Q.1: Using appropriate properties find:
(i) \( – \frac{2}{3}\times\frac{3}{5} + \frac{5}{2} – \frac{3}{5}\times\frac{1}{6} \)
Prerequisites to solve these questions:
1. Properties of Addition of Rational Numbers 2. Distributivity of multiplication over addition 3. Least common Multiple
{ By using commutativity property of addition of rational numbers, \(\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}\)}
=\( – \frac{2}{3} \times \frac{3}{5} – \frac{3}{5} \times \frac{1}{6} + \frac{5}{2} \)
=\( (\frac{3}{5} \times -\frac{2}{3})\) – \( (\frac{3}{5} \times \frac{1}{6})\) + \( \frac{5}{2} \)
Now, by taking \(\frac{3}{5}\) common from the first two terms we get,
=\(\frac{3}{5} \times (- \frac{2}{3} + \frac{1}{6}) + \frac{5}{2} \)
{ By Distributivity of multiplication over addition, \(\frac{a}{b} \times (\frac{c}{d} + \frac{e}{f}) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f}\) }
=\(\frac{3}{5}\times\left( {\frac{{-2 \times 2 + 1}}{6}} \right) + \frac{5}{2} \) { Since, LCM of 3 and 6 is 6 and by adding two unlike fractions }
=\(\frac{3}{5} \times \left( {\frac{-5}{6}} \right) + \frac{5}{2}\)
=\( – \frac{3}{6} + \frac{5}{2}\) =\(\frac{{ – 3 + 5 \times 3}}{6}\) { Since, LCM of 2 and 6 is 6 }
=\(\frac{{ – 3 + 15}}{6}\) = \(\frac{{12}}{6}\) = \(2 \)
(ii) \( \frac{2}{5} \times ( – \frac{3}{7}) – \frac{1}{6} \times \frac{3}{2} + \frac{1}{{14}} \times \frac{2}{5} \)
{ By using commutativity property of addition of rational numbers, \(\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}\)}
\(=(\frac{2}{5} \times ( – \frac{3}{7}) + \frac{1}{{14}}) \times (\frac{2}{5} – \frac{1}{6} \times \frac{3}{2}) \){ By Distributivity of multiplication over addition, \(\frac{a}{b} \times (\frac{c}{d} + \frac{e}{f}) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f}\)}
\(=\frac{2}{5}( – \frac{3}{7} + \frac{1}{{14}}) – \frac{1}{4} \)\( = \frac{2}{5}(\frac{{ – 5}}{{14}}) – \frac{1}{4} \) { Since, LCM of 7 and 14 is 14 }
= \(\frac{ – 1}{7} – \frac{1}{4}\)
\(= \frac{ – 4 – 7}{28}\) \(= \frac{{ – 11}}{{28}}\) [ Since, LCM of 7 and 4 is 28 ]