Secondary 3 Elementary Mathematics Numbers Ratio Proportion Quiz

Answer Key: Secondary 3 Elementary Mathematics Quiz - Numbers Ratio Proportion

1.
Numerator: $(2x^3 y^{-2})^4 = 2^4 x^{12} y^{-8} = 16 x^{12} y^{-8}$
Denominator: $8 x^5 y^{-6}$
Expression: $\frac{16 x^{12} y^{-8}}{8 x^5 y^{-6}} = \frac{16}{8} x^{12-5} y^{-8 - (-6)} = 2 x^7 y^{-2}$
Answer: $2x^7 y^{-2}$ or $\frac{2x^7}{y^2}$
[2 marks: 1 for coefficients/indices rules, 1 for final simplified form]

2.
$\left( \frac{27}{8} \right)^{-\frac{2}{3}} = \left( \frac{8}{27} \right)^{\frac{2}{3}} = \left( \sqrt[3]{\frac{8}{27}} \right)^2$
$\sqrt[3]{\frac{8}{27}} = \frac{2}{3}$
$\left( \frac{2}{3} \right)^2 = \frac{4}{9}$
Answer: $\frac{4}{9}$
[2 marks: 1 for handling negative/fractional index, 1 for correct evaluation]

3.
Ratio = $\frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}}$
$= \frac{1.67}{9.11} \times 10^{-27 - (-31)}$
$= 0.183315... \times 10^4$
$= 1.83315... \times 10^3$
Answer: $1.83 \times 10^3$
[2 marks: 1 for correct operation, 1 for standard form and sig figs]

4.
Inside square root: $\frac{16 a^4 b^{-2}}{a^2 b^6} = 16 a^{4-2} b^{-2-6} = 16 a^2 b^{-8}$
Square root: $\sqrt{16 a^2 b^{-8}} = 4 a^1 b^{-4}$
Answer: $4ab^{-4}$ or $\frac{4a}{b^4}$
[2 marks: 1 for simplifying inside root, 1 for final answer]

5.
$3^{2x-1} = (3^3)^{x+2}$
$3^{2x-1} = 3^{3(x+2)}$
Equating indices: $2x - 1 = 3(x + 2)$
$2x - 1 = 3x + 6$
$-1 - 6 = 3x - 2x$
$x = -7$
Answer: $x = -7$
[2 marks: 1 for equating indices correctly, 1 for solving linear equation]

6.
$\frac{A}{B} = \frac{4.5 \times 10^8}{1.5 \times 10^{-3}}$
$= \frac{4.5}{1.5} \times 10^{8 - (-3)}$
$= 3 \times 10^{11}$
Answer: $3 \times 10^{11}$
[2 marks: 1 for division of coefficients, 1 for index arithmetic]

7.
Multiply numerator and denominator by conjugate $(2+\sqrt{3})$:
$\frac{5(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} = \frac{10 + 5\sqrt{3}}{4 - 3} = \frac{10 + 5\sqrt{3}}{1}$
Answer: $10 + 5\sqrt{3}$
[3 marks: 1 for conjugate method, 1 for denominator simplification, 1 for final form]

8.
Let original boys = $5x$, girls = $4x$.
New boys = $5x - 12$, New girls = $4x + 12$.
Ratio: $\frac{5x - 12}{4x + 12} = \frac{1}{2}$
$2(5x - 12) = 1(4x + 12)$
$10x - 24 = 4x + 12$
$6x = 36 \Rightarrow x = 6$
Original boys = $5(6) = 30$.
Answer: 30
[3 marks: 1 for setting up equation, 1 for solving x, 1 for final answer]

9.
(a) $y = \frac{k}{x^2}$. When $x=3, y=20$:
$20 = \frac{k}{3^2} \Rightarrow k = 20 \times 9 = 180$.
Equation: $y = \frac{180}{x^2}$
(b) When $x=5$:
$y = \frac{180}{5^2} = \frac{180}{25} = 7.2$
Answer: (a) $y = \frac{180}{x^2}$, (b) $7.2$
[3 marks: 1 for finding k, 1 for equation, 1 for substitution]

10.
(a) Scale $1:50,000$. Map distance $8.4$ cm.
Actual distance = $8.4 \times 50,000 = 420,000$ cm.
Convert to km: $420,000 \div 100,000 = 4.2$ km.
(b) Area scale = $(1:50,000)^2 = 1 : 2,500,000,000$.
Map area $12$ cm$^2$.
Actual area = $12 \times 2,500,000,000 = 30,000,000,000$ cm$^2$.
Convert to km$^2$: $1 \text{ km}^2 = (10^5 \text{ cm})^2 = 10^{10} \text{ cm}^2$.
Actual area = $\frac{3 \times 10^{10}}{10^{10}} = 3$ km$^2$.
Answer: (a) 4.2 km, (b) 3 km$^2$
[4 marks: 1 for linear calc, 1 for unit conversion (a), 1 for area scale concept, 1 for final area]

