[tex] \frac{ {a}^{2} - 16}{ {a}^{2} } \times \frac{a}{a - 4} = \frac{(a - 4)(a + 4)}{ {a}^{2} } \times \frac{a}{a - 4} = \frac{a + 4}{a} = (\frac{1}{20} + 4) \div \frac{1}{20} = \frac{81}{20} \times 20 = 81[/tex]
[tex] \frac{ {b}^{2} }{16 {a}^{2} - 49 {b}^{2} } \div \frac{b}{4a - 7b} = \frac{ {b}^{2} }{(4a - 7b)(4a + 7b)} \times \frac{4a - 7b}{b} = \frac{b}{4a + 7b} = \frac{8}{9} \div (4 \times \frac{2}{3} + 7 \times \frac{8}{9} ) = \\ \\ = \frac{8}{9} \div ( \frac{8}{3} + \frac{56}{9} ) = \frac{8}{9} \div ( \frac{24}{9} + \frac{56}{9} ) = \frac{8}{9} \div ( \frac{80}{9} ) = \frac{8}{9} \times \frac{9}{80} = \frac{1}{10} [/tex]
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[tex] \frac{ {a}^{2} - 16}{ {a}^{2} } \times \frac{a}{a - 4} = \frac{(a - 4)(a + 4)}{ {a}^{2} } \times \frac{a}{a - 4} = \frac{a + 4}{a} = (\frac{1}{20} + 4) \div \frac{1}{20} = \frac{81}{20} \times 20 = 81[/tex]
[tex] \frac{ {b}^{2} }{16 {a}^{2} - 49 {b}^{2} } \div \frac{b}{4a - 7b} = \frac{ {b}^{2} }{(4a - 7b)(4a + 7b)} \times \frac{4a - 7b}{b} = \frac{b}{4a + 7b} = \frac{8}{9} \div (4 \times \frac{2}{3} + 7 \times \frac{8}{9} ) = \\ \\ = \frac{8}{9} \div ( \frac{8}{3} + \frac{56}{9} ) = \frac{8}{9} \div ( \frac{24}{9} + \frac{56}{9} ) = \frac{8}{9} \div ( \frac{80}{9} ) = \frac{8}{9} \times \frac{9}{80} = \frac{1}{10} [/tex]