Number - Proportion

Solutions

Question 1

a)

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b)

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c)

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d)

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e)

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f)

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g)

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h)

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Question 2

a) $\begin{align}y&\propto x\\y&=kx\\\text{when }y=10,\,x=5\,\,\,\,\,\,\,10&=k\times5\\\frac{10}{5}&=k\\k&=2\\\text{therefore }y&=2x\end{align}$

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b) $\begin{align}a&\propto b\\a&=kb\\\text{when }a=20,\,b=4\,\,\,\,\,\,\,20&=k\times4\\\frac{20}{4}&=k\\k&=5\\\text{therefore }a&=5b\end{align}$

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c) $\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=1\,\,\,\,\,\,\,2&=k\times1^2\\k&=2\\\text{therefore }y&=2x^2\end{align}$

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d) $\begin{align}f&\propto g^2\\f&=kg^2\\\text{when }f=45,\,g=3\,\,\,\,\,\,\,45&=k\times3^2\\45&=k\times9\\\frac{45}{9}&=k\\k&=5\\\text{therefore }f&=5g^2\end{align}$

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e)

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f) $\begin{align}m&\propto n^3\\m&=kn^3\\\text{when }m=3,\,n=3\,\,\,\,\,\,\,3&=k\times3^3\\3&=k\times27\\\frac{3}{27}&=k\\k&=\frac19\\\text{therefore }m&=\frac19n^3\end{align}$

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g) $\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=4\,\,\,\,\,\,\,2&=k\times4^2\\2&=k\times16\\\frac{2}{16}&=k\\k&=\frac18\\\text{therefore }y&=\frac18x^2\end{align}$

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h) $\begin{align}y&\propto\sqrt x\\y&=\sqrt x\\\text{when }y=10,\,x=4\,\,\,\,\,\,\,10&=k\times\sqrt4\\10&=k\times2\\\frac{10}{2}&=k\\k&=5\\\text{therefore }y&=5\sqrt x\end{align}$

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i) $\begin{align}h&\propto\sqrt g\\h&=k\sqrt g\\5&=k\sqrt{25}\\k&=1\\h&=\sqrt g\end{align}$

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j)

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k)

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Question 3

a) $\begin{align}y&\propto x\\y&=kx\\\text{when }y=6,\,x=2\,\,\,\,\,\,\,6&=k\times2\\\frac{6}{2}&=k\\k&=3\\\text{therefore }y&=3x\\x=10\,\,\,\,\,\,\,y&=3\times10\\y&=30\end{align}$

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b)

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c) $\begin{align}a&\propto b\\a&=kb\\\text{when }a=3,\,b=8\,\,\,\,\,\,\,3&=k\times8\\k&=\frac38\\\text{therefore }a&=\frac38b\\a=5\,\,\,\,\,\,\,5&=\frac38\times b\\5\div\frac38&=b\\5\times\frac83&=b\\b&=\frac{40}{3}\end{align}$

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d)

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e)

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f) $\begin{align}p&\propto q^3\\p&=kq^3\\\text{when }p=9,\,q=3\,\,\,\,\,\,\,9&=k\times3^3\\9&=k\times27\\\frac{9}{27}&=k\\k&=\frac13\\\text{therefore }p&=\frac13q^3\\p=243\,\,\,\,\,\,\,243&=\frac13\times q^3\\243\times3&=q^3\\729&=q^3\\\sqrt[3]{729}&=q\\q&=9\end{align}$

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g)

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h)

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Question 4

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Question 5

a) $\begin{align}y&\propto \frac{1}{x}\\y&=\frac{k}{x}\\\text{when }y=\frac25,\,x=5\,\,\,\,\,\,\,\frac25&=\frac k5\\\frac{2}{5}\times5&=k\\k&=2\\\text{therefore }y&=\frac2x\end{align}$

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b) $\begin{align}A&\propto \frac{1}{B}\\A&=\frac{k}{B}\\\text{when }A=10,\,B=\frac12\,\,\,\,\,\,\,10&=\frac {k}{\frac12}\\10\times\frac12&=k\\k&=5\\\text{therefore }A&=\frac5B\end{align}$

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c) $\begin{align}g&\propto \frac{1}{h^2}\\g&=\frac{k}{h^2}\\\text{when }g=3,\,h=2\,\,\,\,\,\,\,3&=\frac {k}{2^2}\\3\times2^2&=k\\3\times4&=k\\k&=12\\\text{therefore }g&=\frac{12}{h}\end{align}$

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d)

