Number - Proportion
Question 1
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is directly proportional to $x$
a
$y\propto x$, $y=kx$
Question ID: 10070010010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010010
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $m$ is directly proportional to $n$
a
$m\propto n$, $m=kn$
Question ID: 10070010020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010020
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $a$ is directly proportional to the square of $b$
a
$a\propto b^2$, $a=kb^2$
Question ID: 10070010030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010030
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $p$ is directly proportional to the cube of $q$
a
$p\propto q^3$, $p=kq^3$
Question ID: 10070010040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010040
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $u$ is directly proportional to the root of $v$
a
$u\propto \sqrt v$, $u=k\sqrt v$
Question ID: 10070010050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $g$ is directly proportional to the cube root of $h$
a
$g\propto \sqrt[3] h$, $g=k\sqrt[3] h$
Question ID: 10070010060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010060
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $c$ is directly proportional to the $5$th power of $d$
a
$c\propto d^5$, $c=kd^5$
Question ID: 10070010070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010070
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $s$ is directly proportional to the $7$th root of $t$
a
$s\propto \sqrt[7] t$, $s=k\sqrt[7] t$
Question ID: 10070010080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070010080
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070010080
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 2
By finding the constant of proportionality form an equation linking the variables
a) $y$ is directly proportional to $x$. When $y=10, x=5$.
a
$y=2x$
Question ID: 10070020010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto x\\y&=kx\\\text{when }y=10,\,x=5\,\,\,\,\,\,\,10&=k\times5\\\frac{10}{5}&=k\\k&=2\\\text{therefore }y&=2x\end{align}$
Question ID: 10070020010
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $a$ is directly proportional to $b$. When $a=20, b=4$.
a
$a=5b$
Question ID: 10070020020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}a&\propto b\\a&=kb\\\text{when }a=20,\,b=4\,\,\,\,\,\,\,20&=k\times4\\\frac{20}{4}&=k\\k&=5\\\text{therefore }a&=5b\end{align}$
Question ID: 10070020020
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $y$ is directly proportional to the square of $x$. When $y=2, x=1$.
a
$y=2x^2$
Question ID: 10070020030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=1\,\,\,\,\,\,\,2&=k\times1^2\\k&=2\\\text{therefore }y&=2x^2\end{align}$
Question ID: 10070020030
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $f$ is directly proportional to the square of $g$. When $f=45, g=3$.
a
$f=5g^2$
Question ID: 10070020040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}f&\propto g^2\\f&=kg^2\\\text{when }f=45,\,g=3\,\,\,\,\,\,\,45&=k\times3^2\\45&=k\times9\\\frac{45}{9}&=k\\k&=5\\\text{therefore }f&=5g^2\end{align}$
Question ID: 10070020040
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $y$ is directly proportional to the cube of $x$. When $y=24, x=2$.
a
$y=3x^3$
Question ID: 10070020050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070020050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070020050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $m$ is directly proportional to the cube of $n$. When $m=3, n=3$.
a
$n=\frac19n^3$
Question ID: 10070020060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}m&\propto n^3\\m&=kn^3\\\text{when }m=3,\,n=3\,\,\,\,\,\,\,3&=k\times3^3\\3&=k\times27\\\frac{3}{27}&=k\\k&=\frac19\\\text{therefore }m&=\frac19n^3\end{align}$
Question ID: 10070020060
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $y$ is directly proportional to the square of $x$. When $y=2, x=4$.
a
$y=\frac18x^2$
Question ID: 10070020070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=4\,\,\,\,\,\,\,2&=k\times4^2\\2&=k\times16\\\frac{2}{16}&=k\\k&=\frac18\\\text{therefore }y&=\frac18x^2\end{align}$
Question ID: 10070020070
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $y$ is directly proportional to the square root of $x$. When $y=10, x=4$.
a
$y=5\sqrt x$
Question ID: 10070020080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto\sqrt x\\y&=\sqrt x\\\text{when }y=10,\,x=4\,\,\,\,\,\,\,10&=k\times\sqrt4\\10&=k\times2\\\frac{10}{2}&=k\\k&=5\\\text{therefore }y&=5\sqrt x\end{align}$
Question ID: 10070020080
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070020080
Submit Yours! Copy Question ID and
click hereVideo by:
d
i) $h$ is directly proportional to the square root of $g$. When $h=5, g=25$.
