Secondary 1 Mathematics Numbers Ratio Proportion Quiz

Secondary 1 Mathematics Quiz - Numbers Ratio Proportion (Answer Key)

Total Marks: 40

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Section A: Short Answer Questions (Questions 1–10, 2 marks each)

1. Express the ratio $48 : 72$ in its simplest form. [2]

Answer: $2 : 3$

Working:

• Find 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: 1 mark for correct HCF or dividing by a common factor, 1 mark for final answer $2:3$.

Common mistake: Stopping at $4:6$ or $8:12$ (not fully simplified).

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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]

Answer: 25

Working:

• Ratio boys : girls = $5 : 7$
• 7 parts = 35 girls → 1 part = $35 \div 7 = 5$
• Boys = 5 parts = $5 \times 5 = 25$

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

Marking: 1 mark for finding value of 1 part or setting up proportion, 1 mark for answer 25.

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

Answer: 1.6 km

Working:

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

Marking: 1 mark for correct multiplication ($6.4 \times 25\,000$), 1 mark for correct conversion to km and answer 1.6.

Common mistake: Forgetting to convert cm to km (answer 160,000 or 1600 m).

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4. Divide $180$ in the ratio $4 : 5$. [2]

Answer: $80$ and $100$

Working:

• Total parts = $4 + 5 = 9$
• 1 part = $180 \div 9 = 20$
• First share = $4 \times 20 = 80$
• Second share = $5 \times 20 = 100$
• Check: $80 + 100 = 180$ ✓

Marking: 1 mark for total parts = 9 and 1 part = 20, 1 mark for both correct shares (80, 100).

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5. A car travels $240$ km on $15$ litres of petrol. How far can it travel on $22$ litres of petrol? [2]

Answer: 352 km

Working:

• Distance per litre = $240 \div 15 = 16$ km/litre
• Distance on 22 litres = $16 \times 22 = 352$ km

Alternative (proportion): $\frac{240}{15} = \frac{x}{22} \Rightarrow x = 240 \times \frac{22}{15} = 352$

Marking: 1 mark for finding rate (16 km/l) or setting up proportion, 1 mark for answer 352.

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6. The ratio $x : y = 3 : 5$ and $y : z = 2 : 7$. Find $x : y : z$ in its simplest form. [2]

Answer: $6 : 10 : 35$

Working:

• $x : y = 3 : 5 = 6 : 10$ (multiply by 2 to make $y = 10$)
• $y : z = 2 : 7 = 10 : 35$ (multiply by 5 to make $y = 10$)
• Combine: $x : y : z = 6 : 10 : 35$

Marking: 1 mark for making $y$ common (LCM of 5 and 2 = 10), 1 mark for correct combined ratio $6:10:35$.

Common mistake: Not making $y$ the same value in both ratios before combining.

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7. A recipe uses flour and sugar in the ratio $5 : 2$. If $350$ g of flour is used, how much sugar is needed? [2]

Answer: 140 g

Working:

• Flour : Sugar = $5 : 2$
• 5 parts = 350 g → 1 part = $350 \div 5 = 70$ g
• Sugar = 2 parts = $2 \times 70 = 140$ g

Alternative: $\frac{\text{sugar}}{350} = \frac{2}{5} \Rightarrow \text{sugar} = 350 \times \frac{2}{5} = 140$

Marking: 1 mark for 1 part = 70 or proportion setup, 1 mark for answer 140 g.

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8. The scale of a floor plan is $1 : 100$. A rectangular room measures $4.5$ cm by $3.2$ cm on the plan. Find the actual area of the room in square metres. [2]

Answer: 14.4 m²

Working:

• Actual length = $4.5 \times 100 = 450$ cm = 4.5 m
• Actual breadth = $3.2 \times 100 = 320$ cm = 3.2 m
• Actual area = $4.5 \times 3.2 = 14.4$ m²

Alternative (area scale): Area scale = $1 : 100^2 = 1 : 10\,000$

• Plan area = $4.5 \times 3.2 = 14.4$ cm²
• Actual area = $14.4 \times 10\,000 = 144\,000$ cm² = 14.4 m²

Marking: 1 mark for correct actual dimensions or area scale method, 1 mark for answer 14.4 m².

Common mistake: Using length scale for area (giving 1440 m²) or forgetting to convert cm² to m².

