\begin{align*} & 30 \text{ cows last } 20 \text{ days} \\ & 10 \text{ cows last } 20 \times 3 = 60 \text{ days } \phantom{00000} [\text{Less cows, food last longer}] \\ & 40 \text{ cows last } {60 \over 4} = 15 \text{ days} \end{align*}

\begin{align*} & 4 \text{ workers manufacture } 80 \text{ chairs in } 12 \text{ days} \\ & 24 \text{ workers manufacture } 80 \text{ chairs in } 12 \div 6 = 2 \text{ days} \phantom{000000} [\text{More workers, less time needed}] \\ & 24 \text{ workers manufacture } 300 \text{ chairs in } 2 \times 3.75 = 7.5 \text{ days} \phantom{00} [\text{More days to manufacture more chairs; } 80 \times 3.75 = 300] \end{align*}

\begin{align*} I & = {PRT \over 100} \\ \\ \text{Interest for \$800} & = {(800)(1.2)(2) \over 100} \\ & = \$ 19.20 \\ \\ \text{Interest for \$700} & = {(700)(1.2)(1) \over 100} \\ & = \$ 8.40 \\ \\ \text{Total amount} & = 800 + 700 + 19.20 + 8.40 \\ & = \$ 1 \phantom{.} 527.60 \end{align*}

(i)

\begin{align*} \text{Amount in NZ\$} & = 1800 \times 1.03 \\ & = \text{NZ\$} 1854 \end{align*}

(ii)

\begin{align*} \text{NZ\$}1.05 & = \$ 1 \\ \text{NZ\$}1 & = \$ {1 \over 1.05} = \$ {20 \over 21} \\ \\ \text{Amount received} & = 350 \times {20 \over 21} \\ & \approx \$ 333. 33 \end{align*}

\begin{align*} \text{Taxable income} & = 240 \phantom{.} 000 - 1000 - 3(4000) - 2(5500) - 48 \phantom{.} 000 \\ & = \$ 168 \phantom{.} 000 \\ \\ \text{Income tax} & = 13 \phantom{.} 950 + (168 \phantom{.} 000 - 160 \phantom{.} 000) \times {18 \over 100} \\ & = \$ 15 \phantom{.} 390 \end{align*}

(i)

\begin{align*} \text{Deposit} & = 1400 \times {1 \over 4} \\ & = \$ 350 \\ \\ \text{Balance (before interest)} & = 1400 - 350 \\ & = \$ 1050 \\ \\ \text{Interest, } I & = {PRT \over 100} \\ & = { (1050)(6)(2) \over 100} \\ & = \$ 126 \\ \\ \text{Balance (after interest)} & = 1050 + 126 \\ & = \$ 1176 \\ \\ \text{Monthly instalment} & = {1176 \over 24} \phantom{000000} [2 \text{ years} = 24 \text{ months}] \\ & = \$ 49 \end{align*}

(ii)

\begin{align*} \text{Extra amount paid} & = \text{Interest} = \$ 126 \end{align*}

\begin{align*} \text{Cost per gram (A)} & = {1.65 \over 3 \times 80} \\ & = \$ 0.006 \phantom{.} 875 \text{ per gram} \\ \\ \text{Cost per gram (B)} & = {2 \times 2 - 1 \over 5 \times 60 \times 2} \\ & = \$ 0.005 \text{ per gram} \\ \\ \text{Cost per gram (C)} & = {1.80 \over 5 \times 60} \\ & = \$ 0.006 \text{ per gram} \\ \\ \text{Cost per gram (D)} & = {2 \times 2 - 1.10 \over 5 \times 60 \times 2} \\ & = \$ 0.004 \phantom{.} 833 \text{ per gram} \\ \\ \therefore \text{Ali } & \text{should buy brand D} \end{align*}

(i)

\begin{align*} \text{Volume of sphere, } V & = {4 \over 3} \pi r^3 \phantom{000000} [\text{Formula provided}] \\ \\ \implies y & = {4 \over 3} \pi x^3 \\ y & \propto x^3 \\ \\ \therefore n & = 3 \end{align*}

(ii)

\begin{align*} \text{Distance} & = \text{Speed} \times \text{Time} \\ \\ \text{Since } & \text{distance is fixed,} \\ k & = y \times x \\ {k \over x} & = y \\ k x^{-1} & = y \phantom{000000} \left[ a^{-n} = {1 \over a^n} \right] \\ \\ \implies y & \propto x^{-1} \\ \\ n & = -1 \end{align*}

\begin{align*} y & = {k \over x} \\ \\ \text{When } & x = 8 \text{ and } y = 2.5, \\ 2.5 & = {k \over 8} \\ {5 \over 2} & = {k \over 8} \\ 8(5) & = 2k \phantom{000000} [\text{Cross-multiply}] \\ 40 & = 2k \\ {40 \over 2} & = k \\ 20 & = k \\ \\ y & = {20 \over x} \\ \\ \text{When } & x = 10, \\ y & = {20 \over 10} \\ y & = 2 \\ \\ \text{When } & y = 0.8, \\ 0.8 & = {20 \over x} \\ 0.8x & = 20 \\ x & = {20 \over 0.8} \\ x & = 25 \end{align*}

(a)(i)

\begin{align*} V & = k r^2 \\ \\ \text{When } & r = 5 \text{ and } V = 240, \\ 240 & = k(5)^2 \\ 240 & = k(25) \\ {240 \over 25} & = k \\ 9.6 & = k \\ \\ \therefore V & = 9.6 r^2 \end{align*}

(a)(ii)

\begin{align*} V & = 9.6 r^2 \\ \\ \text{When } & r = 8, \\ V & = 9.6 (8)^2 \\ & = 614.4 \\ & \approx 614 \text{ cm}^3 \text{ (to nearest whole number)} \end{align*}

(b)

\begin{align*} V & = 9.6 r^2 \\ \\ \text{Let radius of Q} & = x \\ \\ \text{Volume of } Q & = 9.6 x^2 \\ \\ \text{Radius of } P & = x \times {125 \over 100} \\ & = 1.25x \\ \\ \text{Volume of } P & = 9.6 (1.25x)^2 \\ & = 9.6(1.5625x^2) \\ & = 15x^2 \\ \\ { \text{Volume of } P \over \text{Volume of } Q } & = {15x^2 \over 9.6 x^2} \\ & = {15 \over 9.6} \\ & = {25 \over 16} \\ \\ \therefore & \phantom{.} 25: 16 \end{align*}

\begin{align*} y & = kx^2 \end{align*} \begin{align*} \text{When } & x = 2, &&& \text{When } & x = 5, \\ y & = k(2)^2 &&& y & = k(5)^2 \\ y & = k(4) &&& y & = k(25) \\ y & = 4k &&& y & = 25k \end{align*}
\begin{align*} 25k -4k & = 32 \\ 21k & = 32 \\ k & = {32 \over 21} \\ \\ \therefore y & = {32 \over 21}x^2 \\ \\ \text{When } & x = 3, \\ y & = {32 \over 21}(3)^2 \\ y & = 13{5 \over 7} \end{align*}

(a)

\begin{align*} F & = {k \over d^2} \\ \\ \text{When } & d = 8 \text{ and } F = 10, \\ 10 & = {k \over 8^2} \\ 10 & = {k \over 64} \\ 64(10) & = k \\ 640 & = k \\ \\ F & = {640 \over d^2} \end{align*}

(b)

\begin{align*} F & = {640 \over d^2} \\ \\ \text{When } & F = 25, \\ 25 & = {640 \over d^2} \\ 25d^2 & = 640 \\ d^2 & = {640 \over 25} \\ d & = \sqrt{640 \over 25} \\ d & = 5.0596 \\ d & \approx 5.06 \text{ cm} \end{align*}

(c)

\begin{align*} F & = {640 \over d^2} \\ \\ \text{When } d \text{ is doubled, } & F \text{ is } {1 \over 2} \times {1 \over 2} = {1 \over 4} \text{ of initial value} \\ \\ \text{New value of } F & = 12 \times {1 \over 3} \\ & = 3 \text{ N} \end{align*}

