Secondary 4 Elementary Mathematics Numbers Ratio Proportion Quiz

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Secondary 4 Elementary Mathematics Quiz - Numbers Ratio Proportion

Name: _________________________ Class: _________________________ Date: _________________________ Score: ______ / 40

Duration: 45 minutes Total Marks: 40

Instructions:

• Answer ALL questions.
• Show all working clearly.
• Give non-exact answers correct to 3 significant figures unless stated otherwise.
• Calculators are allowed.
• Marks are indicated in brackets.

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Section A: Short Answer (10 marks)

Answer all questions in the spaces provided.

1. Express $3.45 \times 10^{-4}$ as an ordinary decimal.

[1 mark]

Answer: _________________________

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2. Simplify $\dfrac{12a^5b^{-2}}{3a^2b^3}$, giving your answer with positive indices.

[2 marks]

Answer: _________________________

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3. Evaluate $64^{\frac{2}{3}}$.

[1 mark]

Answer: _________________________

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4. The ratio of boys to girls in a school is $5:7$. If there are 420 girls, find the total number of students in the school.

[2 marks]

Answer: _________________________

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5. A map is drawn to a scale of $1:25000$. Two towns are 8.4 cm apart on the map. Find the actual distance between the towns in kilometres.

[2 marks]

Answer: _________________________ km

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6. Divide $840 in the ratio$3:5:6$.

[2 marks]

Answer: $_________________________

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Section B: Structured Questions (18 marks)

Show all working clearly.

7. (a) Express $0.0000056$ in standard form.

[1 mark]

Answer: _________________________

(b) Evaluate $\dfrac{4.8 \times 10^7}{1.2 \times 10^{-2}}$, giving your answer in standard form.

[2 marks]

Answer: _________________________

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8. The cost of a meal is \72$before GST. GST is charged at$9%$.

(a) Calculate the amount of GST payable.

[1 mark]

Answer: \_________________________$

(b) Find the total cost of the meal including GST.

[1 mark]

Answer: \_________________________$

(c) A service charge of $10\%$ is added to the pre-GST cost. Calculate the final total including both service charge and GST.

[2 marks]

Answer: \_________________________$

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9. A machine produces 240 components in 5 hours.

(a) Find the rate of production in components per hour.

[1 mark]

Answer: _________________________ components per hour

(b) How many components can the machine produce in 8 hours at the same rate?

[1 mark]

Answer: _________________________ components

(c) How long would it take to produce 600 components at this rate? Give your answer in hours and minutes.

[2 marks]

Answer: _________________________

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10. The value of a car depreciates by $15\%$ each year. The car was bought for \45000$.

(a) Calculate the value of the car after 1 year.

[1 mark]

Answer: \_________________________$

(b) Calculate the value of the car after 2 years.

[2 marks]

Answer: \_________________________$

(c) Find the percentage decrease in value over the 2 years.

[2 marks]

Answer: _________________________ $\%$

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11. $y$ is inversely proportional to the square of $x$. When $x = 4$, $y = 9$.

(a) Find an equation connecting $y$ and $x$.

[2 marks]

Answer: _________________________

(b) Find the value of $y$ when $x = 6$.

[1 mark]

Answer: _________________________

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Section C: Problem Solving (12 marks)

Show all working and reasoning clearly.

12. A sum of money is divided among Ali, Ben, and Chen in the ratio $2:3:5$. Ben receives \180$ more than Ali.

(a) Find the total sum of money.

[3 marks]

Answer: \_________________________$

(b) Chen gives $20\%$ of his share to charity. How much does Chen have left?

[2 marks]

Answer: \_________________________$

---

13. The population of a town is $48000$. The population increases by $8\%$ in the first year and then decreases by $5\%$ in the second year.

(a) Find the population after the first year.

[1 mark]

Answer: _________________________

(b) Find the population after the second year.

[2 marks]

Answer: _________________________

(c) Calculate the overall percentage change in population over the two years.

[2 marks]

Answer: _________________________ $\%$

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14. A rectangle has length $l$ cm and width $w$ cm. The length is increased by $20\%$ and the width is decreased by $10\%$.

(a) Express the new length and new width in terms of $l$ and $w$.

[1 mark]

Answer: New length = _________________________ New width = _________________________

(b) Find the percentage change in the area of the rectangle.