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9. $y$ is directly proportional to $x$. When $x = 6$, $y = 18$. Find the value of $y$ when $x = 15$. [2]

Answer: 45

Working:

• $y \propto x \Rightarrow y = kx$
• $18 = k \times 6 \Rightarrow k = 3$
• Equation: $y = 3x$
• When $x = 15$, $y = 3 \times 15 = 45$

Alternative (proportion): $\frac{y_1}{x_1} = \frac{y_2}{x_2} \Rightarrow \frac{18}{6} = \frac{y}{15} \Rightarrow y = 45$

Marking: 1 mark for finding $k = 3$ or setting up proportion, 1 mark for answer 45.

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10. It takes $8$ workers $12$ days to complete a job. How many days will it take $6$ workers to complete the same job, assuming they work at the same rate? [2]

Answer: 16 days

Working:

• Total work = $8 \times 12 = 96$ worker-days
• Days for 6 workers = $96 \div 6 = 16$ days

Alternative (inverse proportion): $W \times D = \text{constant} \Rightarrow 8 \times 12 = 6 \times D \Rightarrow D = 16$

Marking: 1 mark for total work = 96 worker-days or inverse proportion setup, 1 mark for answer 16.

Common mistake: Using direct proportion ($8/12 = 6/x \Rightarrow x = 9$, incorrect).

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Section B: Structured Questions (Questions 11–16, 3 marks each)

11. A sum of money is shared among Ali, Bala, and Cindy in the ratio $3 : 5 : 7$. [3]

(a) What fraction of the sum does Bala receive? [1]
(b) If Cindy receives $84$ more than Ali, find the total sum of money. [2]

Answer (a): $\frac{5}{15} = \frac{1}{3}$

Answer (b): $630$

Working (a):

• Total parts = $3 + 5 + 7 = 15$
• Bala's share = 5 parts
• Fraction = $\frac{5}{15} = \frac{1}{3}$

Working (b):

• Cindy's parts = 7, Ali's parts = 3
• Difference = $7 - 3 = 4$ parts = $84$
• 1 part = $84 \div 4 = 21$
• Total sum = 15 parts = $15 \times 21 = 315$

Wait, check: $15 \times 21 = 315$, not 630. Let me recalculate.

• 4 parts = $84$ → 1 part = $21$
• Total = 15 parts = $15 \times 21 = 315$

Correct Answer (b): $315$

Marking (a): 1 mark for $\frac{1}{3}$ or $\frac{5}{15}$. Marking (b): 1 mark for difference = 4 parts = $84$ → 1 part = $21$, 1 mark for total = $315$.

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12. The ratio of the number of red marbles to blue marbles in a bag is $4 : 9$. After adding $12$ red marbles and removing $6$ blue marbles, the ratio becomes $2 : 3$. How many red marbles were in the bag at first? [3]

Answer: 24

Working:

• Let initial red = $4x$, initial blue = $9x$
• After changes: red = $4x + 12$, blue = $9x - 6$
• New ratio: $\frac{4x + 12}{9x - 6} = \frac{2}{3}$
• Cross-multiply: $3(4x + 12) = 2(9x - 6)$
• $12x + 36 = 18x - 12$
• $36 + 12 = 18x - 12x$
• $48 = 6x$
• $x = 8$
• Initial red = $4x = 4 \times 8 = 32$

Wait, let me check: $4(8) + 12 = 44$, $9(8) - 6 = 66$, ratio $44:66 = 2:3$ ✓ But answer says 24? Let me re-read: "How many red marbles were in the bag at first?" → $4x = 32$.

Correct Answer: 32

Marking: 1 mark for setting up $4x$ and $9x$, 1 mark for correct equation $\frac{4x+12}{9x-6} = \frac{2}{3}$, 1 mark for solving $x=8$ and initial red = 32.

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13. A map is drawn to a scale of $1 : 50\,000$. [3]

(a) Express this scale in the form $1$ cm represents ____ km. [1]
(b) A forest reserve has an actual area of $12.5$ km². Find its area on the map in cm². [2]

Answer (a): 0.5 km (or $\frac{1}{2}$ km)

Answer (b): 5 cm²

Working (a):

• $1 : 50\,000$ means 1 cm = 50,000 cm
• 50,000 cm = $50\,000 \div 100\,000 = 0.5$ km

Working (b):

• Area scale = $1 : (50\,000)^2 = 1 : 2.5 \times 10^9$
• Actual area = $12.5$ km² = $12.5 \times (100\,000)^2$ cm² = $12.5 \times 10^{10}$ cm² = $1.25 \times 10^{11}$ cm²
• Map area = $\frac{1.25 \times 10^{11}}{2.5 \times 10^9} = 50$ cm²

Wait, recalculate: $12.5$ km² = $12.5 \times (10^5)^2$ cm² = $12.5 \times 10^{10}$ cm² = $1.25 \times 10^{11}$ cm² Scale factor for area = $(50\,000)^2 = 2.5 \times 10^9$ Map area = $1.25 \times 10^{11} \div 2.5 \times 10^9 = 50$ cm²

Correct Answer (b): 50 cm²

Marking (a): 1 mark for 0.5 km. Marking (b): 1 mark for area scale = $1 : 2.5 \times 10^9$ or correct conversion, 1 mark for answer 50 cm².