(a)

\begin{align*} y & = k \sqrt{x} \phantom{000000} [\text{Direct proportion since as } x \text{ increases, } y \text{ increases}] \\ \\ \text{When } & x = 4 \text{ and } y = 6, \\ 6 & = k \sqrt{4} \\ 6 & = 2k \\ 3 & = k \\ \\ \text{When } & x = 9 \text{ and } y = 9, \\ 9 & = k \sqrt{9} \\ 9 & = 3k \\ 3 & = k \\ \\ \text{When } & x = 16 \text{ and } y = 12, \\ 12 & = k \sqrt{16} \\ 12 & = 4k \\ 3 & = k \\ \\ \text{When } & x = 25 \text{ and } y = 15, \\ 15 & = k \sqrt{25} \\ 15 & = 5k \\ 3 & = k \\ \\ \therefore y & = 3 \sqrt{x} \end{align*}

(b)

\begin{align*} y & = 3 \sqrt{x} \\ \\ y & \propto \sqrt{x} \end{align*}

(c)

\begin{align*} y & = 3 \sqrt{x} \\ \\ \text{When } & y = 24, \\ 24 & = 3 \sqrt{x} \\ {24 \over 3} & = \sqrt{x} \\ 8 & = \sqrt{x} \\ 8^2 & = x \\ 64 & = x \end{align*}

(a)

\begin{align*} 1 \text{ cm} & : 20 \phantom{.} 000 \text{ cm} \\ 1 \text{ cm} & : 200 \text{ m} \phantom{000000} [1 \text{ m} = 100 \text{ cm} ] \\ 1 \text{ cm} & : 0.2 \text{ km} \phantom{00000.} [1 \text{ km} = 1000 \text{ m}] \\ \\ n & = 0.2 \end{align*}

(b)

\begin{align*} 1 \text{ cm} & : 0.2 \text{ km} \\ 18 \text{ cm} & : 3.6 \text{ km} \\ \\ \text{Actual distance} & = 3.6 \text{ km} \end{align*}

(c)

\begin{align*} 1 \text{ cm} & : 0.2 \text{ km} \\ 1^2 \text{ cm}^2 & : 0.2^2 \text{ km}^2 \\ 1 \text{ cm}^2 & : 0.04 \text{ km}^2 \\ 80 \text{ cm}^2 & : 3.2 \text{ km}^2 \\ \\ \text{Area on map} & = 80 \text{ cm}^2 \end{align*}

(a)

\begin{align*} 1 \text{ cm} & : 4 \text{ km} \\ 6 \text{ cm} & : 24 \text{ km} \\ \\ \text{Actual distance} & = 24 \text{ km} \end{align*}

(b)

\begin{align*} 1 \text{ cm} & : 4 \text{ km} \phantom{000000} [\text{First map}] \\ 1^2 \text{ cm}^2 & : 4^2 \text{ km}^2 \\ 1 \text{ cm}^2 & : 16 \text{ km}^2 \\ 40 \text{ cm}^2 & : 640 \text{ km}^2 \\ \\ \text{Actual area} & = 640 \text{ km}^2 \\ \\ 0.4 \text{ cm}^2 & : 640 \text{ km}^2 \phantom{000000} [\text{Second map}] \\ 1 \text{ cm}^2 & : 1600 \text{ km}^2 \\ \sqrt{1} \text{ cm} & : \sqrt{1600} \text{ km} \\ 1 \text{ cm} & : 40 \text{ km} \\ 1 \text{ cm} & : 40 \phantom{.} 000 \text{ m} \phantom{00000.} [1 \text{ km} = 1000 \text{ m}] \\ 1 \text{ cn} & : 4 \phantom{.} 000 \phantom{.} 000 \text{ cm} \phantom{00} [1 \text{ m} = 100 \text{ cm} ] \\ \\ \therefore n & = 4 \phantom{.} 000 \phantom{.} 000 \end{align*}

(a)

\begin{align*} 1 \text{ cm} & : 20 \text{ cm} \\ 1 \text{ cm} & : 0.2 \text{ m} \phantom{000000} [1 \text{ m} = 100 \text{ cm}] \\ 24 \text{ cm} & : 4.8 \text{ m} \\ \\ \text{Length of lorry} & = 4.8 \text{ m} \end{align*}

