(a)

(b)(i)

$$ 24 $$

(b)(ii)

\begin{align*} \text{No. of students who scored more than 80 marks} & = 120 - \text{No. of students who scored at most 80 marks} \\ & = 120 - 105 \\ & = 15 \end{align*}

(b)(iii)

\begin{align*} \text{No. of students who scored at most 40 marks} & = 35 \\ \\ \text{No. of students who scored more than 40 marks but not more than 90 marks} & = 115 - 35 \\ & = 80 \end{align*}

(a)

(b)(i)

\begin{align*} \text{No. of students who achieved Gold} & = 230 - 184 \\ & = 46 \end{align*}

(b)(ii) To achieve silver award, students need to do 8, 9, 10 or 11 pull-ups

\begin{align*} \text{No. of students who achieved Silver} & = 184 - 132 \\ & = 52 \end{align*}

(b)(iii) To achieve bronze award, students need to do 6 or 7 pull-ups

\begin{align*} \text{No. of students who achieved Bronze} & = 132 - 69 \\ & = 63 \end{align*}

(i)

$$ 26 $$

(ii)

\begin{align*} \text{No. of students whose masses are less than or equal to 68.6 kg} & = 76 \\ \\ \text{No. of students whose masses are more than 68.6 kg} & = 100 - 76 \\ & = 24 \end{align*}

(iii)

\begin{align*} \text{No. of students whose masses are less than or equal to 64.4 kg} & = 20 \\ \\ \text{No. of students whose masses are more than 64.4 kg} & = 100 - 20 \\ & = 80 \\ \\ \text{% of students whose masses are more than 64.4 kg} & = {80 \over 100} \times 100 \\ & = 80 \% \end{align*}

(i)

$$ 33 $$

(ii)

\begin{align*} \text{No. of underweight loaves} & = 3 \\ \\ \text{No. of loaves having masses less than or equal to 453.6 g} & = 47 \\ \\ \text{No. of overweight loaves} & = 50 - 47 \\ & = 3 \\ \\ \text{Total no. of rejected loaves} & = 3 + 3 \\ & = 6 \end{align*}

(iii)

\begin{align*} \text{No. of loaves of bread having masses more than } x \text{ g} & = {3 \over 10} \times 50 \\ & = 15 \\ \\ \text{No. of loaves of bread having masses less than or equals to } x \text{ g} & = 50 - 15 \\ & = 35 \\ \\ \text{From the curve, } x & = 450.6 \end{align*}

(a)(i)

$$ \text{Soil A: 130, Soil B: 90} $$

(a)(ii)

\begin{align*} \text{No. of earthworms with lengths greater than 76 mm (Soil } A ) & = 500 - 455 \\ & = 45 \\ \\ \text{% of earthworms with lengths greater than 76 mm (Soil } A ) & = {45 \over 500} \times 100 \\ & = 9 \% \\ \\ \\ \text{No. of earthworms with lengths greater than 76 mm (Soil } B ) & = 500 - 485 \\ & = 15 \\ \\ \text{% of earthworms with lengths greater than 76 mm (Soil } B ) & = {15 \over 500} \times 100 \\ & = 3 \% \end{align*}

(a)(iii)

\begin{align*} \text{18% of earthworms} & = 500 \times 18\% \\ & = 90 \\ \\ \text{Value of } a \text{ (Soil } A ) & = 41 \\ \\ \text{Value of } a \text{ (Soil } B ) & = 46 \end{align*}

(b)

$$ \text{Soil A (longest is 100 mm)} $$

(c)(i)

\begin{align*} \text{No. of earthworms having length greater than 60 mm} & = 500 - 320 \\ & = 180 \\ \\ \text{% of 'satisfactory' earthworms in Soil } A & = {180 \over 500} \times 100 \\ & = 36 \% \end{align*}

(c)(ii)

\begin{align*} \text{No. of earthworms having length greater than 60 mm} & = 500 - 380 \\ & = 120 \\ \\ \text{% of 'satisfactory' earthworms in Soil } A & = {120 \over 500} \times 100 \\ & = 24 \% \end{align*}

(i)

$$ \text{57 (Mid-point between 56 and 58)} $$

(ii)

\begin{align*} \text{No. of students who failed} & = 26 \\ \\ \text{Fraction of total number of students who failed} & = {26 \over 120} \\ & = {13 \over 60} \end{align*}

(iii)

\begin{align*} \text{No. of students who obtained at least } a \text{ marks} & = 120 \times 27.5\% \\ & = 33 \\ \\ \text{No. of students who obtained less than } a \text{ marks} & = 120 - 33 \\ & = 87 \\ \\ \text{From the curve, } a & = 51 \end{align*}

(i)

$$ \text{13 bicycles} $$

(ii)

\begin{align*} \text{No. of bicycles that travelled at a speed less than 29 km/h} & = 80 \\ \\ \text{No. of bicycles that travelled at a speed greater than or equals to 29 km/h} & = 100 - 80 \\ & = 20 \\ \\ \text{Fraction of total number bicycles that travelled at a speed greater than or equals to 29 km/h} & = {20 \over 100} \\ & = {1 \over 5} \end{align*}

(iii)

\begin{align*} \text{No. of bicycles that travelled at a speed less than } v \text{ km/h} & = 100 \times 40\% \\ & = 40 \\ \\ \text{From the graph, } v & = 23 \end{align*}

(i)

(ii)

$$ 80 \le x < 100 $$

(a)(i)

\begin{align*} \text{No. of tomatoes with masses more than 56 g} & = 80 - 36 \\ & = 44 \\ \\ \text{% of Grade A tomatoes} & = {44 \over 80} \times 100 \\ & = 55 \% \end{align*}

(a)(ii)

\begin{align*} \text{No. of Grade C tomatoes} & = 80 \times 15\% \\ & = 12 \\ \\ \text{From the curve, } y & = 52 \end{align*}

(a)(iii)

\begin{align*} \text{No. of Grade B tomatoes} & = 80 - 44 - 12 \\ & = 24 \end{align*}

(b)(i)

(b)(ii)

\begin{align*} \text{Estimated mean mass of a tomato} & = { \sum fx \over \sum f} \\ & = { (2 \times 42.5) + (6 \times 47.5) + (18 \times 52.5) + (44 \times 57.5) + (10 \times 62.5) \over 80} \\ & = 55.875 \text{ kg} \end{align*}

(i)

$$ \text{Examination B is the most challenging as very few students obtained high scores.} $$

(ii)

$$ \text{Examination C is the least challenging as majority of the students obtained the same score.} $$