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14. $P$ is inversely proportional to the square of $Q$. When $Q = 4$, $P = 9$. [3]

(a) Find the equation connecting $P$ and $Q$. [1]
(b) Find $P$ when $Q = 6$. [2]

Answer (a): $P = \frac{144}{Q^2}$ or $P Q^2 = 144$

Answer (b): 4

Working (a):

• $P \propto \frac{1}{Q^2} \Rightarrow P = \frac{k}{Q^2}$
• $9 = \frac{k}{4^2} = \frac{k}{16} \Rightarrow k = 144$
• Equation: $P = \frac{144}{Q^2}$

Working (b):

• $P = \frac{144}{6^2} = \frac{144}{36} = 4$

Marking (a): 1 mark for $P = \frac{k}{Q^2}$ and $k = 144$. Marking (b): 1 mark for substituting $Q=6$ into equation, 1 mark for answer 4.

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15. A paint mixture contains red, blue, and yellow paint in the ratio $3 : 4 : 5$ by volume. [3]

(a) What percentage of the mixture is blue paint? [1]
(b) If $1.2$ litres of red paint is used, find the total volume of the mixture in litres. [2]

Answer (a): $33\frac{1}{3}\%$ or $33.3\%$

Answer (b): 4.8 litres

Working (a):

• Total parts = $3 + 4 + 5 = 12$
• Blue = 4 parts
• Percentage = $\frac{4}{12} \times 100\% = \frac{1}{3} \times 100\% = 33\frac{1}{3}\%$

Working (b):

• Red = 3 parts = 1.2 L → 1 part = $1.2 \div 3 = 0.4$ L
• Total = 12 parts = $12 \times 0.4 = 4.8$ L

Marking (a): 1 mark for $33\frac{1}{3}\%$ or equivalent. Marking (b): 1 mark for 1 part = 0.4 L, 1 mark for total = 4.8 L.

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16. The time $T$ hours taken to complete a task is inversely proportional to the number of workers $W$. It takes $5$ workers $18$ hours to complete the task. [3]

(a) Find the equation connecting $T$ and $W$. [1]
(b) How many workers are needed to complete the task in $6$ hours? [2]

Answer (a): $T = \frac{90}{W}$ or $TW = 90$

Answer (b): 15

Working (a):

• $T \propto \frac{1}{W} \Rightarrow T = \frac{k}{W}$
• $18 = \frac{k}{5} \Rightarrow k = 90$
• Equation: $T = \frac{90}{W}$

Working (b):

• $6 = \frac{90}{W} \Rightarrow W = \frac{90}{6} = 15$

Marking (a): 1 mark for $T = \frac{k}{W}$ and $k = 90$. Marking (b): 1 mark for substituting $T=6$, 1 mark for answer 15.

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Section C: Application Questions (Questions 17–20)

17. A rectangular field has length and breadth in the ratio $7 : 4$. The perimeter of the field is $440$ m. [4]

(a) Find the length and breadth of the field. [2]
(b) Find the area of the field in hectares. ($1$ hectare $= 10\,000$ m²) [2]

Answer (a): Length = 140 m, Breadth = 80 m

Answer (b): 1.12 hectares

Working (a):

• Let length = $7x$, breadth = $4x$
• Perimeter = $2(7x + 4x) = 22x = 440$
• $x = 440 \div 22 = 20$
• Length = $7 \times 20 = 140$ m
• Breadth = $4 \times 20 = 80$ m

Working (b):

• Area = $140 \times 80 = 11\,200$ m²
• In hectares = $11\,200 \div 10\,000 = 1.12$ hectares

Marking (a): 1 mark for $22x = 440$ and $x = 20$, 1 mark for length = 140 m, breadth = 80 m. Marking (b): 1 mark for area = 11,200 m², 1 mark for 1.12 hectares.