a
$h=\sqrt g$
Question ID: 10070020090
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}h&\propto\sqrt g\\h&=k\sqrt
g\\5&=k\sqrt{25}\\k&=1\\h&=\sqrt g\end{align}$
Question ID: 10070020090
Submit Yours! Copy Question ID and
click hereSolution by: Aaron
v
Question ID: 10070020090
Submit Yours! Copy Question ID and
click hereVideo by:
d
j) $c$ is directly proportional to the cube root of $d$. When $c=2, d=64$.
a
$c=\frac12\sqrt[3] d$
Question ID: 10070020100
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070020100
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070020100
Submit Yours! Copy Question ID and
click hereVideo by:
d
k) $w$ is directly proportional to the $5$th root of $z$. When $w=3, z=32$.
a
$w=\frac32\sqrt[5] z$
Question ID: 10070020110
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070020110
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070020110
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 3
By finding the constant of proportionality, answer the following questions
a) $y$ is directly proportional to $x$. When $y=6, x=2$. Find $y$ when $x=10$
a
$y=30$
Question ID: 10070030010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto x\\y&=kx\\\text{when }y=6,\,x=2\,\,\,\,\,\,\,6&=k\times2\\\frac{6}{2}&=k\\k&=3\\\text{therefore }y&=3x\\x=10\,\,\,\,\,\,\,y&=3\times10\\y&=30\end{align}$
Question ID: 10070030010
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070030010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $y$ is directly proportional to $x$. When $y=2, x=20$. Find $y$ when $x=150$
a
$y=15$
Question ID: 10070030020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070030020
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070030020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $a$ is directly proportional to $b$. When $a=3, b=8$. Find $b$ when $a=5$
a
$b=\frac{40}{3}$ or $b=13\frac13$
Question ID: 10070030030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}a&\propto b\\a&=kb\\\text{when }a=3,\,b=8\,\,\,\,\,\,\,3&=k\times8\\k&=\frac38\\\text{therefore }a&=\frac38b\\a=5\,\,\,\,\,\,\,5&=\frac38\times b\\5\div\frac38&=b\\5\times\frac83&=b\\b&=\frac{40}{3}\end{align}$
Question ID: 10070030030
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070030030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $y$ is directly proportional to the square of $x$. When $y=16, x=4$. Find $y$ when $x=9$
a
$y=81$
Question ID: 10070030040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070030040
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070030040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $r$ is directly proportional to the square of $s$. When $r=15, s=5$. Find $r$ when $s=15$
a
$r=135$
Question ID: 10070030050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070030050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070030050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $p$ is directly proportional to the cube of $q$. When $p=9, q=3$. Find $q$ when $p=243$
a
$q=9$
Question ID: 10070030060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}p&\propto q^3\\p&=kq^3\\\text{when }p=9,\,q=3\,\,\,\,\,\,\,9&=k\times3^3\\9&=k\times27\\\frac{9}{27}&=k\\k&=\frac13\\\text{therefore }p&=\frac13q^3\\p=243\,\,\,\,\,\,\,243&=\frac13\times q^3\\243\times3&=q^3\\729&=q^3\\\sqrt[3]{729}&=q\\q&=9\end{align}$
Question ID: 10070030060
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070030060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $y$ is directly proportional to the square root $x$. When $y=18, x=36$. Find $y$ when $x=64$
a
$y=24$
Question ID: 10070030070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070030070
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070030070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $y$ is directly proportional to the cube root $x$. When $y=1, x=27$. Find $y$ when $x=64$
a
$y=\frac43$ or $y=1\frac13$
Question ID: 10070030080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070030080
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070030080
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 4
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is inversely proportional to $x$
a
$y\propto \frac1x$, $y=\frac kx$
Question ID: 10070040010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040010
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $P$ is inversely proportional to $Q$
a
$P\propto \frac1Q$, $P=\frac kQ$
Question ID: 10070040020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040020
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $y$ is inversely proportional to the square of $x$
a
$P\propto \frac{1}{x^2}$, $P=\frac {k}{x^2}$
Question ID: 10070040030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040030
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $a$ is inversely proportional to the cube of $b$
a
$a\propto \frac{1}{b^2}$, $a=\frac {k}{b^3}$
Question ID: 10070040040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040040
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $h$ is inversely proportional to the square root of $g$
a
$h\propto \frac{1}{\sqrt g}$, $h=\frac {k}{\sqrt g}$
Question ID: 10070040050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $E$ is inversely proportional to the cube root of $F$
a
$E\propto \frac{1}{\sqrt[3] F}$, $E=\frac {k}{\sqrt[3] F}$
Question ID: 10070040060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040060
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $y$ is inversely proportional to the fourth power of $x$
a
$y\propto \frac{1}{x^4}$, $y=\frac {k}{x^4}$
Question ID: 10070040070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040070
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $s$ is inversely proportional to the ninth root of $t$
a
$s\propto \frac{1}{\sqrt[9] t}$, $s=\frac {k}{\sqrt[9] t}$
Question ID: 10070040080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070040080
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070040080
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 5
By finding the constant of proportionality form an equation linking the variables
a) $y$ is inversely proportional to $x$. When $y=\frac{2}{5}, x=5$.