[2 marks]

Answer: _________________________ $\%$

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15. Three partners invest in a business in the ratio $4:5:7$. The total profit for the year is \96000$.

(a) How much profit does each partner receive?

[2 marks]

Answer: Partner 1: \$Partner 2:$$$Partner 3:$$_________________________$

(b) The partner with the largest share reinvests $35\%$ of their profit. How much do they reinvest?

[1 mark]

Answer: \_________________________$

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16. A car travels 180 km on 15 litres of petrol.

(a) Express the fuel consumption in km per litre.

[1 mark]

Answer: _________________________ km/litre

(b) How far can the car travel on 22 litres of petrol?

[1 mark]

Answer: _________________________ km

(c) If the price of petrol is \2.40$ per litre, find the cost of petrol for a journey of 420 km.

[2 marks]

Answer: \_________________________$

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17. $P$ is directly proportional to the cube of $Q$. When $Q = 2$, $P = 40$.

(a) Find the formula connecting $P$ and $Q$.

[2 marks]

Answer: _________________________

(b) Find $P$ when $Q = 5$.

[1 mark]

Answer: _________________________

(c) Find $Q$ when $P = 1080$.

[2 marks]

Answer: _________________________

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18. A shop offers a discount of $25\%$ on all items during a sale. A customer buys a jacket with a marked price of \160$and a pair of shoes with a marked price of$$90$.

(a) Calculate the sale price of the jacket.

[1 mark]

Answer: \_________________________$

(b) Calculate the total amount the customer pays for both items.

[1 mark]

Answer: \_________________________$

(c) The shop makes a profit of $20\%$ on the cost price of the jacket even after the discount. Find the cost price of the jacket.

[2 marks]

Answer: \_________________________$

---

19. A solution contains acid and water in the ratio $3:8$. There is 480 ml of water in the solution.

(a) Find the amount of acid in the solution.

[1 mark]

Answer: _________________________ ml

(b) Find the total volume of the solution.

[1 mark]

Answer: _________________________ ml

(c) How much water must be added to change the ratio of acid to water to $1:4$?

[2 marks]

Answer: _________________________ ml

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20. The density of a substance is given by the formula $\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}$. A block of metal has a mass of 2.4 kg and a volume of 300 cm³.

(a) Calculate the density of the metal in g/cm³.

[1 mark]

Answer: _________________________ g/cm³

(b) Another block of the same metal has a mass of 6 kg. Find its volume.

[1 mark]

Answer: _________________________ cm³

(c) The metal is an alloy of copper and zinc in the ratio $7:3$ by mass. Find the mass of copper in the 2.4 kg block.

[2 marks]

Answer: _________________________ kg

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END OF QUIZ

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Secondary 4 Elementary Mathematics Quiz - Numbers Ratio Proportion

ANSWER KEY AND MARKING SCHEME

Total Marks: 40

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Section A: Short Answer (10 marks)

1. Express $3.45 \times 10^{-4}$ as an ordinary decimal. [1 mark]

Answer: $0.000345$

Marking: 1 mark for correct answer.

---

2. Simplify $\dfrac{12a^5b^{-2}}{3a^2b^3}$, giving your answer with positive indices. [2 marks]

Answer: $\dfrac{4a^3}{b^5}$

Working: $\dfrac{12a^5b^{-2}}{3a^2b^3} = 4a^{5-2}b^{-2-3} = 4a^3b^{-5} = \dfrac{4a^3}{b^5}$

Marking: 1 mark for correct coefficient and $a^3$; 1 mark for correct handling of $b$ with positive index. Accept equivalent forms.

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3. Evaluate $64^{\frac{2}{3}}$. [1 mark]

Answer: $16$

Working: $64^{\frac{2}{3}} = (64^{\frac{1}{3}})^2 = 4^2 = 16$

Marking: 1 mark for correct answer.

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4. The ratio of boys to girls in a school is $5:7$. If there are 420 girls, find the total number of students in the school. [2 marks]

Answer: $720$

Working: $7$ parts $= 420$ girls $1$ part $= 60$ Total parts $= 5 + 7 = 12$ Total students $= 12 \times 60 = 720$

Marking: 1 mark for finding value of 1 part; 1 mark for correct total.

