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Secondary 4 Additional Mathematics Numbers Ratio Proportion Quiz

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Secondary 4 Additional Mathematics Quiz - Numbers Ratio Proportion

Name: _________________________ Class: _________________________ Date: _________________________ Score: ______ / 50

Duration: 45 minutes Total Marks: 50

Instructions:

Answer ALL questions in the spaces provided.
Show all working clearly. Marks are awarded for method as well as final answers.
Unless otherwise stated, give non-exact answers correct to 3 significant figures.
This quiz covers the Numbers, Ratio & Proportion topic from the Additional Mathematics syllabus.

Section A: Direct Proportion and Variation (Questions 1–5)

10 marks | Answer all questions.

1. The variable $y$ is directly proportional to the square of $x$ . Given that $y = 36$ when $x = 3$ , find:

(a) an equation connecting $y$ and $x$ , [2 marks]

(b) the value of $y$ when $x = 5$ . [1 mark]

2. The distance $d$ metres travelled by a falling object is directly proportional to the square of the time $t$ seconds. An object falls 78.4 m in 4 seconds.

(a) Express $d$ in terms of $t$ . [2 marks]

(b) Find the distance fallen in the first 7 seconds. [1 mark]

3. The cost $\$ C $of printing a magazine is partly constant and partly varies directly as the number of pages$ n $. The cost of printing a 40-page magazine is \$ 2.50, and the cost of printing a 100-page magazine is $4.30.

(a) Express $C$ in terms of $n$ . [3 marks]

(b) Find the cost of printing a 150-page magazine. [1 mark]

Section B: Inverse Proportion and Joint Variation (Questions 4–8)

12 marks | Answer all questions.

4. The variable $y$ is inversely proportional to $x$ . Given that $y = 8$ when $x = 3$ , find:

(a) an equation connecting $y$ and $x$ , [2 marks]

(b) the value of $x$ when $y = 12$ . [1 mark]

5. The electrical resistance $R$ ohms of a wire of fixed length varies inversely as the square of its diameter $d$ mm. A wire of diameter 2 mm has a resistance of 15 ohms.

(a) Find an equation connecting $R$ and $d$ . [2 marks]

(b) Find the resistance of a wire of the same material and length but with diameter 3 mm. [1 mark]

6. The variable $z$ varies directly as $x$ and inversely as the square of $y$ . Given that $z = 10$ when $x = 5$ and $y = 2$ , find:

(a) an equation connecting $z$ , $x$ , and $y$ , [2 marks]

(b) the value of $z$ when $x = 8$ and $y = 4$ . [1 mark]

7. The volume $V$ of a given mass of gas varies directly as the absolute temperature $T$ and inversely as the pressure $P$ . When $T = 300$ and $P = 80$ , the volume $V = 60$ .

(a) Express $V$ in terms of $T$ and $P$ . [2 marks]

(b) Find the volume when $T = 360$ and $P = 90$ . [1 mark]

Section C: Ratio Problems (Questions 8–12)

12 marks | Answer all questions.

8. Three numbers $a$ , $b$ , and $c$ are in the ratio $3 : 5 : 7$ . The sum of the numbers is 105. Find the value of each number. [3 marks]

9. The ratio of boys to girls in a school is $7 : 5$ . There are 240 more boys than girls. Find the total number of students in the school. [3 marks]

10. A sum of money is divided among three people in the ratio $2 : 3 : 5$ . The largest share is $350 more than the smallest share. Find the total sum of money. [3 marks]

11. The ratio of the number of red marbles to blue marbles in a bag is $5 : 3$ . After 12 red marbles are removed and 8 blue marbles are added, the ratio becomes $1 : 1$ . Find the original number of red marbles. [3 marks]

Section D: Rates and Proportional Reasoning (Questions 12–16)

10 marks | Answer all questions.

12. A machine produces 240 components in 5 hours. At the same rate, how many components can it produce in 8 hours? [2 marks]

13. A car travels 180 km in $2\frac{1}{4}$ hours at a constant speed. How far will it travel in $3\frac{1}{2}$ hours at the same speed? [2 marks]

14. 12 workers can complete a task in 15 days. How many workers are needed to complete the same task in 9 days, assuming all workers work at the same rate? [2 marks]

15. A tap can fill a tank in 6 hours. Due to a leak at the bottom, it takes 8 hours to fill the tank. How long would the leak take to empty a full tank? [2 marks]

16. A contractor estimates that 20 men can build a wall in 30 days. After 12 days, 8 men leave the job. How many more days will it take the remaining men to complete the wall? [2 marks]

Section E: Challenging Problems (Questions 17–20)

6 marks | Answer all questions.

17. The force of attraction $F$ between two magnets varies inversely as the square of the distance $d$ between them. When the distance is 3 cm, the force is 12 units. Find the force when the distance is reduced to 2 cm. [2 marks]

18. The time $T$ taken for a journey varies directly as the distance $D$ and inversely as the speed $S$ . A journey of 240 km at a speed of 60 km/h takes 4 hours. How long will a journey of 300 km take at a speed of 75 km/h? [2 marks]

19. The illumination $I$ from a light source varies inversely as the square of the distance $d$ from the source. At a distance of 5 m, the illumination is 40 lux. At what distance will the illumination be 10 lux? [1 mark]

20. A number $N$ is divided into three parts in the ratio $2 : 3 : 4$ . If the second part is 45, find the value of $N$ . [1 mark]

END OF QUIZ

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Answers

Secondary 4 Additional Mathematics Quiz - Numbers Ratio Proportion - ANSWERS

Total Marks: 50

Section A: Direct Proportion and Variation (Questions 1–3)