a
$y=\frac2x$
Question ID: 10070050010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto \frac{1}{x}\\y&=\frac{k}{x}\\\text{when }y=\frac25,\,x=5\,\,\,\,\,\,\,\frac25&=\frac k5\\\frac{2}{5}\times5&=k\\k&=2\\\text{therefore }y&=\frac2x\end{align}$
Question ID: 10070050010
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070050010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $A$ is inversely proportional to $B$. When $A=10, B=\frac12$.
a
$A=\frac5B$
Question ID: 10070050020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}A&\propto \frac{1}{B}\\A&=\frac{k}{B}\\\text{when }A=10,\,B=\frac12\,\,\,\,\,\,\,10&=\frac {k}{\frac12}\\10\times\frac12&=k\\k&=5\\\text{therefore }A&=\frac5B\end{align}$
Question ID: 10070050020
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070050020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $g$ is inversely proportional to the square of $h$. When $g=3, h=2$.
a
$g=\frac{12}{h^2}$
Question ID: 10070050030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}g&\propto \frac{1}{h^2}\\g&=\frac{k}{h^2}\\\text{when }g=3,\,h=2\,\,\,\,\,\,\,3&=\frac {k}{2^2}\\3\times2^2&=k\\3\times4&=k\\k&=12\\\text{therefore }g&=\frac{12}{h}\end{align}$
Question ID: 10070050030
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070050030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $y$ is inversely proportional to the square of $x$. When $y=18, x=\frac13$.
a
$y=\frac{2}{x^2}$
Question ID: 10070050040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070050040
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070050040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $P$ is inversely proportional to the cube of $Q$. When $P=\frac34, Q=2$.
a
$P=\frac{6}{Q^3}$
Question ID: 10070050050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070050050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070050050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $S$ is inversely proportional to the square root of $T$. When $S=5, T=16$.
a
$S=\frac{20}{\sqrt T}$
Question ID: 10070050060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070050060
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070050060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $u$ is inversely proportional to the cube root of $w$. When $u=1, w=27$.
a
$u=\frac{3}{\sqrt[3] w}$
Question ID: 10070050070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070050070
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070050070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $Y$ is inversely proportional to the sixth root of $X$. When $Y=\frac{1}{6}, X=64$.
a
$Y=\frac{\frac13}{\sqrt[6] X}$ or $Y=\frac{1}{3\sqrt[6]X}$
Question ID: 10070050080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070050080
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070050080
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 6
By finding the constant of proportionality, answer the following questions
a) $y$ is inversely proportional to $x$. When $y=4, x=3$. Find $y$ when $x=4$
a
$y=3$
Question ID: 10070060010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto \frac1x\\y&=\frac kx\\\text{when }y=4,\,x=3\,\,\,\,\,\,\,4&=\frac k3\\4\times3&=k\\k&=12\\\text{therefore }y&=\frac{12}{x}\\x=4\,\,\,\,\,\,\,y&=\frac{12}{4}\\y&=3\end{align}$
Question ID: 10070060010
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070060010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $C$ is inversely proportional to $D$. When $C=\frac12, D=4$. Find $C$ when $D=7$
a
$C=\frac27$
Question ID: 10070060020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}C&\propto \frac1D\\C&=\frac kD\\\text{when }C=\frac12,\,D=4\,\,\,\,\,\,\,\frac12&=\frac k4\\\frac12\times4&=k\\k&=2\\\text{therefore }C&=\frac{2}{D}\\D=7,\,\,\,\,\,\,\,C&=\frac{2}{7}\end{align}$
Question ID: 10070060020
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070060020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) $y$ is inversely proportional to the square of $x$. When $y=3, x=4$. Find $y$ when $x=2$
a
$y=12$
Question ID: 10070060030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}y&\propto \frac{1}{x^2}\\y&=\frac{k}{x^2}\\\text{when }y=3,\,x=4\,\,\,\,\,\,\,3&=\frac {k}{4^2}\\3\times4^2&=k\\3\times16&=k\\k&=48\\\text{therefore }y&=\frac{48}{x^2}\\x=2,\,\,\,\,\,\,\,y&=\frac{48}{2^2}\\y&=\frac{48}{4}\\y&=12\end{align}$
