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Secondary 4 Elementary Mathematics Numbers Ratio Proportion Quiz

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Questions

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Secondary 4 Elementary Mathematics Quiz - Numbers Ratio Proportion

Name: __________________________
Class: __________________________
Date: __________________________
Score: _________ / 40

Duration: 50 Minutes
Total Marks: 40

Instructions:

  1. Answer all questions.
  2. Write your answers in the spaces provided.
  3. Show all necessary working clearly. No marks will be given for correct answers without working.
  4. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
  5. The use of an approved scientific calculator is expected.

Section A: Standard Form and Indices (10 Marks)

1. Write 0.0004050.000405 in standard form. [1]

Answer: __________________________

2. Evaluate (278)−23\left( \frac{27}{8} \right)^{-\frac{2}{3}} without using a calculator. Give your answer as a fraction in its simplest form. [2]

Working:

Answer: __________________________

3. Simplify the expression 3x4y−29x−1y3\frac{3x^4 y^{-2}}{9x^{-1} y^3}. Give your answer with positive indices only. [2]

Working:

Answer: __________________________

4. The mass of a proton is approximately 1.67×10−271.67 \times 10^{-27} kg. The mass of an electron is approximately 9.11×10−319.11 \times 10^{-31} kg. Calculate how many times heavier a proton is than an electron. Give your answer correct to 3 significant figures. [2]

Working:

Answer: __________________________

5. Given that 2x=322^x = 32 and 3y=1273^y = \frac{1}{27}, find the value of x−yx - y. [3]

Working:

Answer: __________________________

Section B: Ratio and Proportion (10 Marks)

6. Divide $840 between Alice, Bob, and Charlie in the ratio 3:4:53 : 4 : 5. Calculate the amount received by Charlie. [2]

Working:

Answer: __________________________

7. yy is directly proportional to the square of xx. When x=4x = 4, y=48y = 48. (a) Find an equation connecting yy and xx. [2]

Working:

Answer: __________________________

(b) Calculate the value of yy when x=10x = 10. [1]

Answer: __________________________

8. The ratio of boys to girls in a club is 5:45 : 4. After 12 boys join the club, the ratio of boys to girls becomes 3:23 : 2. Find the original number of boys in the club. [3]

Working:

Answer: __________________________

9. AA varies inversely as the cube root of BB. When B=8B = 8, A=15A = 15. (a) Find the value of AA when B=64B = 64. [2]

Working:

Answer: __________________________

(b) Find the value of BB when A=5A = 5. [2]

Working:

Answer: __________________________

Section C: Percentages and Applications (10 Marks)

10. A map is drawn to a scale of 1:50,0001 : 50,000. (a) The actual distance between two towns is 12 km. Calculate the distance between the two towns on the map, in centimetres. [2]

Working:



Answer: __________________________

(b) The area of a forest reserve on the map is $20 \text{ cm}^2$. Calculate the actual area of the forest reserve in $\text{km}^2$. [2]

Working:



Answer: __________________________

11. A shopkeeper buys a watch for $120 and sells it for $150. Calculate his percentage profit. [2]

Working:



Answer: __________________________

12. During a sale, the price of a laptop is reduced by 20%. The sale price is $960. Calculate the original price of the laptop. [2]

Working:



Answer: __________________________

13. The population of a town increases by 5% each year. The current population is 40,000. Calculate the population of the town after 3 years. Give your answer correct to the nearest whole number. [3]

Working:



Answer: __________________________

14. Mr. Tan invests $5,000 in a bank account that pays 4% per annum compound interest. Calculate the total amount in the account at the end of 2 years. [2]

Working:



Answer: __________________________

Section D: Mixed Applications (10 Marks)

15. In an examination, 60% of the candidates passed. If 180 candidates failed, calculate the total number of candidates who sat for the examination. [2]

Working:



Answer: __________________________

16. The price of petrol increases by 10% and then decreases by 10%. Calculate the overall percentage change in the price of petrol. [2]

Working:



Answer: __________________________

17. A company's revenue was $2.5 million in 2022 and $2.8 million in 2023. Calculate the percentage increase in revenue from 2022 to 2023. [2]

Working:



Answer: __________________________

18. The ratio of the lengths of two ropes is 7:37 : 3. The longer rope is 14 meters long. (a) Find the length of the shorter rope. [1]

Answer: __________________________

(b) If 2 meters are cut from the longer rope and added to the shorter rope, find the new ratio of the longer rope to the shorter rope in its simplest form. [2]

Working:



Answer: __________________________

19. PP is inversely proportional to Q2Q^2. When Q=3Q = 3, P=20P = 20. (a) Express PP in terms of QQ. [2]

Working:



Answer: __________________________

(b) Find the value of $P$ when $Q = 6$. [1]

Answer: __________________________

20. A rectangular field has dimensions 1.2×1021.2 \times 10^2 m by 8.5×1018.5 \times 10^1 m. (a) Calculate the area of the field in square meters. Give your answer in standard form. [2]

Working:



Answer: __________________________

(b) If the cost of fencing is \$15 per meter, calculate the total cost to fence the perimeter of the field. Give your answer correct to 3 significant figures. [2]

Working:



Answer: __________________________

End of Quiz

Answers

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Secondary 4 Elementary Mathematics Quiz - Numbers Ratio Proportion (Answer Key)

1. 4.05×10−44.05 \times 10^{-4} [1]

  • Move decimal point 4 places to the right.

2. 49\frac{4}{9} [2]

  • (278)−23=(827)23\left( \frac{27}{8} \right)^{-\frac{2}{3}} = \left( \frac{8}{27} \right)^{\frac{2}{3}} [1]
  • =(8273)2=(23)2=49= \left( \sqrt[3]{\frac{8}{27}} \right)^2 = \left( \frac{2}{3} \right)^2 = \frac{4}{9} [1]

3. x53y5\frac{x^5}{3y^5} [2]

  • Coefficients: 39=13\frac{3}{9} = \frac{1}{3}
  • xx terms: x4−(−1)=x5x^{4 - (-1)} = x^5
  • yy terms: y−2−3=y−5=1y5y^{-2 - 3} = y^{-5} = \frac{1}{y^5}
  • Combined: x53y5\frac{x^5}{3y^5}

4. 18301830 (or 1.83×1031.83 \times 10^3) [2]

  • 1.67×10−279.11×10−31=1.679.11×104\frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} = \frac{1.67}{9.11} \times 10^{4}
  • =0.183315...×104=1833.15...= 0.183315... \times 10^4 = 1833.15...
  • Correct to 3 s.f.: 18301830

5. 88 [3]

  • 2x=32⇒2x=25⇒x=52^x = 32 \Rightarrow 2^x = 2^5 \Rightarrow x = 5 [1]
  • 3y=127⇒3y=3−3⇒y=−33^y = \frac{1}{27} \Rightarrow 3^y = 3^{-3} \Rightarrow y = -3 [1]
  • x−y=5−(−3)=5+3=8x - y = 5 - (-3) = 5 + 3 = 8 [1]

6. $350 [2]

  • Total parts = 3+4+5=123 + 4 + 5 = 12
  • Value of 1 part = 840/12=70840 / 12 = 70
  • Charlie's share (5 parts) = 5×70=3505 \times 70 = 350

7. (a) y=3x2y = 3x^2 [2]

  • y=kx2y = kx^2
  • 48=k(42)⇒48=16k⇒k=348 = k(4^2) \Rightarrow 48 = 16k \Rightarrow k = 3
  • Equation: y=3x2y = 3x^2

(b) 300300 [1]

  • y=3(10)2=3(100)=300y = 3(10)^2 = 3(100) = 300

8. 60 [3]

  • Let original boys = 5u5u, girls = 4u4u.
  • New boys = 5u+125u + 12. Girls remain 4u4u.
  • New ratio: 5u+124u=32\frac{5u + 12}{4u} = \frac{3}{2}
  • 2(5u+12)=3(4u)2(5u + 12) = 3(4u)
  • 10u+24=12u10u + 24 = 12u
  • 2u=24⇒u=122u = 24 \Rightarrow u = 12
  • Original boys = 5(12)=605(12) = 60.

9. (a) 7.57.5 [2]

  • A=kB3A = \frac{k}{\sqrt[3]{B}}
  • 15=k83⇒15=k2⇒k=3015 = \frac{k}{\sqrt[3]{8}} \Rightarrow 15 = \frac{k}{2} \Rightarrow k = 30
  • When B=64B = 64: A=30643=304=7.5A = \frac{30}{\sqrt[3]{64}} = \frac{30}{4} = 7.5

(b) 216216 [2]

  • 5=30B35 = \frac{30}{\sqrt[3]{B}}
  • B3=305=6\sqrt[3]{B} = \frac{30}{5} = 6
  • B=63=216B = 6^3 = 216

10. (a) 2424 cm [2]

  • Scale 1:50,0001 : 50,000.
  • Actual distance = 12 km=1,200,000 cm12 \text{ km} = 1,200,000 \text{ cm}.
  • Map distance = 1,200,00050,000=1205=24 cm\frac{1,200,000}{50,000} = \frac{120}{5} = 24 \text{ cm}.

(b) 5 km25 \text{ km}^2 [2]

  • Area scale = (Linear Scale)2=(1:50,000)2=1:2,500,000,000(\text{Linear Scale})^2 = (1 : 50,000)^2 = 1 : 2,500,000,000.
  • Alternatively: 1 cm1 \text{ cm} on map = 0.5 km0.5 \text{ km} actual.
  • 1 cm21 \text{ cm}^2 on map = (0.5 km)2=0.25 km2(0.5 \text{ km})^2 = 0.25 \text{ km}^2 actual.
  • Actual Area = 20×0.25=5 km220 \times 0.25 = 5 \text{ km}^2.

11. 25%25\% [2]

  • Profit = 150−120=30150 - 120 = 30.
  • Percentage Profit = 30120×100%=14×100%=25%\frac{30}{120} \times 100\% = \frac{1}{4} \times 100\% = 25\%.

12. $1200 [2]

  • Let original price be PP.
  • 80%80\% of P=960P = 960.
  • 0.8P=9600.8 P = 960.
  • P=9600.8=1200P = \frac{960}{0.8} = 1200.

13. 46,30546,305 [3]

  • Population = 40,000×(1+0.05)340,000 \times (1 + 0.05)^3
  • =40,000×(1.05)3= 40,000 \times (1.05)^3
  • =40,000×1.157625= 40,000 \times 1.157625
  • =46,305= 46,305.

14. $5,408 [2]

  • Amount = 5000×(1+0.04)25000 \times (1 + 0.04)^2
  • =5000×(1.04)2= 5000 \times (1.04)^2
  • =5000×1.0816= 5000 \times 1.0816
  • =5408= 5408.

15. 450450 [2]

  • Percentage failed = 100%−60%=40%100\% - 60\% = 40\%.
  • 40%40\% of Total = 180180.
  • Total = 1800.4=450\frac{180}{0.4} = 450.

16. 1%1\% decrease [2]

  • Let original price be 100100.
  • After 10% increase: 100×1.1=110100 \times 1.1 = 110.
  • After 10% decrease: 110×0.9=99110 \times 0.9 = 99.
  • Change = 99−100=−199 - 100 = -1.
  • Percentage change = −1%-1\% (1% decrease).

17. 12%12\% [2]

  • Increase = 2.8−2.5=0.32.8 - 2.5 = 0.3 million.
  • Percentage Increase = 0.32.5×100%\frac{0.3}{2.5} \times 100\%.
  • =325×100%=3×4%=12%= \frac{3}{25} \times 100\% = 3 \times 4\% = 12\%.

18. (a) 66 m [1]

  • Ratio 7:37:3. Longer part (7u7u) = 1414.
  • u=2u = 2.
  • Shorter part (3u3u) = 3×2=63 \times 2 = 6 m.

(b) 4:34 : 3 [2]

  • New longer rope = 14−2=1214 - 2 = 12 m.

  • New shorter rope = 6+2=86 + 2 = 8 m.

  • New ratio = 12:812 : 8.

  • Simplify by dividing by 4: 3:23 : 2. Wait, ratio asked is longer to shorter. Longer is 12, Shorter is 8. Ratio 12:8=3:212:8 = 3:2.

    Re-reading question: "new ratio of the longer rope to the shorter rope". After transfer: Rope 1 (was longer): 14−2=1214 - 2 = 12. Rope 2 (was shorter): 6+2=86 + 2 = 8. Is Rope 1 still the longer one? Yes, 12>812 > 8. Ratio 12:8=3:212 : 8 = 3 : 2.

    Correction: The question asks for the ratio of the longer rope to the shorter rope. Current lengths: 12m and 8m. Longer is 12, Shorter is 8. Ratio 12:8=3:212:8 = 3:2.

    Answer: 3:23 : 2

19. (a) P=180Q2P = \frac{180}{Q^2} [2]

  • P=kQ2P = \frac{k}{Q^2}
  • 20=k32⇒20=k9⇒k=18020 = \frac{k}{3^2} \Rightarrow 20 = \frac{k}{9} \Rightarrow k = 180
  • Equation: P=180Q2P = \frac{180}{Q^2}

(b) 55 [1]

  • P=18062=18036=5P = \frac{180}{6^2} = \frac{180}{36} = 5

20. (a) 1.02×104 m21.02 \times 10^4 \text{ m}^2 [2]

  • Area = (1.2×102)×(8.5×101)(1.2 \times 10^2) \times (8.5 \times 10^1)
  • =(1.2×8.5)×(102×101)= (1.2 \times 8.5) \times (10^2 \times 10^1)
  • =10.2×103= 10.2 \times 10^3
  • Standard form: 1.02×1041.02 \times 10^4

(b) $6,150 [2]

  • Perimeter = 2×(1.2×102+8.5×101)2 \times (1.2 \times 10^2 + 8.5 \times 10^1)
  • =2×(120+85)= 2 \times (120 + 85)
  • =2×205=410= 2 \times 205 = 410 m.
  • Cost = 410×15410 \times 15
  • =6150= 6150.
  • Correct to 3 s.f.: 61506150 (or 6.15×1036.15 \times 10^3).