UPSKILL MATH PLUS
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Learn moreDistributive property of multiplication over addition: \(a(b + c) = ab + ac\) where \(a\), \(b\) and \(c\) are rational numbers.
Distributive property of multiplication over subtraction: \(a(b - c) = ab - ac\) where \(a\), \(b\) and \(c\) are rational numbers.
Now let's see the distributive property of multiplication over addition.
Distribution of multiplication over addition:
If \(a\), \(b\) and \(c\) are three integers, then:
\(a ×\) (\(b + c\)) \(=\) (\(a × b\)) \(+\) (\(a × c\))
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 \(a ×\) (\(b + c\)) \(=\) (\(a × b\)) \(+\) (\(a × c\)).
Distribution of multiplication over subtraction:
If \(a\), \(b\) and \(c\) are three integers, then:
\(a ×\) (\(b - c\)) \(=\) (\(a × b\)) \(-\) (\(a × c\))
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 \(12\)]
\(= \frac{-9}{2} \times \left(\frac{-19}{12} \right)\)
\(= \frac{171}{24}\)
\(= \frac{57}{8}\) ---- (\(1\))
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 \(8\)]
\(= \frac{57}{8}\) ---- (\(2\))
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 \(a ×\) (\(b - c\)) \(=\) (\(a × b\)) \(-\) (\(a × c\)).
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