Question ID: 10070060030
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070060030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) $a$ is inversely proportional to the cube of $b$. When $a=6, b=1$. Find $a$ when $b=2$
a
$a=\frac34$
Question ID: 10070060040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070060040
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070060040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) $g$ is inversely proportional to the square root of $h$. When $g=4, h=49$. Find $g$ when $h=16$
a
$g=7$
Question ID: 10070060050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070060050
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070060050
Submit Yours! Copy Question ID and
click hereVideo by:
d
f) $R$ is inversely proportional to the cube root of $S$. When $R=1, S=8$. Find $R$ when $S=27$
a
$R=\frac23$
Question ID: 10070060060
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070060060
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070060060
Submit Yours! Copy Question ID and
click hereVideo by:
d
g) $f$ is inversely proportional to the cube of $g$. When $f=\frac12, g=2$. Find $g$ when $f=32$
a
$g=\frac12$
Question ID: 10070060070
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
$\begin{align}f&\propto \frac{1}{g^3}\\f&=\frac{k}{g^3}\\\text{when }f=\frac12,\,g=2\,\,\,\,\,\,\,\frac12&=\frac {k}{2^3}\\\frac12\times2^3&=k\\\frac12\times8&=k\\k&=4\\\text{therefore }f&=\frac{4}{g^3}\\f=32,\,\,\,\,\,\,\,32&=\frac{4}{g^3}\\32\times g^3&=4\\g^3&=\frac{4}{32}\\g^3&=\frac18\\g&=\sqrt[3]{\frac18}\\g&=\frac12\end{align}$
Question ID: 10070060070
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070060070
Submit Yours! Copy Question ID and
click hereVideo by:
d
h) $W$ is inversely proportional to the square root of $X$. When $W=\sqrt8, X=2$. Find $W$ when $X=4$
a
$W=1$
Question ID: 10070060080
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070060080
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070060080
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 7
By forming an equation with a constant of proportionality, answer the following
a) $£5$ can be exchanged for $\$7$. Find how much $£20$ can buy in $\$$.
a
$\$28$
Question ID: 10070070010
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
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 $\$28$.
Question ID: 10070070010
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070070010
Submit Yours! Copy Question ID and
click hereVideo by:
d
b) $5$ people can dig $8$ holes in an hour. They all work at the same rate. How many complete holes can $12$ people dig in an hour?
a
$19$ complete holes (rounded from $19.2$ holes)
Question ID: 10070070020
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
Question ID: 10070070020
Submit Yours! Copy Question ID and
click hereSolution by:
v
Question ID: 10070070020
Submit Yours! Copy Question ID and
click hereVideo by:
d
c) The $y$ coordinate on a graph is directly proportional to the square of the $x$ coordinate. The point $(20,2)$ lies on the graph. Find the equation of the graph and hence find the $y$ coordinate when $x=5$.
a
$y=5x^2$ and $y=125$
Question ID: 10070070030
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
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 $x=5$.
Question ID: 10070070030
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070070030
Submit Yours! Copy Question ID and
click hereVideo by:
d
d) The amount of flour put into a bread mix is directly proportional to the cube root of the volume of the bread, once baked. When I put in $50$g of flour, I get a bread with volume $6859$cm$^3$. How much flour is needed to make a bread with volume $10,000$cm$^3$? Give you answer to the nearest gram.
a
$43$g to the nearest gram.
Question ID: 10070070040
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
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.
Question ID: 10070070040
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070070040
Submit Yours! Copy Question ID and
click hereVideo by:
d
e) Gravitational force is inversely proportional to the square of distance from the earth. At $6000$km from the centre of the earth (so on the earth surface) a person feels a gravitational force of $500$N (Newtons, which is a measure of force). How much gravitational force do they feel $90,000$km away?
