(a)

\begin{align*} 2 & | \underline{1320} \\ 2 & | \underline{660} \\ 2 & | \underline{330} \\ 3 & | \underline{165} \\ 5 & | \underline{55} \\ 11 & | \underline{11} \\ & | \underline{1} \\ \\ 1320 & = 2^3 \times 3 \times 5 \times 11 \\ \\ \therefore x & = 3, y = 1 \end{align*}

(b)

\begin{align*} a & = 2^3 \times 5 \times 11 \\ \\ \text{LCM} = 1320 & = 2^3 \times 3 \times 5 \times 11 \\ \\ \text{HCF} = 20 & = 2^2 \times 5 \\ \\ b & = 2^2 \times 3 \times 5 \phantom{000000} [\text{Must have factors } 2^2 \text{ and } 5 \text{ (due to HCF)}] \\ & = 60 \phantom{0000000000000}[\text{Must have factor } 3 \text{ (due to LCM)}] \end{align*}

(c)

\begin{align*} 1320 & = 2^3 \times 3 \times 5 \times 11 \\ \\ l & = 2 \times 5 = 10 \\ \\ b & = 2^2 \times 3 = 12 \\ \\ h & = 11 \\ \\ \text{Dimensions: } & 10 \text{ cm by 12 cm by 11 cm} \end{align*}

(a)

\begin{align*} 12 & = 2^2 \times 3 \\ \\ 18 & = 2 \times 3^2 \\ \\ \text{LCM} & = 2^2 \times 3^2 \\ & = 36 \end{align*}

(b)

\begin{align*} \text{HCF} & = 26 = 2 \times 13 \phantom{000000} [y \text{ must have factors 2 and 13}] \\ \\ y & = 2 \times 11 \times 13 = 286 \\ \\ y & = 2 \times 3^2 \times 13 = 234 \end{align*}

(a)

\begin{align*} 2 & | \underline{1008} \\ 2 & | \underline{504} \\ 2 & | \underline{252} \\ 2 & | \underline{126} \\ 3 & | \underline{63} \\ 3 & | \underline{21} \\ 7 & | \underline{7} \\ & | \underline{1} \end{align*}
$$ 1008 = 2^4 \times 3^2 \times 7 $$

(b)(i)

\begin{align*} M & = a^x \times b^2 \times 7 \\ \\ N & = a^{ x+ 1} \times b^3 \times 7 \\ \\ \text{HCF} = 1008 & = 2^4 \times 3^2 \times 7 \\ \\ a & = 2, x = 4, b = 3 \end{align*}

(b)(ii)

\begin{align*} M & = a^x \times b^2 \times 7 \\ & = 2^4 \times 3^2 \times 7 \\ \\ N & = a^{ x+ 1} \times b^3 \times 7 \\ & = 2^5 \times 3^3 \times 7 \\ \\ \text{LCM} & = 2^5 \times 3^3 \times 7 \\ & = 6048 \end{align*}

(a)

\begin{align*} 1550 & \approx 1600 \text{ (to nearest hundred)} \\ \\ 1649 & \approx 1600 \text{ (to nearest hundred)} \\ \\ \text{Least number} & = 1550 \\ \\ \text{Greatest number} & = 1649 \end{align*}

(b)

\begin{align*} \text{Least amount each person gets} & = { \text{Least amount of prize money} \over \text{Greatest number of receivers}} \\ \\ \text{Least amount of prize money} & = \$ 49 \phantom{.} 500 \phantom{000000} [ 49 \phantom{.} 500 \approx 50 \phantom{.} 000 \text{ (to nearest thousand)}] \\ \\ \text{Greatest number of receivers} & = 1649 \times {10 \over 100} \\ & = 164.9 \\ & \approx 164 \\ \\ \text{Least amount each person gets} & = { 49 \phantom{.} 500 \over 164} \\ & = 301.829 \\ & \approx \$ 301.83 \end{align*}

(a)

\begin{align*} 8 \times 10^3 & = 8000 \\ \\ 7 \times 10^2 & = 700 \\ \\ 6 \times 10^0 & = 6 \\ \\ 3 \times 10^{-2} & = 0.03 \\ \\ \therefore m & = 0, n = -2 \end{align*}