(b)

\begin{align*} 1 \text{ cm} & : 0.2 \text{ m} \\ 1^2 \text{ cm}^2 & : 0.2^2 \text{ m}^2 \phantom{000000} \left[ {A_1 \over A_2} = \left(l_1 \over l_2\right)^2 \right] \\ 1 \text{ cm}^2 & : 0.04 \text{ m}^2 \\ 250 \text{ cm}^2 & : 10 \text{ m}^2 \\ \\ \text{Area of platform of model} & = 250 \text{ cm}^2 \end{align*}

(c)

\begin{align*} 1 \text{ cm} & : 0.2 \text{ m} \\ 1^3 \text{ cm}^3 & : 0.2^3 \text{ m}^3 \phantom{000000} \left[ {V_1 \over V_2} = \left(l_1 \over l_2\right)^3 \right] \\ 1 \text{ cm}^3 & : 0.008 \text{ m}^3 \\ 30 \text{ cm}^3 & : 0.24 \text{ m}^3 \\ \\ \text{Volume of fuel} & = 0.24 \text{ m}^3 \\ & = 240 \phantom{.} 000 \text{ cm}^3 \phantom{000000} [1 \text{ m}^3 = 1 \phantom{.} 000 \phantom{.} 000 \text{ cm}^3] \\ & = 240 \text{ litres} \phantom{00000000.} [1 \text{ litres} = 1000 \text{ cm}^3] \end{align*}

(i)

\begin{align*} 210 \text{ m/s} & = { 210 \text{ m} \over 1 \text{ second}} \\ & = { {210 \over 1000} \text{ km} \over {1 \over 3600} \text{ hour} } \\ & = 756 \text{ km/h} \end{align*}

(ii)

\begin{align*} \text{Time taken} & = {\text{Distance} \over \text{Speed}} \\ & = {3240 \text{ km} \over 756 \text{ km/h} } \\ & = 4{2 \over 7} \text{ h} \\ & = 4 \text{ hours } {2 \over 7} \times 60 \text{ minutes} \\ & \approx 4 \text{ hours } 17 \text{ minutes} \end{align*}

\begin{align*} \text{Circumference of circle} & = \pi d \\ \text{Circumference of wheel} & = \pi (58.4) \\ & = 58.4 \pi \text{ cm} \\ \\ \text{Distance covered by bicycle in one minute} & = 25 \div 60 \phantom{000000} [25 \text{ km/h} \implies 25 \text{ km in 1 hour}] \\ & = {5 \over 12} \text{ km} \\ & = 416{2 \over 3} \text{ m} \\ & = 41 \phantom{.} 666 {2 \over 3} \text{ cm} \\ \\ \text{Number of revolutions} & = { 41 \phantom{.} 666 {2 \over 3} \text{ cm} \over 58.4 (3.142) \text{ cm} } \\ & = 227.07 \\ & \approx 227 \end{align*}

\begin{align*} \text{Speed} & = { \text{Distance} \over \text{Time} } \\ \\ \text{Initial speed} & = { 57 \text{ km} \over 2{3 \over 4} \text{ h} } \\ & = 20{8 \over 11} \text{ km/h} \\ \\ \text{Required speed} & = { 117 - 57 \over 6.5 - 2{3 \over 4} } \\ & = 16 \text{ km/h} \\ \\ \text{Required reduction} & = 20{8 \over 11} - 16 \\ & = 4{8 \over 11} \text{ km/h} \end{align*}

(i)

\begin{align*} 298 \phantom{.} 000 \text{ km/h} & = { 298 \phantom{.} 000 \text{ km} \over 1 \text{ h} } \\ & = { 298 \phantom{.} 000 \times 1000 \text{ m} \over 3600 \text{ s} } \\ & = 82777{7 \over 9} \\ & \approx 8.2800 \\ & = 8.28 \times 10^4 \text{ m/s} \end{align*}

(ii)

\begin{align*} \text{Time taken} & = { \text{Distance} \over \text{Speed} } \\ & = { 372.5 \text{ m} \over 82777 {7 \over 9} \text{ m/s}} \\ & = 0.0045 \text{ s} \\ \\ \text{Time difference} & = 0.0045 \times 2 \phantom{000000} [\text{Account for reflection}] \\ & = 0.009 \\ & = 9 \times 10^{-3} \text{ s} \end{align*}