# Division of Rational Numbers

To learn division of rational numbers let us recall how to divide a fraction by another fraction. We know division of fractions is the inverse of multiplication.

Similarly, in case of rational number also, division is the inverse of multiplication as defined below:

Division: If m and n two rational numbers such that n ≠ 0, then the result of dividing m by n is the rational number obtained on multiplying m by the  reciprocal of n.

When x is divided by y, we write m ÷ n. Thus m ÷ n = m × 1/n.

If w/x and y/z are two rational numbers such that y/z ≠ 0, then

w/x ÷ y/z = w/x × (y/z)^-1 = w/x × z/y

Dividend: The number to be divided is called the dividend.

Divisor: The number which divides the dividend is called the divisor.

Quotient: When dividend is divided by the divisor, the result of the division is called the quotient.

If w/x is divided by y/z, then w/x is the dividend,  y/z is the divisor and w/x ÷ y/z = w/x × z/y is the quotient.

Note: It should be noted that division by 0 is not defined.

Examples on division of rational numbers:

1. Divide:

(i) 9/16 by 5/8

(ii) -6/25 by 3/5

(iii) 11/24 by -5/8

(iv) -9/40 by -3/8

Solution:

(i) 9/16 ÷ 5/8

= 9/16 × 8/5

= (9 × 8)/(16 × 5)

= 72/80

= 9/10

(ii) -6/25 ÷ 3/5

= -6/25 × 5/3

= {(-6) × 5}/(25 × 3)

= -30/75

= -2/5

(iii) 11/24 ÷ (-5)/8

= 11/24 × 8/(-5)

= (11 × 8)/{24 × (-5)}

= 88/-120

= -11/15

(iv) -9/40 ÷ (-3)/8

= (-9)/40 × 8/(-3)

= {(-9) × 8}/(40 × (-3))

= -72/-120

= 3/5

2. The product of two numbers is -28/27. If one of the numbers is -4/9, find the other.

Solution:

Let the other number be x.

x × (-4)/9 = -28/27

x = (-28)/27 ÷ (-4)/9

x = (-28)/27 × 9/-4

x = {(-28) × 9}/{27 × (-4)}

x = -(28 × 9)/-(27 × 4)

x = (287 × 91 )/(273 × 41 )

x = 7/3

Hence, the other number is 7/3.

3. Fill in the blanks: 27/16 ÷ (_____) = -15/8

Solution:

Let 27/16 ÷ (a/b) = -15/8.

27/16 × b/a = -15/8

b/a = -15/8 × 16/27 = -10/9

a/b = 9/-10 = -9/10

Hence, the missing number is -9/10.

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

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers