A Level H2 Mathematics Numbers Ratio Proportion Quiz

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A-Level Maths H2 Quiz - Numbers Ratio Proportion

Name: ________________________
Class: ________________________
Date: ________________________
Score: ________________________

Duration: 60 minutes
Total Marks: 40
Topic: Numbers, Ratio & Proportion

Instructions:

• Answer all 20 questions.
• Show your working clearly where applicable.
• Use a calculator where necessary.
• Write your answers in the spaces provided.

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Section A: Basic Ratio and Proportion (Questions 1–5)

1. [2 marks]
A recipe uses flour, sugar and butter in the ratio $4 : 2 : 1$ by mass. If a baker uses $350\text{ g}$ of sugar, find the total mass of the three ingredients used.

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2. [2 marks]
Given that $x : y = 3 : 5$ and $y : z = 10 : 7$, find the ratio $x : y : z$ in simplest form.

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3. [2 marks]
If $a$ is directly proportional to $b$ and $a = 18$ when $b = 6$, find the value of $a$ when $b = 11$.

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4. [2 marks]
The variables $p$ and $q$ are inversely proportional. Given $p = 8$ when $q = 3$, find $q$ when $p = 12$.

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5. [2 marks]
A sum of \720$is divided among three people in the ratio$5 : 3 : 4$. How much does the person with the smallest share receive?

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Section B: Algebraic Proportion and Applications (Questions 6–10)

6. [2 marks]
$y$ is inversely proportional to the square of $x$. Given $y = 4$ when $x = 2$, write an equation connecting $y$ and $x$.

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7. [2 marks]
The cost $C$ of printing $n$ books is partly fixed (\50$) and partly proportional to$n$. If printing$100$books costs$$250$, find the cost of printing$250$ books.

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8. [3 marks]
A liquid mixture contains acid and water in the ratio $3 : 8$. How many litres of acid must be added to $44$ litres of this mixture to make the new ratio $5 : 8$?

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9. [3 marks]
A car travels $240\text{ km}$ in $3$ hours at a constant speed. Another car travels the same distance at a speed that is $20\%$ faster. Find the time taken by the second car, in hours and minutes.

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10. [2 marks]
If $\frac{2x + 1}{3x - 2} = \frac{5}{4}$, solve for $x$.

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Section C: Multi-Step Ratio and Proportion Problems (Questions 11–15)

11. [3 marks]
Three workers A, B and C can complete a task in $10$, $15$ and $30$ days respectively working alone. If they work together, how long will they take to finish the task? Give your answer in days as a fraction.

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12. [3 marks]
A map is drawn to a scale of $1 : 25\,000$. If the distance between two towns on the map is $8\text{ cm}$, find the actual distance in kilometres.

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13. [3 marks]
The number of students in two classes is in the ratio $7 : 5$. After $10$ students move from the larger class to the smaller class, the ratio becomes $3 : 2$. Find the original number of students in each class.

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14. [3 marks]
A quantity $Q$ is directly proportional to $r^3$. When $r = 2$, $Q = 16$. Find $Q$ when $r = 5$.

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15. [3 marks]
A shop sells a product at a profit of $25\%$ on cost. During a sale, the price is reduced by $20\%$ of the selling price. What is the overall percentage profit or loss on the original cost?

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Section D: Challenging Synthesis (Questions 16–20)

16. [2 marks]
If $a : b = 2 : 3$ and $b : c = 4 : 5$, find $a : c$.

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17. [3 marks]
A tank can be filled by pipe A in $6$ hours and by pipe B in $4$ hours. If both pipes are opened together but pipe B is turned off after $1$ hour, how much longer will pipe A take to fill the rest of the tank?

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18. [3 marks]
The ratio of boys to girls in a school is $5 : 4$. If $10\%$ of the boys and $20\%$ of the girls wear glasses, what fraction of the whole school wears glasses?

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19. [3 marks]
$x$ varies directly as $y$ and inversely as $z$. When $y = 12$ and $z = 3$, $x = 8$. Find $x$ when $y = 18$ and $z = 6$.

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20. [3 marks]
A sum of money is divided among P, Q and R such that P gets twice as much as Q and R gets \40$more than Q. If the total is$$400$, find how much each receives.

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A-Level Maths H2 Quiz - Numbers Ratio Proportion (Answer Key)

Topic: Numbers, Ratio & Proportion
Total Marks: 40

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Section A: Basic Ratio and Proportion

1. [2 marks]
Ratio flour : sugar : butter = $4 : 2 : 1$.
Sugar corresponds to $2$ parts $= 350\text{ g}$ → $1$ part $= 175\text{ g}$.
Total parts $= 4 + 2 + 1 = 7$ → total mass $= 7 \times 175 = 1225\text{ g}$.
Answer: $1225\text{ g}$.
Teaching note: Divide total known mass by its ratio number to get one part, then multiply by total parts.

2. [2 marks]
$x : y = 3 : 5 = 6 : 10$ (multiply by 2 to match $y=10$).
$y : z = 10 : 7$.
Thus $x : y : z = 6 : 10 : 7$.
Answer: $6 : 10 : 7$.
Common mistake: Not making the $y$ values equal before combining.

3. [2 marks]
$a \propto b \Rightarrow a = kb$.
$18 = k \times 6 \Rightarrow k = 3$.
When $b = 11$, $a = 3 \times 11 = 33$.
Answer: $33$.

4. [2 marks]
$p \propto \frac{1}{q} \Rightarrow pq = k$.
$8 \times 3 = 24 = k$.
When $p = 12$, $12q = 24 \Rightarrow q = 2$.
Answer: $2$.

