For the multiplication of rational numbers, we take the numerators multiplication and the denominators multiplication and divide the numerator multiplication by denominator multiplication. Simplify the obtained result to get the multiplication of rational numbers. In this article we will cover the multiplication of rational numbers with basics of rational number. Also, we will discuss how to multiply rational numbers and solve some examples related to it.

What are Rational Numbers?

Numbers of the type p/q, where p and q are integers and q ≠ 0, are known as rational numbers.

p is the numerator, and q is the denominator. Rational numbers are represented by Q.

Rational number examples: 2/7, 5/9, 10/3, etc.

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Multiplication of Rational Numbers

Rational numbers, when multiplied results in a rational number. It is the ratio of the multiplication of numerators of given rational numbers by the multiplication of denominators of given rational numbers. Simplify the obtained number to get the product of given rational numbers. The multiplication of rational numbers follows commutative property as well as associative property.

Multiplication of Rational Numbers Formula

Let two rational numbers w/x and y/z then, the multiplication of two rational numbers is given by:

(w / x) × (y / z) = (w × y) / (x × z)

How to Multiply Rational Numbers

The steps to find rational numbers multiplication are given below.

• Firstly, find the product of all the numbers in the numerators of given rational numbers.

• Secondly, find the product of all the numbers in the denominators of given rational numbers.

• To find the overall multiplication of rational numbers put the resultant number obtained in first step in the numerator and the resultant obtained with second step in the denominator.

• Then, find the simplified form of obtained number.

• The number obtained represents the multiplication of rational numbers.

Solved Examples on Multiplication of Rational Numbers

Example 1: Multiply the two rational numbers: 15/4, -8/5.

Solution:

Multiply numerators of both rational numbers = 15 × (-8) = -120

Multiply denominators of both rational numbers = 4 × 5 = 20

(15/4) × (-8/5) = [15 × (-8)] / [4 × 5]

(15/4) × (-8/5) = -120 / 20

Simplify the above number we get

(15/4) × (-8/5) = -6 / 1

(15/4) × (-8/5) = -6

Example 2: Find the length of rectangle given the area of rectangle is 63/ 50 sq. units and breadth of rectangle is 3/ 100 units.

Solution:

Area of rectangle = length × breadth

Let L be the length of rectangle.

63 / 50 = L × (3/10)

L = (63/50) × (10/3)

L = 21 / 5 units

The length of given rectangle is 21/ 5 units.

Example 3: If the product of two rational numbers is -24/25 and one rational Number is – 6/5. Find the second rational number.

Solution:

Let second number be y.

The multiplication of given rational numbers is:

(-6/5) × y = -24 / 25

y = (-24/25) × (-5/6)

y = 4 / 5

So, the second number is 4/5.

Example 4: Prove that: (1/9 × 5/ 3) = 5/3 × 1/9.

Solution:

LHS = (1/9 × 5/ 3)

(1/9 × 5/ 3) = (1 × 5) / (9 × 3) = 5 / 27

RHS = 5/3 × 1/9

5/3 × 1/9 = (5 × 1) / (3 × 9) = 5/ 27

LHS = RHS

Hence Proved

Example 5: Prove the associative law for multiplication of rational numbers (-21/2 × 3/ 5) × -7/ 6 = -21/2 × (3/ 5 × -7/ 6).

Solution:

LHS = (-21/2 × 3/ 5) × -7/ 6

(-21/2 × 3/ 5) × -7/ 6 = [(-21 × 3) / (2 × 5)] × -7/ 6

(-21/2 × 3/ 5) × -7/ 6 = (-63 / 10) × -7/ 6

(-21/2 × 3/ 5) × -7/ 6 = [(-63 × -7) / (10 × 6)]

(-21/2 × 3/ 5) × -7/ 6 = 441 / 60

RHS = -21/2 × (3/ 5 × -7/ 6)

-21/2 × (3/ 5 × -7/ 6) = -21/2 × [(3 × -7) / (6 × 5)]

-21/2 × (3/ 5 × -7/ 6) = -21/2 × (-21/ 30)

-21/2 × (3/ 5 × -7/ 6) = (-21 × -21) / (2 × 30)

-21/2 × (3/ 5 × -7/ 6) = 441 / 60

LHS = RHS

Hence Proved

Example 6: Calculate the multiplication of rational numbers given the multiplication of their numerators is 10 and the multiplication of denominators is -32.

Solution:

To calculate the multiplication of rational numbers we use formula:

Multiplication of Rational numbers = Multiplication of numerators / Multiplication of denominators

Multiplication of Rational numbers = 10 / (-32)

Multiplication of Rational numbers = -10 / 32

On simplifying we get,

Multiplication of Rational numbers = -5 / 16

Practice Questions on Multiplication of Rational Numbers

Q1. Multiply the rational numbers: -4/5 and 2/3.

Q2. If the first number be -3/5 and multiplication of two rational numbers is 6/35, then find the second number.

Q3. Find the breadth of rectangle given that length of rectangle is 50 / 9 units and area of rectangle is 1000/99 sq. units.

Q4. Prove that: (7/3 × 2/ 9) = 2/9 × 7/3.

Q5. Prove the associative law for multiplication of rational numbers (-2/13 × 4/ 7) × 5/ 4 = -2/13 × (4/ 7 × 5/ 4).

Q6. Calculate the multiplication of rational numbers given the product of their numerators is -12 and the product of denominators is 25.

FAQs on Multiplication of Rational Numbers

How Can You Multiply Rational Numbers?

To multiply rational numbers, we first multiply all numerators of rational numbers and put it in the numerator and then we multiply all denominators and put it in the denominator. The obtained result gives the multiplication of rational numbers.

Which Rational Number has No Reciprocal?

The rational number 0 has no reciprocal.

Can You Multiply Two Rational Numbers?

Yes, we can multiply two rational number which gives a rational number as result of multiplication.

What is the Multiplicative of Rational Numbers?

The multiplicative of rational numbers is 1.