Secondary 2 Mathematics Numbers Ratio Proportion Quiz

Secondary 2 Mathematics Quiz - Numbers Ratio Proportion

Answer Key

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Section A: Basic Concepts [10 marks]

1. Express $2\frac{3}{8}$ as a percentage.

Answer: 237.5%

Working: $2\frac{3}{8} = \frac{19}{8} = 2.375 = 237.5\%$

Mark scheme: M1 for converting to decimal, A1 for correct percentage

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2. Find the smallest positive integer $k$ such that $\frac{180}{k}$ is a perfect square.

Answer: 5

Working:

• $180 = 2^2 \times 3^2 \times 5$
• For perfect square, all prime powers must be even
• Need to divide by 5 to make $5^1$ become $5^0$
• $k = 5$

Mark scheme: M1 for prime factorization, A1 for correct value

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3. If $y$ is directly proportional to $x$ and $y = 15$ when $x = 5$, find the value of $y$ when $x = 8$.

Answer: 24

Working:

• $y = kx$, so $15 = k \times 5$, therefore $k = 3$
• When $x = 8$: $y = 3 \times 8 = 24$

Mark scheme: M1 for finding constant, A1 for correct value

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4. A map has a scale of 1 : 50000. What is the actual distance in km if the map distance is 8 cm?

Answer: 4 km

Working:

• Actual distance = $8 \times 50000 = 400000$ cm = 4 km

Mark scheme: M1 for correct calculation, A1 for correct distance with units

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5. Express the ratio 0.6 : 1.5 : 2.4 in its simplest form.

Answer: 2 : 5 : 8

Working:

• Multiply by 10: 6 : 15 : 24
• Divide by HCF(6,15,24) = 3: 2 : 5 : 8

Mark scheme: M1 for eliminating decimals, A1 for correct simplified ratio

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Section B: Proportional Relationships [10 marks]

6. $p$ is inversely proportional to the square of $q$. When $p = 8$, $q = 3$.

(a) Find an equation connecting $p$ and $q$. [2]

Answer: $p = \frac{72}{q^2}$

Working:

• $p = \frac{k}{q^2}$
• $8 = \frac{k}{3^2} = \frac{k}{9}$
• $k = 72$
• Therefore $p = \frac{72}{q^2}$

Mark scheme: M1 for correct form $p = \frac{k}{q^2}$, A1 for correct constant

(b) Find the value of $p$ when $q = 6$. [1]

Answer: $p = 2$

Working: $p = \frac{72}{6^2} = \frac{72}{36} = 2$

Mark scheme: A1 for correct value

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7. In a survey of 240 students, 35% chose Mathematics as their favourite subject.

(a) How many students chose Mathematics? [1]

Answer: 84 students

Working: $240 \times 0.35 = 84$

Mark scheme: A1 for correct number

(b) If the number of students who chose Science was $\frac{2}{3}$ of those who chose Mathematics, how many students chose Science? [2]

Answer: 56 students

Working:

• Students who chose Mathematics = 84
• Students who chose Science = $\frac{2}{3} \times 84 = 56$

Mark scheme: M1 for correct setup, A1 for correct answer

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8. The ratio of boys to girls in a school is 7 : 8. If there are 168 boys, find:

(a) The number of girls [2]

Answer: 192 girls

Working:

• Boys : Girls = 7 : 8
• If 7 parts = 168, then 1 part = 24
• Girls = 8 parts = $8 \times 24 = 192$

Mark scheme: M1 for finding value of one part, A1 for correct number of girls

(b) The total number of students [1]

Answer: 360 students

Working: Total = 168 + 192 = 360

Mark scheme: A1 for correct total

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9. $z$ varies directly as $x$ and inversely as the square of $y$. When $x = 12$ and $y = 2$, $z = 18$.

Answer: 8

Working:

• $z = \frac{kx}{y^2}$
• $18 = \frac{k \times 12}{2^2} = \frac{12k}{4} = 3k$, so $k = 6$
• When $x = 8, y = 3$: $z = \frac{6 \times 8}{3^2} = \frac{48}{9} = \frac{16}{3} = 5.33$ (to 3 s.f.)

Mark scheme: A1 for correct value

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10. Three quantities $a$, $b$ and $c$ are in the ratio $3 : 5 : 7$. If $b - a = 14$, find the value of $c$.

Answer: 49

Working:

• Let $a = 3k, b = 5k, c = 7k$
• $b - a = 5k - 3k = 2k = 14$
• So $k = 7$
• Therefore $c = 7k = 7 \times 7 = 49$

Mark scheme: M1 for setting up with parameter, A1 for correct value

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Section C: Applications [10 marks]

11. A recipe for 6 people requires 450g of flour and 300ml of milk.

(a) How much flour is needed for 10 people? [1]

Answer: 750g

Working: $\frac{450 \times 10}{6} = 750$g

Mark scheme: A1 for correct amount

(b) If only 200ml of milk is available, for how many people can the recipe be made? [2]

Answer: 4 people

Working:

• 300ml serves 6 people
• 200ml serves $\frac{200 \times 6}{300} = 4$ people

Mark scheme: M1 for correct setup, A1 for correct number of people

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12. The time $T$ taken for a pendulum to complete one swing is directly proportional to the square root of its length $L$.