1. (a) $y = kx^2$ . $36 = k(3)^2 \Rightarrow 36 = 9k \Rightarrow k = 4$ . Equation: $y = 4x^2$ [2 marks] (b) $y = 4(5)^2 = 4(25) = 100$ [1 mark]

2. (a) $d = kt^2$ . $78.4 = k(4)^2 \Rightarrow 78.4 = 16k \Rightarrow k = 4.9$ . Equation: $d = 4.9t^2$ [2 marks] (b) $d = 4.9(7)^2 = 4.9(49) = 240.1$ m [1 mark]

3. (a) $C = a + bn$ . $2.50 = a + 40b$ and $4.30 = a + 100b$ . Subtract: $1.80 = 60b \Rightarrow b = 0.03$ . Then $a = 2.50 - 40(0.03) = 2.50 - 1.20 = 1.30$ . Equation: $C = 1.30 + 0.03n$ [3 marks] (b) $C = 1.30 + 0.03(150) = 1.30 + 4.50 = \$ 5.80$ [1 mark]

Section B: Inverse Proportion and Joint Variation (Questions 4–7)

4. (a) $y = \frac{k}{x}$ . $8 = \frac{k}{3} \Rightarrow k = 24$ . Equation: $y = \frac{24}{x}$ [2 marks] (b) $12 = \frac{24}{x} \Rightarrow x = \frac{24}{12} = 2$ [1 mark]

5. (a) $R = \frac{k}{d^2}$ . $15 = \frac{k}{2^2} \Rightarrow 15 = \frac{k}{4} \Rightarrow k = 60$ . Equation: $R = \frac{60}{d^2}$ [2 marks] (b) $R = \frac{60}{3^2} = \frac{60}{9} = 6.67$ ohms (3 s.f.) [1 mark]

6. (a) $z = \frac{kx}{y^2}$ . $10 = \frac{k(5)}{2^2} \Rightarrow 10 = \frac{5k}{4} \Rightarrow k = 8$ . Equation: $z = \frac{8x}{y^2}$ [2 marks] (b) $z = \frac{8(8)}{4^2} = \frac{64}{16} = 4$ [1 mark]

7. (a) $V = \frac{kT}{P}$ . $60 = \frac{k(300)}{80} \Rightarrow 60 = \frac{300k}{80} \Rightarrow 60 = 3.75k \Rightarrow k = 16$ . Equation: $V = \frac{16T}{P}$ [2 marks] (b) $V = \frac{16(360)}{90} = \frac{5760}{90} = 64$ [1 mark]

Section C: Ratio Problems (Questions 8–11)

8. Let parts be $3x, 5x, 7x$ . Sum: $15x = 105 \Rightarrow x = 7$ . Numbers: $a = 21$ , $b = 35$ , $c = 49$ [3 marks]

9. Let boys $= 7x$ , girls $= 5x$ . Difference: $7x - 5x = 240 \Rightarrow 2x = 240 \Rightarrow x = 120$ . Total students $= 12x = 12(120) = 1440$ [3 marks]

10. Shares: $2x, 3x, 5x$ . Largest - smallest: $5x - 2x = 350 \Rightarrow 3x = 350 \Rightarrow x = \frac{350}{3}$ . Total sum $= 10x = 10(\frac{350}{3}) = \frac{3500}{3} = \$ 1166.67$ (or exact fraction) [3 marks]

11. Original red $= 5x$ , blue $= 3x$ . After changes: red $= 5x - 12$ , blue $= 3x + 8$ . Ratio: $\frac{5x - 12}{3x + 8} = \frac{1}{1} \Rightarrow 5x - 12 = 3x + 8 \Rightarrow 2x = 20 \Rightarrow x = 10$ . Original red $= 5(10) = 50$ [3 marks]

Section D: Rates and Proportional Reasoning (Questions 12–16)

12. Rate $= \frac{240}{5} = 48$ components/hour. In 8 hours: $48 \times 8 = 384$ components [2 marks]

13. Speed $= \frac{180}{2.25} = 80$ km/h. Distance in 3.5 hours: $80 \times 3.5 = 280$ km [2 marks]

14. Total work $= 12 \times 15 = 180$ worker-days. Workers needed $= \frac{180}{9} = 20$ workers [2 marks]

15. Filling rate $= \frac{1}{6}$ tank/hour. Net rate with leak $= \frac{1}{8}$ tank/hour. Leak rate $= \frac{1}{6} - \frac{1}{8} = \frac{4}{24} - \frac{3}{24} = \frac{1}{24}$ tank/hour. Time to empty $= 24$ hours [2 marks]

16. Total work $= 20 \times 30 = 600$ man-days. Work done in 12 days $= 20 \times 12 = 240$ man-days. Remaining work $= 600 - 240 = 360$ man-days. Remaining men $= 12$ . Days needed $= \frac{360}{12} = 30$ more days [2 marks]

Section E: Challenging Problems (Questions 17–20)

17. $F = \frac{k}{d^2}$ . $12 = \frac{k}{3^2} \Rightarrow k = 108$ . When $d = 2$ : $F = \frac{108}{2^2} = \frac{108}{4} = 27$ units [2 marks]

18. $T = \frac{kD}{S}$ . $4 = \frac{k(240)}{60} \Rightarrow 4 = 4k \Rightarrow k = 1$ . For journey: $T = \frac{1(300)}{75} = 4$ hours [2 marks]

19. $I = \frac{k}{d^2}$ . $40 = \frac{k}{5^2} \Rightarrow k = 1000$ . When $I = 10$ : $10 = \frac{1000}{d^2} \Rightarrow d^2 = 100 \Rightarrow d = 10$ m [1 mark]

20. Parts: $2x, 3x, 4x$ . Second part $3x = 45 \Rightarrow x = 15$ . $N = 2x + 3x + 4x = 9x = 9(15) = 135$ [1 mark]

END OF ANSWER KEY