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Question 6

a) $\begin{align}y&\propto \frac1x\\y&=\frac kx\\\text{when }y=4,\,x=3\,\,\,\,\,\,\,4&=\frac k3\\4\times3&=k\\k&=12\\\text{therefore }y&=\frac{12}{x}\\x=4\,\,\,\,\,\,\,y&=\frac{12}{4}\\y&=3\end{align}$

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b) $\begin{align}C&\propto \frac1D\\C&=\frac kD\\\text{when }C=\frac12,\,D=4\,\,\,\,\,\,\,\frac12&=\frac k4\\\frac12\times4&=k\\k&=2\\\text{therefore }C&=\frac{2}{D}\\D=7,\,\,\,\,\,\,\,C&=\frac{2}{7}\end{align}$

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c) $\begin{align}y&\propto \frac{1}{x^2}\\y&=\frac{k}{x^2}\\\text{when }y=3,\,x=4\,\,\,\,\,\,\,3&=\frac {k}{4^2}\\3\times4^2&=k\\3\times16&=k\\k&=48\\\text{therefore }y&=\frac{48}{x^2}\\x=2,\,\,\,\,\,\,\,y&=\frac{48}{2^2}\\y&=\frac{48}{4}\\y&=12\end{align}$

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d)

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f)

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g) $\begin{align}f&\propto \frac{1}{g^3}\\f&=\frac{k}{g^3}\\\text{when }f=\frac12,\,g=2\,\,\,\,\,\,\,\frac12&=\frac {k}{2^3}\\\frac12\times2^3&=k\\\frac12\times8&=k\\k&=4\\\text{therefore }f&=\frac{4}{g^3}\\f=32,\,\,\,\,\,\,\,32&=\frac{4}{g^3}\\32\times g^3&=4\\g^3&=\frac{4}{32}\\g^3&=\frac18\\g&=\sqrt[3]{\frac18}\\g&=\frac12\end{align}$

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h)

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Question 7

a) Let P be the amount of £ and let D be the amount of $\$$.
$£$ are directly proportional to $\$$.

$\begin{align}P&\propto D\\P&=kD\\\text{when }P=5,\,D=7\,\,\,\,\,\,\,5&=k\times7\\k&=\frac57\\\text{therefore }P&=\frac{5}{7}D\\P=20,\,\,\,\,\,\,\,20&=\frac{5}{7}D\\20\div\frac57&=D\\20\times\frac75&=D\\D&=28\end{align}$

So $£20$ can buy $\$28$.

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b)

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c) Let $y$ be the $y$ coordinate and let $x$ be the $x$ coordinate.

$\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=20,\,x=2\,\,\,\,\,\,\,20&=k\times2^2\\20\div2^2&=k\\20\div4&=k\\k&=5\\\text{therefore the equation of the graph is }\,\,\,y&=5x^2\\x=5,\,\,\,\,\,\,\,y&=5\times5^2\\y&=5\times25\\y&=125\end{align}$

So the $y$ coordinate is $125$ when $x=5$.

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d) Let $F$ be the amount of flour and let $V$ be the volume of bread, once baked.

$\begin{align}F&\propto \sqrt[3]V\\F&=k\sqrt[3]V\\\text{when }F=40,\,V=8000\,\,\,\,\,\,\,40&=k\times\sqrt[3]{8000}\\40&=k\times20\\40\div20&=k\\k&=2\\\text{therefore }\,\,\,F&=2\sqrt[3]V\\V=10,000\,,\,\,\,\,\,\,\,F&=2\times\sqrt[3]{10,000}\\F&=43.08869\dots\\F&=43\text{ to nearest gram}\end{align}$

So the amount of flour needed to make a bread with volume $10,000$cm$^3$ is $43$g to the nearest gram.

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e) Let $D$ be the distance to the centre of the Earth and let $G$ be the gravitational force.

$\begin{align}G&\propto \frac{1}{D^2}\\G&=\frac{k}{D^2}\\\text{when }D=6000,\,G=500\,\,\,\,\,\,\,500&=\frac{k}{6000^2}\\500\times6000^2&=k\\500\times36000000&=k\\k&=18000000000 \text{ or }1.8\times10^{10}\\\text{therefore }\,\,\,G&=\frac{1.8\times10^{10}}{D^2}\\D=90,000\,,\,\,\,\,\,\,\,G&=\frac{1.8\times10^{10}}{90,000^2}\\G&=2\frac29\end{align}$

So the gravitational force felt at $90,000$km is $2\frac29$N

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