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5. A map is drawn to a scale of $1:25000$. Two towns are 8.4 cm apart on the map. Find the actual distance between the towns in kilometres. [2 marks]

Answer: $2.1$ km

Working: Actual distance $= 8.4 \times 25000 = 210000$ cm $= 2100$ m $= 2.1$ km

Marking: 1 mark for correct multiplication; 1 mark for correct conversion to km.

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6. Divide \840$in the ratio$3:5:6$. [2 marks]

Answer: \180$,$$300$,$$360$

Working: Total parts $= 3 + 5 + 6 = 14$ $1$ part $= 840 \div 14 = 60$ Shares: $3 \times 60 = 180$, $5 \times 60 = 300$, $6 \times 60 = 360$

Marking: 1 mark for finding value of 1 part; 1 mark for all three correct shares.

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Section B: Structured Questions (18 marks)

7. (a) Express $0.0000056$ in standard form. [1 mark]

Answer: $5.6 \times 10^{-6}$

Marking: 1 mark for correct standard form.

(b) Evaluate $\dfrac{4.8 \times 10^7}{1.2 \times 10^{-2}}$, giving your answer in standard form. [2 marks]

Answer: $4.0 \times 10^9$

Working: $\dfrac{4.8}{1.2} \times 10^{7-(-2)} = 4 \times 10^9$

Marking: 1 mark for correct coefficient; 1 mark for correct power of 10.

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8. The cost of a meal is \72$before GST. GST is charged at$9%$.

(a) Calculate the amount of GST payable. [1 mark]

Answer: \6.48$

Working: $72 \times 0.09 = 6.48$

Marking: 1 mark for correct answer.

(b) Find the total cost of the meal including GST. [1 mark]

Answer: \78.48$

Working: $72 + 6.48 = 78.48$

Marking: 1 mark for correct answer.

(c) A service charge of $10\%$ is added to the pre-GST cost. Calculate the final total including both service charge and GST. [2 marks]

Answer: \85.68$

Working: Service charge $= 72 \times 0.10 = 7.20$ Subtotal with service $= 72 + 7.20 = 79.20$ GST $= 79.20 \times 0.09 = 7.128$ Total = 79.20 + 7.128 = 86.328 \approx \86.33$

Note: Accept \86.33$if GST is applied to (cost + service charge). If GST is applied only to cost, total =$72 + 7.20 + 6.48 = $85.68$. Either interpretation accepted with clear working.

Marking: 1 mark for service charge; 1 mark for correct final total with GST applied appropriately.

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9. A machine produces 240 components in 5 hours.

(a) Find the rate of production in components per hour. [1 mark]

Answer: $48$ components per hour

Working: $240 \div 5 = 48$

Marking: 1 mark for correct rate.

(b) How many components can the machine produce in 8 hours at the same rate? [1 mark]

Answer: $384$ components

Working: $48 \times 8 = 384$

Marking: 1 mark for correct answer.

(c) How long would it take to produce 600 components at this rate? Give your answer in hours and minutes. [2 marks]

Answer: $12$ hours $30$ minutes

Working: Time $= 600 \div 48 = 12.5$ hours $= 12$ hours $30$ minutes

Marking: 1 mark for correct decimal hours; 1 mark for correct conversion to hours and minutes.

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10. The value of a car depreciates by $15\%$ each year. The car was bought for \45000$.

(a) Calculate the value of the car after 1 year. [1 mark]

Answer: \38250$

Working: $45000 \times 0.85 = 38250$

Marking: 1 mark for correct answer.

(b) Calculate the value of the car after 2 years. [2 marks]

Answer: \32512.50$

Working: $38250 \times 0.85 = 32512.50$

Marking: 1 mark for method; 1 mark for correct answer.

(c) Find the percentage decrease in value over the 2 years. [2 marks]

Answer: $27.75\%$

Working: Decrease $= 45000 - 32512.50 = 12487.50$ Percentage decrease $= \dfrac{12487.50}{45000} \times 100\% = 27.75\%$

Alternatively: $(1 - 0.85^2) \times 100\% = (1 - 0.7225) \times 100\% = 27.75\%$

Marking: 1 mark for correct decrease; 1 mark for correct percentage.

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11. $y$ is inversely proportional to the square of $x$. When $x = 4$, $y = 9$.

