Multiplication of Rational Numbers

To learn multiplication of rational numbers let us recall how to multiply two fractions. The product of two given fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the given fractions.

In other words, product of two given fractions = product of their numerators/product of their denominators

Similarly, we will follow the same rule for the product of rational numbers.

Therefore, product of two rational numbers = product of their numerators/product of their denominators.

Thus, if a/b and c/d are any two rational numbers, then

a/b × c/d = a × c/b × d

Solved examples on multiplication of rational numbers:

1. Multiply 2/7 by 3/5

Solution:

2/7 × 3/5

= 2 × 3/7 × 5

= 6/35

2. Multiply 5/9 by (-3/4)

Solution:

5/9 × (-3/4)

= 5 × -3/9 × 4

= -15/36

= -5/12

3. Multiply (-7/6) by 5

Solution:

(-7/6) × 5

= (-7/6) × 5/1

= -7 × 5/6 × 1

= -35/6

4. Find each of the following products:

(i) -3/7 × 14/5

(ii) 13/6 × -18/91

(iii) -11/9 × -51/44

Solution:

(i) -3/7 × 14/5

= {(-3) × 14/(7 × 5)

$Multiplication of Rational Numbers$

= -6/5

(ii) 13/6 × -18/91

= {13 × (-18)}/(6 × 91)

$Multiplication of Rational Numbers$

= -3/7

(iii) -11/9 × 51/44

= {(-11) × (-51)}/(9 × 44)

$Multiplication of Rational Numbers$

= 17/12

5. Verify that:

(i) (-3/16 × 8/15) = (8/15 × (-3)/16)

(ii) 5/6 × {(-4)/5 + (-7)/10} = {5/6 × (-4)/5} + {5/6 × (-7)/10}

Solution:

(i) LHS = ((-3)/16 × 8/15) = {(-3) × 8}/(16 × 15) = -24/240 = -1/10

RHS = (8/15 × (-3)/16) = {8 × (-3)}/(15 × 16) = -24/240 = -1/10

Therefore, LHS = RHS.

Hence, ((-3)/16 × 8/15) = (8/15 × (-3)/16)

(ii) LHS = 5/6 × {-4/7 + (-7)/10} = 5/6 × [{(-8) + (-7)}/10}

= 5/6 × (-15)/10

= 5/6 × (-3)/2 = {5 × (-3)}/(6 × 2) = -15/12 = -5/4

RHS = {5/6 × -4/5} + {5/6 ×(-7)/10}

= {5 × (-4)/(6 × 5) + { 5 × (-7)}/(6 × 10) = -20/30 + (-35)/60

= (-2)/3 + (-7)/12

= {(-8) + (-7) }/ 12 = (-15)/12 = (-5)/4

Therefore, LHS = RHS

Hence, 5/6 × (-4/5 + (-7)/10) = {5/6 × (-4)/5} + (5/6 × (-7)/10)