Secondary 2 Mathematics Numbers Ratio Proportion Quiz

Secondary 2 Mathematics Quiz - Numbers Ratio Proportion (Answer Key)

Total Marks: 40

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Section A: Ratio and Proportion Fundamentals (Questions 1–5, 10 marks)

1. Express the ratio $48 : 72$ in its simplest form.
[1 mark]

Answer: $2 : 3$

Working:

• Find the HCF of 48 and 72: $48 = 2^4 \times 3$, $72 = 2^3 \times 3^2$, HCF $= 2^3 \times 3 = 24$
• Divide both parts by 24: $48 \div 24 = 2$, $72 \div 24 = 3$
• Simplest form: $2 : 3$

Marking Note: 1 mark for correct simplified ratio. Accept $2:3$ or $\frac{2}{3}$.

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2. The ratio of boys to girls in a class is $5 : 7$. If there are 35 girls, how many boys are there?
[2 marks]

Answer: 25 boys

Working:

• Ratio boys : girls = $5 : 7$
• 7 units = 35 girls
• 1 unit = $35 \div 7 = 5$
• Boys = 5 units = $5 \times 5 = 25$

Alternative Method: $\frac{\text{boys}}{35} = \frac{5}{7} \Rightarrow \text{boys} = 35 \times \frac{5}{7} = 25$

Marking Note: 1 mark for finding 1 unit = 5, 1 mark for final answer 25.

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3. A map has a scale of $1 : 25\,000$. The distance between two towns on the map is $6.4$ cm. Find the actual distance between the two towns in kilometres.
[2 marks]

Answer: 1.6 km

Working:

• Scale $1 : 25\,000$ means 1 cm on map = 25,000 cm in reality
• Actual distance = $6.4 \times 25\,000 = 160\,000$ cm
• Convert to km: $160\,000 \div 100\,000 = 1.6$ km

Marking Note: 1 mark for correct multiplication ($6.4 \times 25\,000$), 1 mark for correct unit conversion to km. Common error: forgetting to convert cm to km (giving 160,000 cm or 1.6 km incorrectly as 160 km).

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4. $y$ is directly proportional to the square of $x$. When $x = 3$, $y = 54$. Find the equation connecting $y$ and $x$.
[2 marks]

Answer: $y = 6x^2$

Working:

• $y \propto x^2 \Rightarrow y = kx^2$ where $k$ is a constant
• Substitute $x = 3$, $y = 54$: $54 = k(3)^2 = 9k$
• $k = 54 \div 9 = 6$
• Equation: $y = 6x^2$

Marking Note: 1 mark for writing $y = kx^2$ and finding $k = 6$, 1 mark for final equation $y = 6x^2$. Common error: writing $y = 54x^2$ (not finding $k$ first) or $y = 6x$ (missing the square).

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5. It takes 6 workers 8 hours to paint a house. Assuming all workers work at the same rate, how long would it take 4 workers to paint the same house?
[3 marks]

Answer: 12 hours

Working:

• This is inverse proportion: more workers → less time
• Total work = $6 \times 8 = 48$ worker-hours
• Time for 4 workers = $48 \div 4 = 12$ hours

Alternative Method (Proportion): $6 \times 8 = 4 \times t \Rightarrow t = \frac{6 \times 8}{4} = 12$ hours

Marking Note: 1 mark for recognising inverse proportion / calculating total work (48 worker-hours), 1 mark for correct division, 1 mark for final answer with units. Common error: using direct proportion ($6/8 = 4/t \Rightarrow t = 5.33$ hours).

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Section B: Rate, Speed, and Applied Proportion (Questions 6–12, 18 marks)

6. A car travels $240$ km in $3$ hours. Calculate its average speed in km/h.
[1 mark]

Answer: 80 km/h

Working: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{240}{3} = 80$ km/h

Marking Note: 1 mark for correct answer with units.