(b)

\begin{align*} 8706.03 & = 8.706 \phantom{.} 03 \times 10^3 \end{align*}

(c)(i)

\begin{align*} 8706.03 & \approx 8710 \end{align*}

(c)(ii)

\begin{align*} 8706.03 & \approx 8706.0 \end{align*}

(a)

\begin{align*} -{5 \over 8}, 3.2 \times 10^{-12}, 0.003, {1 \over 300} \end{align*}

(b)(i)

\begin{align*} b + c & = {1 \over 300} + 0.003 \\ & = 0.006 \phantom{.} 333 \\ & = 6.333 \times 10^{-3} \\ & \approx 6.33 \times 10^{-3} \end{align*}

(b)(ii)

\begin{align*} {b \over a} & = { {1 \over 300} \over 3.2 \times 10^{-12} } \\ & = 1 \phantom{.} 041 \phantom{.} 666 \phantom{.} 667 \\ & \approx 1 \phantom{.} 04 \phantom{.} 000 \phantom{.} 000 \\ & = 1.04 \times 10^9 \end{align*}

(b)(iii)

\begin{align*} -a^2 & = - (3.2 \times 10^{-12})^2 \\ & = - 1.024 \times 10^{-23} \\ & \approx -1.02 \times 10^{-23} \end{align*}

(a)

\begin{align*} 60 \text{ cm} & = 0.6 \text{ m} \phantom{000000} [1 \text{ m} = 100 \text{ cm}] \\ \\ \text{Thickness of one sheet} & = {0.6 \over 750} \\ & = 0.000 8 \\ & = 8 \times 10^{-4} \text{ m} \end{align*}

(b)

\begin{align*} 59.5 & \approx 60 \\ \\ \text{Least possible} & = 59.5 \text{ cm} \end{align*}

(c)

\begin{align*} \text{No. of 750 sheets} & = {1.8 \over 0.6} \\ & = 3 \\ \\ \text{No. of sheets} & = 750 \times 3 \\ & = 2250 \end{align*}

(a)

\begin{align*} \text{Percentage discount} & = {240 - 180 \over 240} \times 100 \\ & = 25 \% \end{align*}

(b)

\begin{align*} \text{Usual price} & = 285 \times {100 \over 100 - 25} \\ & = \$ 380 \end{align*}

(a)

\begin{align*} \text{Price after year 1} & = 2400 \times {130 \over 100} \\ & = \$ 3120 \\ \\ \text{Price after year 2} & = 3120 \times {130 \over 100} \\ & = \$4056 \end{align*}

(b)

\begin{align*} \text{Selling price} & = 2400 \times {100 + 125 \over 100} \\ & = \$ 5400 \end{align*}

(c)

\begin{align*} \text{Let width} & = x \\ \\ \text{Height} & = x \times {80 \over 100} \\ & = 0.8x \\ \\ \text{Perimeter} & = 2x + 2(0.8x) \\ 216 & = 2x + 1.6x \\ 216 & = 3.6x \\ {216 \over 3.6} & = x \\ 60 & = x \\ \\ \text{Width} & = 60 \text{ cm} \end{align*}

\begin{align*} \text{Current paying customers} & = 1280 - 1244 \\ & = 36 \\ \\ \text{Required paying customers} & = 1280 \times {20 \over 100} \\ & = 256 \\ \\ \text{Conversions required} & = 256 - 36 \\ & = 220 \end{align*}

\begin{align*} 100 - 36 & = 64 \\ \\ \text{Percentage of water left behind} & = 80 \times {64 \over 100} \\ & = 51.2 \% \\ \\ \text{Volume of bottle} & = 1.28 \times {100 \over 51.2} \\ & = 2.5 \text{ litres} \end{align*}

\begin{align*} \text{Balance owned (before interest)} & = 1498 - 500 \\ & = \$ 998 \\ \\ \text{Balance owned (after interest)} & = 185 \times 6 \\ & = \$ 1110 \\ \\ \text{Interest} & = 1110 - 998 \\ & = \$ 112 \\ \\ \text{Required percentage} & = {112 \over 1498} \times 100 \\ & = 7.4766 \\ & \approx 7.48 \% \end{align*}

(a)

\begin{align*} 100 - 20 & = 80 \\ \\ \text{Percentage of water left} & = 45 \times {80 \over 100} \\ & = 36 \% \\ \\ \text{Capacity of tank} & = 16.2 \times {100 \over 36} \\ & = 45 \text{ litres} \end{align*}