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18. A model of a ship is made to a scale of $1 : 200$. [5]

(a) The length of the model is $1.25$ m. Find the actual length of the ship in metres. [1]
(b) The actual deck area of the ship is $800$ m². Find the deck area of the model in cm². [2]
(c) The actual volume of the ship's hull is $2.4 \times 10^6$ m³. Find the volume of the model's hull in cm³. [2]

Answer (a): 250 m

Answer (b): 200 cm²

Answer (c): 300 cm³

Working (a):

• Scale $1 : 200$ → Actual length = $1.25 \times 200 = 250$ m

Working (b):

• Area scale = $1 : 200^2 = 1 : 40\,000$
• Actual area = $800$ m² = $800 \times 10\,000 = 8\,000\,000$ cm²
• Model area = $8\,000\,000 \div 40\,000 = 200$ cm²

Working (c):

• Volume scale = $1 : 200^3 = 1 : 8\,000\,000$
• Actual volume = $2.4 \times 10^6$ m³ = $2.4 \times 10^6 \times 10^6$ cm³ = $2.4 \times 10^{12}$ cm³
• Model volume = $2.4 \times 10^{12} \div 8 \times 10^6 = 0.3 \times 10^6 = 300\,000$ cm³

Wait, check: $200^3 = 8\,000\,000 = 8 \times 10^6$ $2.4 \times 10^6$ m³ = $2.4 \times 10^6 \times (100)^3$ cm³ = $2.4 \times 10^6 \times 10^6 = 2.4 \times 10^{12}$ cm³ Model volume = $2.4 \times 10^{12} \div 8 \times 10^6 = 0.3 \times 10^6 = 300\,000$ cm³

Correct Answer (c): 300,000 cm³ (or $3 \times 10^5$ cm³)

Marking (a): 1 mark for 250 m. Marking (b): 1 mark for area scale $1:40\,000$ or correct conversion, 1 mark for 200 cm². Marking (c): 1 mark for volume scale $1:8\,000\,000$ or correct conversion, 1 mark for 300,000 cm³.

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19. The cost $C$ of producing $n$ items is given by $C = 500 + 12n$. The items are sold at $20$ each. [4]

(a) Write down an expression for the revenue $R$ from selling $n$ items. [1]
(b) Write down an expression for the profit $P$ from selling $n$ items. [1]
(c) Find the least number of items that must be sold to make a profit. [2]

Answer (a): $R = 20n$

Answer (b): $P = 8n - 500$

Answer (c): 63

Working (a):

• Revenue = selling price × quantity = $20 \times n = 20n$

Working (b):

• Profit = Revenue - Cost = $20n - (500 + 12n) = 20n - 500 - 12n = 8n - 500$

Working (c):

• For profit: $P > 0 \Rightarrow 8n - 500 > 0$
• $8n > 500 \Rightarrow n > 62.5$
• Least integer $n = 63$

Marking (a): 1 mark for $R = 20n$. Marking (b): 1 mark for $P = 8n - 500$ (or $20n - 500 - 12n$). Marking (c): 1 mark for $8n - 500 > 0$ or $8n > 500$, 1 mark for $n = 63$.

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20. A sum of money is invested at simple interest. After $3$ years, the amount is $12\,400$. After $5$ years, the amount is $13\,200$. [5]

(a) Find the interest earned per year. [1]
(b) Find the principal sum invested. [2]
(c) Find the rate of interest per annum. [2]

Answer (a): $400$

Answer (b): $11\,200$

Answer (c): $3\frac{4}{7}\%$ or $3.57\%$ (3 s.f.)

Working (a):

• Interest for 2 years (year 3 to year 5) = $13\,200 - 12\,400 = 800$
• Interest per year = $800 \div 2 = 400$

Working (b):

• Interest for 3 years = $3 \times 400 = 1200$
• Principal = Amount after 3 years - Interest for 3 years = $12\,400 - 1200 = 11\,200$

Working (c):

• Simple interest formula: $I = \frac{PRT}{100}$
• $400 = \frac{11\,200 \times R \times 1}{100}$
• $400 = 112R$
• $R = \frac{400}{112} = \frac{25}{7} = 3\frac{4}{7}\% \approx 3.57\%$

Marking (a): 1 mark for $400$. Marking (b): 1 mark for interest for 3 years = $1200$, 1 mark for principal = $11\,200$. Marking (c): 1 mark for correct substitution into $I = PRT/100$, 1 mark for $R = 25/7\%$ or $3.57\%$.

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End of Answer Key