Multiplication of Rational Numbers

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.