What is Multiplication of Rational Numbers? (Definition, Steps and Examples) - BYJUS

# Multiplication of Rational Numbers

Rational numbers include all natural numbers, whole numbers, and integers. We have learned how to perform multiplication operations on natural numbers, whole numbers, and integers. Here we will apply the same concept to find the product of two rational numbers. ...Read MoreRead Less

## Multiplying Rational Numbers

A fraction with integer values both in the numerator and the denominator (the denominator is not zero) is a rational number. When two rational numbers are multiplied, a third rational number is produced. Use the same rules for signs as you did for integers when multiplying rational numbers. When the signs of the two numbers that are being multiplied are the same, the product is positive and when the signs are different, the product is negative.

For example, $$\frac{-1}{7}\times \frac{1}{3}=\frac{-1}{21}$$

## Solved Multiplication of Rational Numbers Examples

Example 1: Find: -2.6 x 3.6

Solution:

2.6      (1 decimal place)

x 3.6      (+1 decimal place)

____

156

780

____

9.36      (2 decimal places)

Example 2: Find: -0.6 x 6.3

Solution:

6.3      (1 decimal place)

x 0.6      (+1 decimal place)

____

3.78     (2 decimal places)

So,  -0.6 x 6.3 = -3.78

Example 3: Find $$-\frac{4}{5}\left(-\frac{2}{3}\right)=?$$

Solution:

$$\frac{4}{5}\left(\frac{2}{3}\right)=\frac{4 × 2}{5 × 3}$$

$$=\frac{8}{15}$$                                   (Multiplying the numerators and the denominators)

So, $$-\frac{4}{5}\left(-\frac{2}{3}\right)=\frac{8}{15}$$

Example 4: Find $$-\frac{1}{2}\left(-5\frac{2}{3}\right)=?$$

Solution:

$$\frac{1}{2}\left(5\frac{2}{3}\right)=\frac{1}{2}\left(\frac{17}{3}\right)$$                           (Writing the mixed number as an improper fraction)

$$=\frac{17}{6}$$                                     (Multiplying the numerators and the denominators)

So, $$-\frac{1}{2}\left(-5\frac{2}{3}\right)=\frac{17}{6}$$

Example 5: Find $$\left(-\frac{1}{6}\times \frac{4}{5}\right)\times(-6)\times \left(-\frac{1}{2}\right)=?$$

Solution: To find the product, we can use the properties of multiplication.

$$\left(-\frac{1}{6}\times \frac{4}{5}\right)\times (-6)\times \left(-\frac{1}{2}\right)=-6\times \left(-\frac{1}{6} \times \frac{4}{5}\right)\times \left(-\frac{1}{2}\right)$$            (Using the commutative property of multiplication)

$$=\left[-6\times \left(-\frac{1}{6}\right)\right]\times \frac{4}{5}\times \left(-\frac{1}{2}\right)$$                                                     (Using the associative property of multiplication)

$$=1\times \frac{4}{5}\times \left(-\frac{1}{2}\right)$$                                                                         (Using the multiplicative inverse property)

$$=\frac{4}{5}\times \left(-\frac{1}{2}\right)$$                                                                                (Using the multiplication property of one)

$$=\frac{4\times (-1)}{5\times 2}$$                                                                                     (Using multiplication and dividing the common factor 2)

$$=\frac{-2}{5}$$ or $$=-\frac{2}{5}$$                                                                            (Simplified)

Example 6: Find $$-\frac{4}{5}\times 7\frac{3}{5}\times \frac{5}{4}=?$$

Solution: To find the product, we can use the properties of multiplication.

$$-\frac{4}{5}\times 7\frac{3}{5}\times \frac{5}{4}=-\frac{4}{5}\times \frac{5}{4}\times 7\frac{3}{5}$$               (Using the commutative property of multiplication)

$$=\left(-\frac{4}{5}\times \frac{5}{4}\right)\times 7\frac {3}{5}$$                                  (Using the associative property of multiplication)

$$=-1\times 7\frac{3}{5}$$                                              (Using the multiplicative inverse property)

$$=-7\frac{3}{5}$$                                                      (Using the multiplication property of one)

$$=-\frac{35+3}{5}$$                                                  (Writing the mixed number as an improper fraction)

$$=-\frac{38}{5}$$

Example 7: Katie has a lemonade stall. Each lemonade costs $1.5. She sells 1 lemonade every hour. She has been selling lemonade for 4 hours. How much has she earned so far? Solution: Cost of 1 lemonade =$1.5

Number of hours worked = 3.5

She sells a lemonade every hour

Total amount earned = 1.5 × 4

= 6

Amount earned so far is \$6.

$$\left(\frac{1}{2}\right)\times \left(\frac{2}{1}\right)=1$$