5. [2 marks]
Ratio $5 : 3 : 4$, total parts $= 12$.
Smallest share $= 3$ parts.
$3/12 \times 720 = 180$.
Answer: \180$.

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Section B: Algebraic Proportion and Applications

6. [2 marks]
$y \propto \frac{1}{x^2} \Rightarrow y = \frac{k}{x^2}$.
$4 = \frac{k}{2^2} = \frac{k}{4} \Rightarrow k = 16$.
Equation: $y = \frac{16}{x^2}$.
Answer: $y = \frac{16}{x^2}$.

7. [2 marks]
$C = 50 + kn$.
$250 = 50 + 100k \Rightarrow 100k = 200 \Rightarrow k = 2$.
For $n=250$: $C = 50 + 2 \times 250 = 550$.
Answer: \550$.

8. [3 marks]
Original mixture: ratio $3:8$, total $44\text{ L}$ → acid $= \frac{3}{11}\times44 = 12\text{ L}$, water $= 32\text{ L}$.
Let $a$ L acid added. New ratio: $\frac{12+a}{32} = \frac{5}{8}$.
$8(12+a) = 5 \times 32 = 160 \Rightarrow 96 + 8a = 160 \Rightarrow 8a = 64 \Rightarrow a = 8$.
Answer: $8\text{ L}$.
Marking: 1 mark setup, 1 mark equation, 1 mark answer.

9. [3 marks]
Speed car 1 $= 240/3 = 80\text{ km/h}$.
Speed car 2 $= 80 \times 1.2 = 96\text{ km/h}$.
Time $= 240/96 = 2.5\text{ h} = 2\text{ h }30\text{ min}$.
Answer: $2$ hours $30$ minutes.
Marking: 1 mark speeds, 1 mark time, 1 mark conversion.

10. [2 marks]
$\frac{2x+1}{3x-2} = \frac{5}{4}$
$4(2x+1) = 5(3x-2)$
$8x + 4 = 15x - 10$
$14 = 7x \Rightarrow x = 2$.
Answer: $x = 2$.

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Section C: Multi-Step Ratio and Proportion Problems

11. [3 marks]
Rates: A $= 1/10$, B $= 1/15$, C $= 1/30$ per day.
Combined rate $= \frac{1}{10}+\frac{1}{15}+\frac{1}{30} = \frac{3+2+1}{30} = \frac{6}{30} = \frac{1}{5}$.
Time $= 5$ days.
Answer: $5$ days (or $\frac{1}{1/5}$).
Marking: 1 mark rates, 1 mark sum, 1 mark final.

12. [3 marks]
Scale $1 : 25\,000$ → $8\text{ cm}$ on map $= 8 \times 25\,000 = 200\,000\text{ cm}$.
$200\,000\text{ cm} = 2000\text{ m} = 2\text{ km}$.
Answer: $2\text{ km}$.
Marking: 1 mark multiply, 1 mark cm→m, 1 mark m→km.

13. [3 marks]
Let classes be $7k$ and $5k$.
After move: $(7k-10) : (5k+10) = 3 : 2$.
$2(7k-10) = 3(5k+10) \Rightarrow 14k - 20 = 15k + 30 \Rightarrow k = 50$.
Original: $350$ and $250$.
Answer: $350$, $250$.
Marking: 1 mark setup, 1 mark solve, 1 mark states.

14. [3 marks]
$Q \propto r^3 \Rightarrow Q = kr^3$.
$16 = k \times 2^3 = 8k \Rightarrow k = 2$.
When $r=5$, $Q = 2 \times 125 = 250$.
Answer: $250$.

15. [3 marks]
Let cost = \100$. Selling price$= 125$. Reduced by$20%$of$125 = 25$→ sale price$= 100$. Overall profit$= 0%$(break-even). **Answer:**$0%$ profit (no profit no loss).
Marking: 1 mark cost/SP, 1 mark reduction, 1 mark conclusion.

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Section D: Challenging Synthesis

16. [2 marks]
$a : b = 2 : 3 = 8 : 12$; $b : c = 4 : 5 = 12 : 15$ → $a : c = 8 : 15$.
Answer: $8 : 15$.

17. [3 marks]
A rate $= 1/6$, B rate $= 1/4$.
In $1$ h together: $(1/6+1/4) = 5/12$ filled.
Remaining $= 7/12$.
A alone time $= (7/12) / (1/6) = 3.5\text{ h}$.
Answer: $3.5$ hours.
Marking: 1 mark combined, 1 mark remainder, 1 mark time.

18. [3 marks]
Boys $= 5k$, girls $= 4k$.
Glasses: $0.1 \times 5k = 0.5k$ boys, $0.2 \times 4k = 0.8k$ girls.
Total glasses $= 1.3k$ out of $9k$ → fraction $= 13/90$.
Answer: $\frac{13}{90}$.

19. [3 marks]
$x = k\frac{y}{z}$.
$8 = k \times \frac{12}{3} = 4k \Rightarrow k = 2$.
When $y=18, z=6$: $x = 2 \times \frac{18}{6} = 6$.
Answer: $6$.

20. [3 marks]
Let Q get \q$. Then P$= 2q$, R$= q+40$.$2q + q + q + 40 = 400 \Rightarrow 4q = 360 \Rightarrow q = 90$. P$= 180$, Q$= 90$, R$= 130$. **Answer:** P$$180$, Q$$90$, R$$130$.
Marking: 1 mark let, 1 mark equation, 1 mark values.