Verify each of the following: (i) (5/7 x 12/13) x 7/18 = 5/7 x (12/13 x 7/18)

Question

Verify each of the following:

(i) \((\frac{5}{7}\times\frac{12}{13})\times\frac{7}{18}=\frac{5}{7}\times(\frac{12}{13}\times\frac{7}{18})\)

(ii) \(\frac{-13}{24}\times(\frac{-12}{5}\times\frac{35}{36})=(\frac{-13}{24}\times\frac{-12}{5})\times\frac{35}{36}\)

(iii) \((\frac{-9}{5}\times\frac{-10}{3}) \times\frac{21}{-4}=\frac{-9}{5}\times(\frac{-10}{3}\times\frac{21}{-4})\)

Answer 1

(i)

\((\frac{5}{7}\times\frac{12}{13})\times\frac{7}{18}=\frac{5}{7}\times(\frac{12}{13}\times\frac{7}{18})\)

LHS = \((\frac{5}{7}\times\frac{12}{13})\times\frac{7}{18}\)

= \(\frac{5\times12}{7\times13}\times\frac{7}{18}\)

= \(\frac{60}{91}\times\frac{7}{18}\)

= \(\frac{60\times7}{91\times18}\)

= \(\frac{420}{1638}\)

In lowest terms,

\(\frac{420}{1638}= \frac{420 ÷42}{1638 ÷42}=\frac{10}{13}\)

RHS = \(\frac{5}{7}\times(\frac{12}{13}\times\frac{7}{18})\)

= \(\frac{5}{7}\times\frac{12\times7}{13\times18}\)

= \(\frac{5}{7}\times\frac{84}{234}\)

= \(\frac{420}{1638}\)

In lowest terms,

\(\frac{420}{1638}= \frac{420 ÷42}{1638 ÷42}=\frac{10}{13}\)

LHS=RHS

(ii)

\(\frac{-13}{24}\times(\frac{-12}{5}\times\frac{35}{36}=(\frac{-13}{24}\times\frac{-12}{5})\times\frac{35}{36}\)

LHS = \(\frac{-13}{24}\times(\frac{-12}{5}\times\frac{35}{36})\)

= \(\frac{-13}{24}\times\frac{-12\times35}{5\times36}\)

= \(\frac{-13}{24}\times\frac{-420}{180}\)

= \(\frac{60\times7}{24\times180}\)

= \(\frac{5460}{4320}\)

In lowest terms,

\(\frac{546}{432}= \frac{5460 ÷10}{4320 ÷10}=\frac{546}{432}\)

Further,

\(\frac{546}{432}= \frac{546 ÷6}{432 ÷6}=\frac{91}{72}\)

RHS = \((\frac{-13}{24}\times\frac{-12}{5})\times\frac{35}{36}\)

= \(\frac{-13\times-12}{24\times5}\times\frac{35}{36}\)

= \(\frac{156}{120}\times\frac{35}{36}\)

= \(\frac{5460}{4320}\)

In lowest terms,

\(\frac{546}{432}= \frac{5460 ÷10}{4320 ÷10}=\frac{546}{432}\)

Further,

\(\frac{546}{432}= \frac{546 ÷6}{432 ÷6}=\frac{91}{72}\)

LHS=RHS

(iii)

\((\frac{-9}{5}\times\frac{-10}{3}) \times\frac{21}{-4}=\frac{-9}{5}\times(\frac{-10}{3}\times\frac{21}{-4})\)

LHS = \((\frac{-9}{5}\times\frac{-10}{3})\times\frac{21}{-4}\)

= \(\frac{-9\times-10}{5\times3}\times\frac{21}{-4}\)

= \(\frac{90}{15}\times\frac{21}{-4}\)

= \(\frac{90\times21}{15\times-4}\)

= \(\frac{1890}{-60}= \frac{1890\times-1}{-60\times-1}=\frac{-1890}{6}\)

In lowest terms,

\(\frac{-1890}{60}= \frac{-1890\times-10}{60\times-10}=\frac{-189}{6}\)

RHS = \(\frac{-9}{5}\times(\frac{-10}{3}\times\frac{21}{-4})\)

= \(\frac{-9}{5}\times\frac{-10\times21}{3\times-4}\)

= \(\frac{-9}{5}\times\frac{-210}{-12}\)

= \(\frac{1890}{-60}= \frac{1890\times-1}{-60\times-1}=\frac{-1890}{60}\)

In lowest terms,

\(\frac{-1890}{60}= \frac{-1890\times-1}{-60\times-1}=\frac{-1890}{60}\)

Further,

\(\frac{-189}{6}= \frac{-189 ÷3}{6 ÷3}=\frac{-63}{2}\)

LHS=RHS