11.
(a) $p = k \sqrt[3]{q}$. When $p=4, q=8$:
$4 = k \sqrt[3]{8} \Rightarrow 4 = 2k \Rightarrow k=2$.
Equation: $p = 2\sqrt[3]{q}$
(b) When $q=64$:
$p = 2 \sqrt[3]{64} = 2(4) = 8$.
Answer: (a) $p = 2\sqrt[3]{q}$, (b) 8
[3 marks: 1 for finding k, 1 for equation, 1 for substitution]

12.
Total parts = $3 + 4 + 5 = 12$.
Value of 1 part = 840 / 12 = \70$. Alice =$3 \times 70 = $210$. Charlie =$5 \times 70 = $350$. Difference =$350 - 210 = $140. **Answer:** \140
[2 marks: 1 for value of part, 1 for difference]

13.
(a) $R = \frac{kL}{d^2}$
(b) New $L' = 2L$, New $d' = \frac{d}{2}$.
$R' = \frac{k(2L)}{(\frac{d}{2})^2} = \frac{2kL}{\frac{d^2}{4}} = \frac{8kL}{d^2} = 8R$.
Factor is 8.
Answer: (a) $R = \frac{kL}{d^2}$, (b) 8 times
[3 marks: 1 for formula, 1 for substitution, 1 for final factor]

14.
Ratio Cement:Sand:Gravel = $1:2:4$.
Sand corresponds to 2 parts.
2 parts = $150$ kg $\Rightarrow$ 1 part = $75$ kg.
Total parts = $1+2+4=7$.
Total weight = $7 \times 75 = 525$ kg.
Answer: 525 kg
[2 marks: 1 for finding unit weight, 1 for total]

15.
Time at speed $v$: $t_1 = \frac{240}{v}$.
Time at speed $v+10$: $t_2 = \frac{240}{v+10}$.
Difference is 20 mins = $\frac{20}{60} = \frac{1}{3}$ hour.
$\frac{240}{v} - \frac{240}{v+10} = \frac{1}{3}$
Divide by 240: $\frac{1}{v} - \frac{1}{v+10} = \frac{1}{720}$
$\frac{v+10 - v}{v(v+10)} = \frac{1}{720}$
$\frac{10}{v^2+10v} = \frac{1}{720}$
$7200 = v^2 + 10v$
$v^2 + 10v - 7200 = 0$
[3 marks: 1 for time expressions, 1 for equation setup, 1 for algebraic simplification]

16.
(a) $P = 45000(1.05)^3$
$P = 45000(1.157625) \approx 52093.125$
Answer: 52,093 (nearest whole person).
(b) $45000(1.05)^n > 60000$
$(1.05)^n > \frac{60000}{45000} = 1.333...$
Using logs or trial:
$n=6 \Rightarrow 1.340$ (Exceeds)
$n=5 \Rightarrow 1.276$ (Does not exceed)
Answer: 6 years.
Answer: (a) 52,093, (b) 6 years
[3 marks: 1 for compound interest formula, 1 for calculation (a), 1 for solving inequality/trial (b)]

17.
(a) Linear scale factor $k = \frac{15}{6} = 2.5 = \frac{5}{2}$.
Volume scale factor = $k^3 = (\frac{5}{2})^3 = \frac{125}{8}$.
Ratio Small:Large = $8:125$.
(b) $V_{large} = V_{small} \times \frac{125}{8} = 120 \times \frac{125}{8}$.
$120 / 8 = 15$.
$15 \times 125 = 1875$ cm$^3$.
Answer: (a) $8:125$, (b) 1875 cm$^3$
[3 marks: 1 for volume ratio, 1 for calculation setup, 1 for final answer]

18.
Reflex angle = $240^\circ$. Minor angle = $360^\circ - 240^\circ = 120^\circ$.
Area = $\frac{120}{360} \times \pi r^2 = \frac{1}{3} \pi (10)^2 = \frac{100\pi}{3}$.
$\frac{100 \times 3.142}{3} \approx 104.73$
Answer: 105 cm$^2$ (3 s.f.)
[2 marks: 1 for identifying minor angle, 1 for area calculation]

19.
Original cost per student: $\frac{C}{n}$.
New cost per student: $\frac{C}{n+5}$.
Difference: $\frac{C}{n} - \frac{C}{n+5} = 4$.
Answer: $\frac{C}{n} - \frac{C}{n+5} = 4$ (or equivalent $C(n+5) - Cn = 4n(n+5)$)
[2 marks: 1 for expressions, 1 for equation]

20.
Mass of Gold = $100 \times 19.3 = 1930$ g.
Mass of Silver = $50 \times 10.5 = 525$ g.
Total Mass = $1930 + 525 = 2455$ g.
Total Volume = $100 + 50 = 150$ cm$^3$.
Density = $\frac{2455}{150} = 16.366...$
Answer: 16.4 g/cm$^3$
[2 marks: 1 for total mass/volume, 1 for final density]