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

Answer: $y = \dfrac{144}{x^2}$

Working: $y = \dfrac{k}{x^2}$ $9 = \dfrac{k}{4^2} = \dfrac{k}{16}$ $k = 144$ $y = \dfrac{144}{x^2}$

Marking: 1 mark for correct form; 1 mark for correct constant.

(b) Find the value of $y$ when $x = 6$. [1 mark]

Answer: $4$

Working: $y = \dfrac{144}{6^2} = \dfrac{144}{36} = 4$

Marking: 1 mark for correct answer.

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Section C: Problem Solving (12 marks)

12. A sum of money is divided among Ali, Ben, and Chen in the ratio $2:3:5$. Ben receives \180$ more than Ali.

(a) Find the total sum of money. [3 marks]

Answer: \900$

Working: Difference between Ben and Ali $= 3 - 2 = 1$ part $1$ part = \180$Total parts$= 2 + 3 + 5 = 10$Total sum$= 10 \times 180 = $900$

Marking: 1 mark for identifying 1 part = difference; 1 mark for total parts; 1 mark for correct total.

(b) Chen gives $20\%$ of his share to charity. How much does Chen have left? [2 marks]

Answer: \360$

Working: Chen's share = 5 \times 180 = \900$Amount left$= 900 \times 0.80 = $720$

Correction: Chen's share = 5 × 180 = $900. Amount left = 900 × 0.80 = $720.

Marking: 1 mark for Chen's share; 1 mark for correct remaining amount.

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13. The population of a town is $48000$. The population increases by $8\%$ in the first year and then decreases by $5\%$ in the second year.

(a) Find the population after the first year. [1 mark]

Answer: $51840$

Working: $48000 \times 1.08 = 51840$

Marking: 1 mark for correct answer.

(b) Find the population after the second year. [2 marks]

Answer: $49248$

Working: $51840 \times 0.95 = 49248$

Marking: 1 mark for method; 1 mark for correct answer.

(c) Calculate the overall percentage change in population over the two years. [2 marks]

Answer: $2.6\%$ increase

Working: Overall multiplier $= 1.08 \times 0.95 = 1.026$ Percentage change $= (1.026 - 1) \times 100\% = 2.6\%$ increase

Marking: 1 mark for overall multiplier; 1 mark for correct percentage with indication of increase.

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14. A rectangle has length $l$ cm and width $w$ cm. The length is increased by $20\%$ and the width is decreased by $10\%$.

(a) Express the new length and new width in terms of $l$ and $w$. [1 mark]

Answer: New length $= 1.2l$, New width $= 0.9w$

Marking: 1 mark for both correct expressions.

(b) Find the percentage change in the area of the rectangle. [2 marks]

Answer: $8\%$ increase

Working: Original area $= lw$ New area $= 1.2l \times 0.9w = 1.08lw$ Percentage change $= (1.08 - 1) \times 100\% = 8\%$ increase

Marking: 1 mark for new area expression; 1 mark for correct percentage with indication of increase.

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15. Three partners invest in a business in the ratio $4:5:7$. The total profit for the year is \96000$.

(a) How much profit does each partner receive? [2 marks]

Answer: Partner 1: \24000$, Partner 2:$$30000$, Partner 3:$$42000$

Working: Total parts $= 4 + 5 + 7 = 16$ $1$ part $= 96000 \div 16 = 6000$ Partner 1: $4 \times 6000 = 24000$ Partner 2: $5 \times 6000 = 30000$ Partner 3: $7 \times 6000 = 42000$

Marking: 1 mark for value of 1 part; 1 mark for all three correct shares.

(b) The partner with the largest share reinvests $35\%$ of their profit. How much do they reinvest? [1 mark]

Answer: \14700$

Working: $42000 \times 0.35 = 14700$

Marking: 1 mark for correct answer.

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16. A car travels 180 km on 15 litres of petrol.

(a) Express the fuel consumption in km per litre. [1 mark]

Answer: $12$ km/litre

Working: $180 \div 15 = 12$

Marking: 1 mark for correct answer.

(b) How far can the car travel on 22 litres of petrol? [1 mark]

Answer: $264$ km

Working: $12 \times 22 = 264$

Marking: 1 mark for correct answer.