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7. Convert $72$ km/h to m/s.
[2 marks]

Answer: 20 m/s

Working:

• $1$ km = $1000$ m, $1$ hour = $3600$ seconds
• $72$ km/h = $72 \times \frac{1000}{3600} = 72 \times \frac{5}{18} = 20$ m/s

Alternative: $72 \div 3.6 = 20$ m/s

Marking Note: 1 mark for correct conversion factor ($\times \frac{1000}{3600}$ or $\div 3.6$), 1 mark for final answer 20 m/s.

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8. A cyclist travels at a constant speed of $15$ km/h for $2$ hours $20$ minutes. Find the distance travelled in kilometres.
[2 marks]

Answer: 35 km

Working:

• Convert time to hours: $2$ h $20$ min = $2 + \frac{20}{60} = 2\frac{1}{3} = \frac{7}{3}$ hours
• Distance = Speed $\times$ Time = $15 \times \frac{7}{3} = 5 \times 7 = 35$ km

Marking Note: 1 mark for correct time conversion ($2\frac{1}{3}$ h or $\frac{7}{3}$ h), 1 mark for correct distance calculation.

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9. Water flows into a tank at a rate of $12$ litres per minute. The tank has a capacity of $1800$ litres. How long, in hours and minutes, will it take to fill the empty tank completely?
[3 marks]

Answer: 2 hours 30 minutes

Working:

• Time in minutes = $\frac{1800}{12} = 150$ minutes
• Convert to hours and minutes: $150 \div 60 = 2$ remainder $30$
• $= 2$ hours $30$ minutes

Marking Note: 1 mark for time in minutes (150), 1 mark for conversion to hours/minutes, 1 mark for final answer in correct format. Common error: giving 2.5 hours instead of 2 hours 30 minutes.

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10. A recipe for 8 people uses 500 g of flour. How much flour is needed for 14 people? Give your answer in kilograms.
[3 marks]

Answer: 0.875 kg (or $\frac{7}{8}$ kg)

Working:

• Flour per person = $500 \div 8 = 62.5$ g
• Flour for 14 people = $62.5 \times 14 = 875$ g
• Convert to kg: $875 \div 1000 = 0.875$ kg

Alternative (Proportion): $\frac{500}{8} = \frac{x}{14} \Rightarrow x = 500 \times \frac{14}{8} = 500 \times 1.75 = 875$ g $= 0.875$ kg

Marking Note: 1 mark for correct proportion setup, 1 mark for calculation (875 g), 1 mark for conversion to kg (0.875 kg). Accept $\frac{7}{8}$ kg.

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11. The cost of 5 identical pens is $12.50. Find the cost of 12 such pens.
[2 marks]

Answer: $30.00

Working:

• Cost per pen = $12.50 \div 5 =$2.50
• Cost of 12 pens = $2.50 \times 12 =$30.00

Alternative (Proportion): $\frac{12.50}{5} = \frac{x}{12} \Rightarrow x = 12.50 \times \frac{12}{5} = 12.50 \times 2.4 = 30.00$

Marking Note: 1 mark for unit cost or proportion setup, 1 mark for final answer $30.00 (must show 2 decimal places for currency).

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12. A machine can produce 360 bottles in 4 minutes. At this rate, how many bottles can it produce in 1 hour 15 minutes?
[3 marks]

Answer: 6750 bottles

Working:

• Rate = $360 \div 4 = 90$ bottles per minute
• Time = 1 hour 15 minutes = 75 minutes
• Bottles produced = $90 \times 75 = 6750$

Marking Note: 1 mark for rate (90 bottles/min), 1 mark for time conversion (75 min), 1 mark for final answer.

