There are infinite rational numbers between two points hence we cannot practically calculate the number of rational numbers skipped by the frog.

Each part: 1 Mark
(a) Four irrational numbers are $\sqrt{2},\sqrt{3},\sqrt{5}$ and $\pi$
(b) Let the other number be $x$.
 $\Rightarrow \frac{3}{4}=\frac{5}{6}\times x$
 $\Rightarrow x=\frac{3\times 6}{4\times 5}=\frac{9}{10}$

(a) Solution: 1 Mark
(b) Steps: 1 Mark
     Result: 1 Mark
(a) Firstly we simplify the fraction $\frac{9}{24}$ = $\frac{3}{8}$
Now the LCM of denominator 8 and 16 is 16.
$\frac{9}{16}+\frac{3}{8}=\frac{9+6}{16}=\frac{15}{16}$
(b) 
  
Arun walks $\frac{1}{3}$km towards East from place A
Then he walks $1\frac{2}{3}=\frac{5}{3}$km towards West, ie. in the opposite direction.
Arun's new position is $\frac{5}{3}-\frac{1}{3}=\frac{4}{3}=1\frac{1}{3}$
Arun is $1\frac{1}{3}$km towards West from place A.

(a) Explanation: 2 Marks
(b) Answer: 1 Mark
(a) $\frac{22}{7}$ is a rational number because we can write it in a form of $\frac{p}{q}$.

The actual value of $\pi$ is 3.141516........which implies that there is no end to the values after the decimal, the decimal goes on forever without repeating.
Hence we are not able to find the actual value of $\pi$ therefore we can't represent it on a number line and moreover, we can't represent it in the form of $\frac{p}{q}$, that's why it is an irrational number.
(b) Yes, the numbers $\frac{1}{3}$ and $\frac{-5}{2}$ lie on the opposite sides of 0 on the number line as one of them is positive and the other is negative.

 

(a) Answer: 1 Mark
(b) Steps: 1 Mark
     Result: 1 Mark
(a) Each corner has three mutually perpendicular edges, hence forming three right angles in each corner. A room has 8 corners. Hence, there are 24 right angles in the room.
(b) Let the number to be subtracted be x.
 $\Rightarrow \frac{5}{12}+\frac{3}{2}-x=2$
 $\Rightarrow \frac{5+18}{12}-x=2$
 $\Rightarrow \frac{23}{12}-x=2$
 $\Rightarrow x=\frac{23}{12}-2$
$\Rightarrow x=\frac{23-24}{12}$
$\Rightarrow x=\frac{-1}{12}$
$\Rightarrow x=\frac{-1}{12}$

Each part: 2 Marks
(a) First write the rational numbers with positive denominator = $\frac{-7}{10}$, $\frac{5}{-8}$, $\frac{2}{-3}$.

LCM of denominators (LCM of 10, 8, 3 is 120)

$\frac{-7}{10}$ = $\frac{(-7\times 12)}{(10\times 12)}=\frac{-84}{120}$

$\frac{-5}{8}$ = $\frac{(-5\times 15)}{(8\times 15)}=\frac{-75}{120}$

$\frac{-2}{3}$ = $\frac{(2\times 40)}{(3\times 40)}=\frac{-80}{120}$

Comparing the numerators of these numbers, we get:

- 84 < - 80 < - 75

∴ $\frac{-84}{120}$ < $\frac{-80}{120}$ < $\frac{-75}{120}$;
$\frac{-7}{10}$ < $\frac{-2}{3}$ < $\frac{-5}{8}$.
(b)  $\frac{4}{3}$, $\frac{17}{2}$, $\frac{5}{18}$

LCM of denominators (LCM of 3, 2, 18 is 18)

$\frac{4}{3}$ = $\frac{(4\times 6)}{(3\times 6)}=\frac{24}{18}$

$\frac{17}{2}$ = $\frac{(17\times 9)}{(2\times 9)}=\frac{153}{18}$

$\frac{5}{18}$ = $\frac{(5\times 1)}{(18\times 1)}=\frac{5}{18}$

Comparing the numerators of these numbers, we get:

5 < 24 < 153

∴ $\frac{5}{18}$ < $\frac{24}{18}$ < $\frac{153}{18}$;
$\frac{5}{18}$ < $\frac{4}{3}$ < $\frac{17}{2}$.