\(\frac{2}{3}\times\frac{5}{9}=\frac{2\times5}{3\times9}=\frac{10}{27}.\)

\(\frac{-3}{-7},\frac{-5}{-11},\frac{-7}{-13},\frac{-9}{-17},\frac{-7}{-15}\)

Let the total volume = 1; Body floats volume = \(\frac{2}{9}\)
Body submerged volume = \(\frac{1}{1}-\frac{2}{9}\)
\(\because\)  LCM of 1 and 9 = 9
On multiplying numerator and denominator by their LCM, we get
\(\frac{1\times9}{1\times9}=\frac{9}{9}\)
\(\therefore\) Body submerged volume =\(\frac{9}{9}-\frac{2}{9}=\frac{7}{9}\)
Ratio

On multiplying both sides by 9, we get
\(\frac{7}{9}\times9:\frac{2}{9}\times9=7:2\)
In rational number form =\(\frac{7}{2}\)

\(\frac{5}{11}\div\left(\frac{-3}{8}\right)\)=\(\frac{5}{11}\times\left(\frac{8}{-3}\right)\)
=\(\frac{5\times8}{11\times(-3)}\)
=\(\frac{40}{-33}\)
=\(\frac{40}{-33}\)
(b) \(\frac { 7 }{ 5 } \div \left( \frac { -2 }{ 3 } \right) =\frac { 7 }{ 5 } \times \left( \frac { 3 }{ -2 } \right) \)
=\(\frac{7\times3}{5\times(-2)}\)
=-\(\frac{21}{10}\)

(a) \(\frac { 6 }{ 7 } \times \left( -\frac { 3 }{ 5 } \right) =\frac { 6\times (-3) }{ 7\times 5 } \)
=\(\frac{18}{35}\)
(b) \(\left( -\frac { 11 }{ 4 } \right) \times \frac { 5 }{ 7 } =\frac { (-11)\times 5 }{ 4\times 7 } \)
=-\(\frac{55}{28}\)

Given rational numbers are \(\frac { 49 }{ 3 } \) and \(\frac { 7 }{ 3 } \). 
To find,\(\frac { 49 }{ 3 } \)\(\div \)\(\frac { 7 }{ 3 } \).Since the reciprocal of \(\\ \frac { 7 }{ 3 } \) is \(\\ \frac { 3 }{ 7 } \) \(\Rightarrow \frac { 49 }{ 3 } \div \frac { 7 }{ 3 } =\frac { 49 }{ 3 } \times \frac { 3 }{ 7 } =\frac { 49\times 3 }{ 3\times 7 } =7\)

We have,\(\frac { -4 }{ 5 } \div (-3) \) = \(\frac { -4 }{ 5 } \div (-3)=\frac { -4 }{ 5 } \div \frac { (-3) }{ 1 } =\frac { -4 }{ 5 } \times \)Reciprocal of (-3)
\(=\frac { -4 }{ 5 } \times \frac { 1 }{ (-3) } =\frac { (-4)\times 1 }{ 5\times (-3) } =\frac { -4 }{ -15 } \)

\(-2\frac { 1 }{ 9 } -6=-\frac { 73 }{ 9 } =-8\frac { 1 }{ 9 } \)

\(\frac { 30 }{ 12 } \)

 \(\frac { 5 }{ 7 } \) is a positive rational number.

8

Given rational number is  \(\frac { -21 }{ 27 } \) 
For standard /simplest form, \(\frac { -21\div 3 }{ 27\div 3 } =\frac { -7 }{ 9 } \)
\(\frac { -21\div 3 }{ 27\div 3 } =\frac { -7 }{ 9 } \) [∵HCF of 21 and 27 is 3]

The standard form of \(\frac { -21 }{ 27 } \) is \(\frac { -7 }{ 9 } \)

\(\frac { -3 }{ -11 } \) is a positive rational number 
\(\because \) Both 3 and 11 have negative signs 

Given ,  \(\frac { 5}{ -6 } \)
Rational number with positive denominator is \(\frac { 5}{ 6 } \)

Given, \(\frac { -4 }{ -5 } \)
Rational number with positive denominator is \(\frac { 4 }{ 5 } \)