(a) Find an equation connecting $T$ and $L$. [2]

Answer: $T = 0.2\sqrt{L}$

Working:

• $T = k\sqrt{L}$
• $1.6 = k\sqrt{64} = k \times 8$
• $k = 0.2$
• Therefore $T = 0.2\sqrt{L}$

Mark scheme: M1 for correct form, A1 for correct constant

(b) Find the length of a pendulum that takes 2 seconds to complete one swing. [2]

Answer: 100 cm

Working:

• $2 = 0.2\sqrt{L}$
• $\sqrt{L} = 10$
• $L = 100$ cm

Mark scheme: M1 for correct substitution, A1 for correct length

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13. A car travels at a constant speed. The distance travelled is directly proportional to the time taken.

Answer: 3.5 hours

Working:

• Distance ∝ Time, so $\frac{d_1}{t_1} = \frac{d_2}{t_2}$
• $\frac{150}{2.5} = \frac{210}{t}$
• $t = \frac{210 \times 2.5}{150} = 3.5$ hours

Mark scheme: A1 for correct time

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14. The cost of printing books varies partly as a fixed charge and partly as the number of books printed.

Answer: $1000

Working:

• Let cost = $a + bn$ where $n$ is number of books
• $800 = a + 100b$ and $1200 = a + 200b$
• Subtracting: $400 = 100b$, so $b = 4$
• $a = 800 - 400 = 400$
• For 150 books: Cost = $400 + 4(150) = 400 + 600 = 1000$

Mark scheme: M1 for setting up equations, A1 for correct cost

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15. A metal rod expands when heated. The length $L$ cm of the rod at temperature $T°C$ is given by $L = 50 + 0.002T$.

(a) Find the length of the rod at 100°C. [1]

Answer: 50.2 cm

Working: $L = 50 + 0.002(100) = 50 + 0.2 = 50.2$ cm

Mark scheme: A1 for correct length

(b) At what temperature will the rod be 50.5 cm long? [1]

Answer: 250°C

Working: $50.5 = 50 + 0.002T$, so $0.002T = 0.5$, therefore $T = 250°C$

Mark scheme: A1 for correct temperature

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Section D: Problem Solving [10 marks]

16. Two taps A and B can fill a tank. Tap A alone takes 6 hours and tap B alone takes 9 hours.

Answer: 3.6 hours

Working:

• Rate of A = $\frac{1}{6}$ tank/hour, Rate of B = $\frac{1}{9}$ tank/hour
• Combined rate = $\frac{1}{6} + \frac{1}{9} = \frac{3+2}{18} = \frac{5}{18}$ tank/hour
• Time = $\frac{1}{\frac{5}{18}} = \frac{18}{5} = 3.6$ hours

Mark scheme: M1 for finding combined rate, A1 for correct time

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17. The pressure $P$ of a gas is inversely proportional to its volume $V$ when temperature is constant.

(a) Find the pressure when the volume is 12 litres. [2]

Answer: 25 units

Working:

• $P = \frac{k}{V}$, so $15 = \frac{k}{20}$, therefore $k = 300$
• When $V = 12$: $P = \frac{300}{12} = 25$ units

Mark scheme: M1 for finding constant, A1 for correct pressure

(b) Find the volume when the pressure is 10 units. [1]

Answer: 30 litres

Working: $10 = \frac{300}{V}$, so $V = 30$ litres

Mark scheme: A1 for correct volume

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18. In a mixture of concrete, the ratio of cement : sand : gravel is 1 : 3 : 5.

(a) The amount of cement needed [1]

Answer: 9 kg

Working: If sand = 27 kg = 3 parts, then 1 part = 9 kg, so cement = 9 kg

Mark scheme: A1 for correct amount

(b) The total mass of the concrete mixture [2]

Answer: 81 kg

Working:

• Cement = 9 kg, Sand = 27 kg, Gravel = 5 × 9 = 45 kg
• Total = 9 + 27 + 45 = 81 kg

Mark scheme: M1 for finding gravel amount, A1 for correct total

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19. The number of bacteria in a culture doubles every 3 hours.

Answer: 8000 bacteria

Working:

• After 12 hours = 4 periods of 3 hours
• Number = $500 \times 2^4 = 500 \times 16 = 8000$

Mark scheme: A1 for correct number

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20. A photocopier reduces documents in the ratio 4 : 3.

Answer: 337.5 cm²

Working:

• Linear scale factor = $\frac{3}{4}$
• Area scale factor = $(\frac{3}{4})^2 = \frac{9}{16}$
• Reduced area = $600 \times \frac{9}{16} = 337.5$ cm²

Mark scheme: A1 for correct area

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Total: 40 marks