\(Q={3\over4},R={6\over4},S={9\over4},T={-5\over2}\)

Number of above average students=\({1\over6}\) of the class student
Number of average students= \({1\over4}\)of the class student
\(\therefore\) Number of below average students  \(=1-[{1\over6}+{1\over4}]\) of the class student
\(=-[{2+3\over12}]=1-{5\over12}={7\over12}\)of the class student
Since, number of students in the class = 48
\(\therefore\)Number of below average students
\(={7\over12}\times48=28\)
So, Number of below average students= 28

We have, product of two numbers = \({-128\over75}\)
One of the numbers is \(64\over3\)
\(\therefore\)The other number =\({-128\over75}+{64\over3}={-128\over75}\times{3\over64}={-2\over25}{}\)

We have, \({-2\over3}\times{3\over5}+{5\over2}-{3\over5}\times{1\over6}={-2\over3}\times{3\over5}-{3\over5}\times{1\over6}+{5\over2}\) [by associativity]
\(={3\over5}\times{-2\over3}+{3\over5}\times{-1\over6}+{5\over2}\)         [by commutativity]
\(={3\over5}\times({-2\over3}-{1\over6})+{5\over2}\)
[by distributivity, taking \({3\over5}\)  as common factor]
\(={3\over5}\times({-4-1\over6})+{5\over2} \ \ \ \ [\because LCM \ of \ 3 \ and \ 6=6] \)
\(\\ ={3\over5}\times({-5\over6})+{5\over2}={3\over5}\times {-5\over6}+{5\over2}={-1\over2}+{5\over2}={-1+5\over2}\)
\(\\ ={4\over2}=2\)

All properties of operations on rational numbers also hold in case of integers except the following property:
a \(\div\) b is a rational number if b \(\neq\) 0 but a \(\div\)b is not necessarily an integer in case ab \(\in\) J,
All properties of operations on rational numbers also hold in case of whole numbers except the following properties:
(i) If a and b are rational numbers, then (a - b) may or may not be a whole number,
(ii) If a and b are rational numbers, then a\(\div\)b (where b \(\neq\) 0) is not necessarily a whole number,

Yes! the properties of commutativity and associativity made the calculations easier.

\(\frac{1}{2}-\frac{3}{5}=\frac{5-6}{10}=\frac{-1}{10}\)
\(\frac{3}{5}-\frac{1}{2}=\frac{6-5}{10}=\frac{1}{10}\)
\(\because \frac{-1}{10} \ne \frac{1}{10}\)
\(\therefore \frac{1}{2}-\frac{3}{5} \ne \frac{3}{5}-\frac{1}{2}\)

4 + 7 = 11;
Yes; it is a whole number.

\(\therefore\) product of two rational numbers  = \(\frac { -28 }{ 75 } \)
Any one of the ratioanl numbers = \(\frac { 14 }{ 25 } \)
\(\therefore\) the other number = \(\left[ \frac { -28 }{ 75 } \right] \div \frac { 14 }{ 25 } \)
= \(\frac { -28 }{ 75 } \times \frac { 25 }{ 14 } =\frac { -2\times 1 }{ 3\times 1 } =\frac { -2 }{ 3 } \)
Thus, the required rational number is \(\left( \frac { -2 }{ 3 } \right) \)

\(\therefore\) Reciprocal of  \(\frac { 7 }{ 8 } \) is \(\frac { 8 }{ 7 } \)
Reciprocal  of \(\frac { -2 }{ 21 } \) is \(\frac { -21 }{ 2 } \)
\(\therefore\) \(\left[ Reciprocal\ of\frac { 7 }{ 8 } \right] \times \left[ Reciprocal\ of\left( \frac { -2 }{ 21 } \right) \right] \)
= \(\frac { 8 }{ 7 } \times \left( \frac { -21 }{ 2 } \right) =\frac { 4\times (-3) }{ 1\times 1 } =-12\)

Existence of additive identity

The additive Inverse of \({17\over-3}is{17\over13}\)

\(41\over48\)

we have, -1
\(\therefore\)The multiplicative inverse of-1 is -1

We have, x =\(11\over15\)
LHS = -(-x) = -\(({11\over15})=​​​​{11\over15}=X=RHS\)
S0, - (-x) = x is verified for x =\(11\over15\)

(1) \(\rightarrow\) Additive Inverse
(2) \(\rightarrow\) Rational Inverse
(3) \(\rightarrow\) Product
(4) \(\rightarrow\) Reciprocal
(5) \(\rightarrow\) Additive
(6) \(\rightarrow\) Inverse
(7) \(\rightarrow\) Zero

Now, we have, \({1\over25},{1\over32},{1\over40},{1\over20}\)
Now, making same denominator, we get
\({1\over25}={32\over800},{1\over32}={25\over800},{1\over40}={20\over800},{1\over20}={40\over800}\)
(a) Soni hop more than Nancy with distance  
\(={40\over800}-{25\over800}={15\over800}={3\over160}Km\)
(b) Total distance covered by Seema and Megha  
\(={32\over800}+{20\over800}={52\over800}={13\over200}Km\)
(c) Clearly, Nancy walked farther than Megha.
(d) By competition, we make ourself relate with the surrounding in terms of compatibility of our quality. It helps to enrich our strength.

\(-{11\over8}\)

\(22\over7\)