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Section C: Multi-Step Problems and Real-World Applications (Questions 13–20, 12 marks)

13. A sum of money is divided among Ali, Bala, and Cindy in the ratio $3 : 5 : 7$. If Bala receives $120 more than Ali, find the total sum of money.
[3 marks]

Answer: $900

Working:

• Ratio Ali : Bala : Cindy = $3 : 5 : 7$
• Difference between Bala and Ali = $5 - 3 = 2$ units
• 2 units = $120 \Rightarrow 1$ unit = $60
• Total units = $3 + 5 + 7 = 15$ units
• Total sum = $15 \times 60 =$900

Marking Note: 1 mark for finding 1 unit = $60, 1 mark for total units (15), 1 mark for final answer$900.

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14. The ratio of the number of red marbles to blue marbles in a bag is $4 : 9$. After adding 15 red marbles and removing 5 blue marbles, the ratio becomes $1 : 2$. How many red marbles were in the bag at first?
[4 marks]

Answer: 20 red marbles

Working:

• Let initial red marbles = $4x$, blue marbles = $9x$
• After changes: Red = $4x + 15$, Blue = $9x - 5$
• New ratio: $\frac{4x + 15}{9x - 5} = \frac{1}{2}$
• Cross-multiply: $2(4x + 15) = 9x - 5$
• $8x + 30 = 9x - 5$
• $30 + 5 = 9x - 8x$
• $x = 35$
• Initial red marbles = $4x = 4 \times 35 = 140$

Wait, let me recheck: $2(4x + 15) = 9x - 5$ $8x + 30 = 9x - 5$ $35 = x$ Red = $4 \times 35 = 140$

But check: Blue = $9 \times 35 = 315$ After: Red = $140 + 15 = 155$, Blue = $315 - 5 = 310$ Ratio = $155 : 310 = 1 : 2$ ✓

Answer: 140 red marbles

Marking Note: 1 mark for setting up variables ($4x$, $9x$), 1 mark for forming equation from new ratio, 1 mark for solving $x = 35$, 1 mark for final answer (140). Common error: solving for $x$ but forgetting to multiply by 4 for red marbles.

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15. A car travels from Town A to Town B at an average speed of 60 km/h and returns from Town B to Town A at an average speed of 80 km/h. The total journey takes 7 hours. Find the distance between Town A and Town B.
[4 marks]

Answer: 240 km

Working:

• Let distance = $d$ km
• Time from A to B = $\frac{d}{60}$ hours
• Time from B to A = $\frac{d}{80}$ hours
• Total time: $\frac{d}{60} + \frac{d}{80} = 7$
• Common denominator 240: $\frac{4d}{240} + \frac{3d}{240} = 7$
• $\frac{7d}{240} = 7$
• $7d = 1680$
• $d = 240$ km

Check: Time A→B = $240/60 = 4$ h, Time B→A = $240/80 = 3$ h, Total = 7 h ✓

Marking Note: 1 mark for setting up time expressions, 1 mark for forming equation, 1 mark for solving equation correctly, 1 mark for final answer with units. Common error: averaging speeds ($70$ km/h) and multiplying by 7.

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16. $P$ is inversely proportional to the cube of $Q$. When $Q = 2$, $P = 16$. Find the value of $P$ when $Q = 4$.
[3 marks]

Answer: 2

Working:

• $P \propto \frac{1}{Q^3} \Rightarrow P = \frac{k}{Q^3}$
• When $Q = 2$, $P = 16$: $16 = \frac{k}{2^3} = \frac{k}{8} \Rightarrow k = 128$
• Equation: $P = \frac{128}{Q^3}$
• When $Q = 4$: $P = \frac{128}{4^3} = \frac{128}{64} = 2$

Alternative (Ratio Method): $\frac{P_1}{P_2} = \frac{Q_2^3}{Q_1^3} \Rightarrow \frac{16}{P_2} = \frac{4^3}{2^3} = \frac{64}{8} = 8$ $P_2 = \frac{16}{8} = 2$

Marking Note: 1 mark for correct proportionality statement ($P = k/Q^3$), 1 mark for finding $k = 128$, 1 mark for final answer $P = 2$. Common error: using direct proportion or wrong power.

