Division Of Rational Numbers – Rules

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is the integer and the denominator is a non-zero integer.

The steps for dividing rational numbers are as follows:

Dividing rational numbers involves a few straightforward steps.

1.Understanding Rational Numbers

A rational number can be represented as a/b, where: a (the numerator) and b (the denominator) are integers and b≠0.

2. Reciprocal (Multiplicative Inverse)

The reciprocal of a rational number a/b is b/a
​, provided a≠0.

The reciprocal essentially flips the numerator and the denominator.

3. Division as Multiplication

Dividing by a rational number is the same as multiplying by its reciprocal. For instance, to divide

a/b by c/d, we multiply

a/b by the reciprocal of

c/d, which is d/c.

4. Steps for Division

To divide two rational numbers

a/b and c/d :

Find the reciprocal of the divisor c/d. The reciprocal is d/c.

Multiply the dividend

a\b by the reciprocal of the divisor d/c.

5.Multiplication of Rational Numbers

To multiply two rational numbers

a/b and c/d:

Multiply the numerators together: a×c.

Multiply the denominators together: b×d.

6. Simplify the Result

If possible, simplify the resulting fraction by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it.


Let’s go through an example step-by-step.


Divide 3/4 by 2/5.


Reciprocal of 2/5 is 5/2

Multiply 3/4 by 5/2:

Multiply the numerators: 3×5


Multiply the denominators: 4×2


So, (3/4)/(5×2) = 15/8

Simplifying the Result

The fraction 15/8 is already in its simplest form since 15 and 8 have no common factors other than 1.

Thus, 3/4÷2/5 = 15/8

General Rules Summary:

To divide a/b by c/d, multiply a/b by d/c.
Reciprocal of c/d is d/c.

Multiply the numerators and the denominators.
Simplify the resulting fraction if possible.

To divide rational numbers, multiply the first rational number by the reciprocal of the second rational number and simplify the result if possible.

Additional Notes:

Division by zero is undefined. Hence, ensure that the divisor (both the original denominator and the reciprocal numerator) is not zero.
When simplifying fractions, always look for the greatest common divisor (GCD) to reduce the fraction to its simplest form.

Keep track of signs (positive or negative) as we perform the operations, since dividing negative numbers follows the same rules as for integers (a negative divided by a positive is negative, a negative divided by a negative is positive, and so on).
By following these steps and rules, we can accurately divide rational numbers.

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