(c) If the price of petrol is \2.40$ per litre, find the cost of petrol for a journey of 420 km. [2 marks]

Answer: \84.00$

Working: Petrol needed $= 420 \div 12 = 35$ litres Cost = 35 \times 2.40 = \84.00$

Marking: 1 mark for litres needed; 1 mark for correct cost.

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17. $P$ is directly proportional to the cube of $Q$. When $Q = 2$, $P = 40$.

(a) Find the formula connecting $P$ and $Q$. [2 marks]

Answer: $P = 5Q^3$

Working: $P = kQ^3$ $40 = k(2^3) = 8k$ $k = 5$ $P = 5Q^3$

Marking: 1 mark for correct form; 1 mark for correct constant.

(b) Find $P$ when $Q = 5$. [1 mark]

Answer: $625$

Working: $P = 5(5^3) = 5 \times 125 = 625$

Marking: 1 mark for correct answer.

(c) Find $Q$ when $P = 1080$. [2 marks]

Answer: $6$

Working: $1080 = 5Q^3$ $Q^3 = 216$ $Q = \sqrt[3]{216} = 6$

Marking: 1 mark for setting up equation; 1 mark for correct $Q$.

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18. A shop offers a discount of $25\%$ on all items during a sale. A customer buys a jacket with a marked price of \160$and a pair of shoes with a marked price of$$90$.

(a) Calculate the sale price of the jacket. [1 mark]

Answer: \120$

Working: $160 \times 0.75 = 120$

Marking: 1 mark for correct answer.

(b) Calculate the total amount the customer pays for both items. [1 mark]

Answer: \187.50$

Working: Shoes sale price $= 90 \times 0.75 = 67.50$ Total $= 120 + 67.50 = 187.50$

Marking: 1 mark for correct total.

(c) The shop makes a profit of $20\%$ on the cost price of the jacket even after the discount. Find the cost price of the jacket. [2 marks]

Answer: \100$

Working: Sale price $= 120\%$ of cost price $120 = 1.2 \times$ cost price Cost price = 120 \div 1.2 = \100$

Marking: 1 mark for correct equation; 1 mark for correct cost price.

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19. A solution contains acid and water in the ratio $3:8$. There is 480 ml of water in the solution.

(a) Find the amount of acid in the solution. [1 mark]

Answer: $180$ ml

Working: $8$ parts $= 480$ ml $1$ part $= 60$ ml Acid $= 3 \times 60 = 180$ ml

Marking: 1 mark for correct answer.

(b) Find the total volume of the solution. [1 mark]

Answer: $660$ ml

Working: $180 + 480 = 660$ ml

Marking: 1 mark for correct answer.

(c) How much water must be added to change the ratio of acid to water to $1:4$? [2 marks]

Answer: $240$ ml

Working: Let $x$ ml of water be added. New ratio: $\dfrac{180}{480 + x} = \dfrac{1}{4}$ $180 \times 4 = 480 + x$ $720 = 480 + x$ $x = 240$ ml

Marking: 1 mark for setting up proportion; 1 mark for correct answer.

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20. The density of a substance is given by the formula $\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}$. A block of metal has a mass of 2.4 kg and a volume of 300 cm³.

(a) Calculate the density of the metal in g/cm³. [1 mark]

Answer: $8$ g/cm³

Working: Mass in grams $= 2.4 \times 1000 = 2400$ g Density $= \dfrac{2400}{300} = 8$ g/cm³

Marking: 1 mark for correct answer with units.

(b) Another block of the same metal has a mass of 6 kg. Find its volume. [1 mark]

Answer: $750$ cm³

Working: Volume $= \dfrac{\text{Mass}}{\text{Density}} = \dfrac{6000}{8} = 750$ cm³

Marking: 1 mark for correct answer with units.

(c) The metal is an alloy of copper and zinc in the ratio $7:3$ by mass. Find the mass of copper in the 2.4 kg block. [2 marks]

Answer: $1.68$ kg

Working: Total parts $= 7 + 3 = 10$ $1$ part $= 2.4 \div 10 = 0.24$ kg Copper $= 7 \times 0.24 = 1.68$ kg

Marking: 1 mark for value of 1 part; 1 mark for correct mass of copper.

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END OF ANSWER KEY