common divisor of 14 and 35 is 7.

Dividing the numerator and denominator of $\frac{14}{35}$ by 7, we get

$\frac{14}{35}$ = $\frac{14 ÷ 7}{35 ÷ 7}$ = $\frac{2}{5}$

Hence, the standard form of a rational number $\frac{-14}{-35}$ is $\frac{2}{5}$.

(iii) $\frac{27}{-72}$

The denominator of $\frac{27}{-72}$ is negative. So, we first make it positive.

Multiplying the numerator and denominator of $\frac{27}{-72}$ by -1, we have

$\frac{27}{-72}$ = $\frac{27 × (-1)}{(-72) × (-1)}$ = $\frac{-27}{72}$

The greatest common divisor of 27 and 72 is 9.

Dividing the numerator and denominator of $\frac{-27}{72}$ by 9, we get

$\frac{-27}{72}$ = $\frac{(-27) ÷ 9}{72 ÷ 9}$ = $\frac{-3}{8}$

Hence, the standard form of $\frac{27}{-72}$ is $\frac{-3}{8}$.

(iv) $\frac{-55}{-99}$

The denominator of $\frac{-55}{-99}$ is negative. So, we first make it positive.

Multiplying the numerator and denominator of $\frac{-55}{-99}$ by -1, we have

$\frac{-55}{-99}$ = $\frac{(-55) × (-1)}{(-99) × (-1)}$= $\frac{55}{99}$

The greatest common divisor of 55 and 99 is 11.

Dividing the numerator and denominator of by $\frac{55}{99}$ by 11, we get

$\frac{55}{99}$ = $\frac{55 ÷ 11}{99 ÷ 11}$ = $\frac{5}{9}$

Hence, the standard form of $\frac{-55}{-99}$ is $\frac{5}{9}$.

More examples on standard form of a rational number:

2. Express the rational number $\frac{-247}{-228}$ in the standard form:

Solution:

The denominator of $\frac{-247}{-228}$ is negative. So, we first make it positive.

Multiplying the numerator and denominator of $\frac{-247}{-228}$ by -1, we get

$\frac{-247}{-228}$ = $\frac{(-247) × (-1)}{(-228) × (-1)}$ = $\frac{247}{228}$

Now, we find the greatest common divisor of 247 and 228.

247 = 13 × 19 and 228 = 2 × 2 × 3 × 19

Clearly, the greatest common divisor of 228 and 247 is equal to 19.

Dividing the numerator and denominator of $\frac{247}{228}$ by 19, we get

$\frac{247}{228}$ = $\frac{247 ÷ 19}{228 ÷ 19}$ = 13/12

Hence, the standard form of $\frac{-247}{-228}$ is $\frac{13}{12}$.

3. Express the rational number $\frac{299}{-161}$ in the standard form:

Solution:

The denominator of $\frac{299}{-161}$ is negative. So we first make it positive.

Multiplying the numerator and denominator of $\frac{299}{-161}$ by -1, we get

$\frac{299}{-161}$ = $\frac{299 × (-1)}{(-161) × (-1)}$ = $\frac{-299}{161}$

Now, we find the greatest common divisor of 299 and 161:

299 = 13 × 23 and 161 = 7 × 23

Clearly, the greatest common divisor of 299 and 161 is equal to 23.

Dividing the numerator and denominator of $\frac{-299}{161}$
by 23 we get