6. Distributivity of multiplication over addition and subtraction

 

\(a ×\) (\(b + c\)) \(=\) (\(a × b\)) \(+\) (\(a × c\))

 

Let \(a = \frac{-9}{2}\), \(b = \frac{-5}{4}\) and \(c = \frac{1}{3}\).

 

Then, \(\frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)\) \(=\) \(\left(\frac{-9}{2} \times \frac{-5}{4} \right) +\) \(\left(\frac{-9}{2} \times \frac{1}{3} \right)\)

 

Let us verify the property.

 

Consider the LHS.

 

LHS \(= \frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)\)

 

\(= \frac{-9}{2} \times \left(\frac{-15 + 4}{12} \right)\) [LCM of \(3\) and \(4\) is \(12\)]

 

\(= \frac{-9}{2} \times \left(\frac{-11}{12} \right)\)

 

\(= \frac{99}{24}\)

 

\(= \frac{33}{8}\) ---- (\(1\))

 

Now, consider the RHS.

 

RHS \(=\) \(\left(\frac{-9}{2} \times \frac{-5}{4} \right) +\) \(\left(\frac{-9}{2} \times \frac{1}{3} \right)\)

 

\(= \frac{45}{8} + \left(\frac{-9}{6} \right)\)

 

\(= \frac{45}{8} - \frac{3}{2}\)

 

\(= \frac{45 - 12}{8}\) [LCM of \(2\) and \(8\) is \(8\)]

 

\(= \frac{33}{8}\) ---- (\(2\))

 

From equations (\(1\)) and (\(2\)), we have:

 

\(\frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)\) \(=\) \(\left(\frac{-9}{2} \times \frac{-5}{4} \right) +\) \(\left(\frac{-9}{2} \times \frac{1}{3} \right)\)

 

This implies that \(a ×\) (\(b + c\)) \(=\) (\(a × b\)) \(+\) (\(a × c\)).