(b)

\begin{align*} 1 - {2 \over 7} & = {5 \over 7} \\ \\ 100 - 60 & = 40 \\ \\ \text{Fraction of employees by other modes} & = {5 \over 7} \times {40 \over 100} \\ & = {2 \over 7} \\ \\ \text{No. of employees} & = 38 \times {7 \over 2} \\ & = 133 \end{align*}

(c)

\begin{align*} \text{Books owned by Bryan} & = 125 \times {100 - 4 \over 100} \\ & = 120 \\ \\ \text{Books owned by Ayden} & = 120 \times {100 - 10 \over 100} \\ & = 108 \\ \\ \text{Total books} & = 125 + 120 + 108 \\ & = 353 \end{align*}

(a)

\begin{align*} 2 & | \underline{756} \\ 2 & | \underline{378} \\ 3 & | \underline{189} \\ 3 & | \underline{63} \\ 3 & | \underline{21} \\ 7 & | \underline{7} \\ & | \underline{1} \\ \\ \therefore 756 & = 2^2 \times 3^3 \times 7 \end{align*}

(b)

\begin{align*} {756 \over m} & = { 2^2 \times 3^3 \times 7 \over 3 \times 7} \\ & = 2^2 \times 3^2 \phantom{000000} [\text{Perfect square}] \\ \\ \therefore m & = 3 \times 7 = 21 \end{align*}

(c)(i)

\begin{align*} 756k & = 2^2 \times 3^3 \times 7 \times k \\ & = 2^2 \times 3^3 \times 7 \times (2 \times 7^2) \\ & = 2^3 \times 3^3 \times 7^3 \\ \\ \therefore k & = 2 \times 7^2 = 98 \end{align*}

(c)(ii)

\begin{align*} \text{Volume} = 756k & = 2^3 \times 3^3 \times 7^3 \\ \\ \text{Length of cube} & = \sqrt[3]{ 2^3 \times 3^3 \times 7^3 } \\ & = 2 \times 3 \times 7 \\ & = 42 \\ \\ \text{Surface area} & = 6(42 \times 42) \\ & = 10 \phantom{.} 584 \text{ cm}^2 \end{align*}

(d)

\begin{align*} {756 p \over q} & = { 2^2 \times 3^3 \times 7 \times p \over q} \\ & = {2^2 \times 3^3 \times 7 \times 2 \over 7} \\ & = 2^3 \times 3^3 \\ \\ \therefore p & = 2, q = 7 \end{align*}

(a)

\begin{align*} 5 \text{ kg} & = (5 \times 1000) \text{ g} \\ & = 5000 \text{ g} \\ \\ {40 \over 5000} \times 100 & = 0.8 \% \end{align*}

(b)

\begin{align*} \text{World population in 2001} & = (7.84 \times 10^9) \times {100 \over 100 + 26} \\ & = 6 \phantom{.} 222 \phantom{.} 222 \phantom{.} 222 \\ & = 6.222 \phantom{.} 222 \phantom{.} 222 \times 10^9 \\ & \approx 6.22 \times 10^9 \end{align*}

(c)(i)

\begin{align*} \text{Percentage change} & = { \text{Final} - \text{Initial} \over \text{Initial} } \times 100 \\ & = { (8.18 \times 10^9) - (7.84 \times 10^9) \over 7.84 \times 10^9} \times 100 \\ & = 4.3367 \\ & \approx 4.34 \% \end{align*}

(c)(ii)

\begin{align*} \text{Total amount} & = P \left(1 + {r \over 100}\right)^n \phantom{000000} [\text{Compound interest formula}] \\ 8.18 \times 10^9 & = (7.84 \times 10^9) \left(1 + {r \over 100}\right)^4 \\ {8.18 \times 10^9 \over 7.84 \times 10^9} & = \left(1 + {r \over 100}\right)^4 \\ {409 \over 392} & = \left(1 + {r \over 100}\right)^4 \\ \pm \sqrt[4]{409 \over 392} & = 1 + {r \over 100} \\ \\ \text{Since } 1 + {r \over 100} > 0, & \text{ reject } - \sqrt[4]{409 \over 392} \\ \\ 1 + {r \over 100} & = \sqrt[4]{409 \over 392} \\ {r \over 100} & = \sqrt[4]{409 \over 392} - 1 \\ r & = 100 \left( \sqrt[4]{409 \over 392} - 1 \right) \\ r & = 1.0669 \\ r & \approx 1.07 \end{align*}