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17. A rectangular field has length and breadth in the ratio $5 : 3$. The perimeter of the field is 320 m. Find the area of the field in square metres.
[3 marks]

Answer: 6000 m²

Working:

• Let length = $5x$, breadth = $3x$
• Perimeter = $2(5x + 3x) = 16x = 320$
• $x = 320 \div 16 = 20$
• Length = $5 \times 20 = 100$ m, Breadth = $3 \times 20 = 60$ m
• Area = $100 \times 60 = 6000$ m²

Marking Note: 1 mark for setting up length/breadth as $5x$ and $3x$, 1 mark for finding $x = 20$, 1 mark for area calculation (6000 m²).

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18. Two taps, A and B, can fill a tank in 6 hours and 8 hours respectively when turned on separately. If both taps are turned on together, how long will it take to fill the tank? Give your answer in hours and minutes.
[3 marks]

Answer: 3 hours 25 minutes (or $3\frac{5}{12}$ hours)

Working:

• Tap A rate = $\frac{1}{6}$ tank per hour
• Tap B rate = $\frac{1}{8}$ tank per hour
• Combined rate = $\frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24}$ tank per hour
• Time = $1 \div \frac{7}{24} = \frac{24}{7} = 3\frac{3}{7}$ hours
• $\frac{3}{7} \times 60 = 25\frac{5}{7} \approx 25.7$ minutes → 3 hours 26 minutes (or exactly $3\frac{3}{7}$ h)

Wait, let me recalculate: $\frac{3}{7} \times 60 = \frac{180}{7} = 25\frac{5}{7}$ minutes ≈ 25 minutes 43 seconds. Usually rounded to nearest minute: 3 hours 26 minutes. But exact fraction is $3\frac{3}{7}$ hours.

Better to give exact: $3\frac{3}{7}$ hours or 3 hours $25\frac{5}{7}$ minutes.

Marking Note: 1 mark for individual rates, 1 mark for combined rate ($\frac{7}{24}$), 1 mark for correct time calculation and conversion. Accept $3\frac{3}{7}$ h or 3 h 26 min.

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19. The scale of a floor plan is $1 : 100$. On the plan, a rectangular room measures $4.5$ cm by $3.2$ cm. Find the actual area of the room in square metres.
[3 marks]

Answer: 14.4 m²

Working:

• Scale $1 : 100$ means 1 cm on plan = 100 cm = 1 m in reality
• Actual length = $4.5 \times 1 = 4.5$ m
• Actual breadth = $3.2 \times 1 = 3.2$ m
• Actual area = $4.5 \times 3.2 = 14.4$ m²

Alternative (Area Scale Factor):

• Area scale factor = $100^2 = 10\,000$
• Plan area = $4.5 \times 3.2 = 14.4$ cm²
• Actual area = $14.4 \times 10\,000 = 144\,000$ cm² = $14.4$ m²

Marking Note: 1 mark for correct length/breadth conversion, 1 mark for area calculation, 1 mark for correct units (m²). Common error: forgetting to convert cm² to m² (giving 144,000 cm²) or using linear scale factor for area.

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20. A factory produces two types of widgets, Type X and Type Y, in the ratio $7 : 5$. In one week, the factory produces 480 more Type X widgets than Type Y widgets. How many widgets does the factory produce in total that week?
[4 marks]

Answer: 2880 widgets

Working:

• Ratio X : Y = $7 : 5$
• Difference in ratio units = $7 - 5 = 2$ units
• 2 units = 480 widgets
• 1 unit = 240 widgets
• Total units = $7 + 5 = 12$ units
• Total widgets = $12 \times 240 = 2880$

Check: X = $7 \times 240 = 1680$, Y = $5 \times 240 = 1200$, Difference = $1680 - 1200 = 480$ ✓

Marking Note: 1 mark for finding difference in ratio units (2), 1 mark for value of 1 unit (240), 1 mark for total units (12), 1 mark for final answer (2880).

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End of Answer Key