a
$G=2\frac29$N
Question ID: 10070070050
Something wrong? Copy Question ID and
contact usSubmit yours! Copy Question ID and
click hereQuestion by: ada
Answer by: ada
s
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
Question ID: 10070070050
Submit Yours! Copy Question ID and
click hereSolution by: ada
v
Question ID: 10070070050
Submit Yours! Copy Question ID and
click hereVideo by:
d
Question 1
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is directly proportional to $x$
b) $m$ is directly proportional to $n$
c) $a$ is directly proportional to the square of $b$
d) $p$ is directly proportional to the cube of $q$
e) $u$ is directly proportional to the root of $v$
f) $g$ is directly proportional to the cube root of $h$
g) $c$ is directly proportional to the $5$th power of $d$
h) $s$ is directly proportional to the $7$th root of $t$
Question 2
By finding the constant of proportionality form an equation linking the variables
a) $y$ is directly proportional to $x$. When $y=10, x=5$.
b) $a$ is directly proportional to $b$. When $a=20, b=4$.
c) $y$ is directly proportional to the square of $x$. When $y=2, x=1$.
d) $f$ is directly proportional to the square of $g$. When $f=45, g=3$.
e) $y$ is directly proportional to the cube of $x$. When $y=24, x=2$.
f) $m$ is directly proportional to the cube of $n$. When $m=3, n=3$.
g) $y$ is directly proportional to the square of $x$. When $y=2, x=4$.
h) $y$ is directly proportional to the square root of $x$. When $y=10, x=4$.
i) $h$ is directly proportional to the square root of $g$. When $h=5, g=25$.
j) $c$ is directly proportional to the cube root of $d$. When $c=2, d=64$.
k) $w$ is directly proportional to the $5$th root of $z$. When $w=3, z=32$.
Question 3
By finding the constant of proportionality, answer the following questions
a) $y$ is directly proportional to $x$. When $y=6, x=2$. Find $y$ when $x=10$
b) $y$ is directly proportional to $x$. When $y=2, x=20$. Find $y$ when $x=150$
c) $a$ is directly proportional to $b$. When $a=3, b=8$. Find $b$ when $a=5$
d) $y$ is directly proportional to the square of $x$. When $y=16, x=4$. Find $y$ when $x=9$
e) $r$ is directly proportional to the square of $s$. When $r=15, s=5$. Find $r$ when $s=15$
f) $p$ is directly proportional to the cube of $q$. When $p=9, q=3$. Find $q$ when $p=243$
g) $y$ is directly proportional to the square root $x$. When $y=18, x=36$. Find $y$ when $x=64$
h) $y$ is directly proportional to the cube root $x$. When $y=1, x=27$. Find $y$ when $x=64$
Question 4
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is inversely proportional to $x$
b) $P$ is inversely proportional to $Q$
c) $y$ is inversely proportional to the square of $x$
d) $a$ is inversely proportional to the cube of $b$
e) $h$ is inversely proportional to the square root of $g$
f) $E$ is inversely proportional to the cube root of $F$
g) $y$ is inversely proportional to the fourth power of $x$
h) $s$ is inversely proportional to the ninth root of $t$
Question 5
By finding the constant of proportionality form an equation linking the variables
a) $y$ is inversely proportional to $x$. When $y=\frac{2}{5}, x=5$.
b) $A$ is inversely proportional to $B$. When $A=10, B=\frac12$.
c) $g$ is inversely proportional to the square of $h$. When $g=3, h=2$.
d) $y$ is inversely proportional to the square of $x$. When $y=18, x=\frac13$.
e) $P$ is inversely proportional to the cube of $Q$. When $P=\frac34, Q=2$.
f) $S$ is inversely proportional to the square root of $T$. When $S=5, T=16$.
g) $u$ is inversely proportional to the cube root of $w$. When $u=1, w=27$.
h) $Y$ is inversely proportional to the sixth root of $X$. When $Y=\frac{1}{6}, X=64$.
Question 6
By finding the constant of proportionality, answer the following questions
a) $y$ is inversely proportional to $x$. When $y=4, x=3$. Find $y$ when $x=4$
b) $C$ is inversely proportional to $D$. When $C=\frac12, D=4$. Find $C$ when $D=7$
c) $y$ is inversely proportional to the square of $x$. When $y=3, x=4$. Find $y$ when $x=2$
d) $a$ is inversely proportional to the cube of $b$. When $a=6, b=1$. Find $a$ when $b=2$
e) $g$ is inversely proportional to the square root of $h$. When $g=4, h=49$. Find $g$ when $h=16$
f) $R$ is inversely proportional to the cube root of $S$. When $R=1, S=8$. Find $R$ when $S=27$
g) $f$ is inversely proportional to the cube of $g$. When $f=\frac12, g=2$. Find $g$ when $f=32$
h) $W$ is inversely proportional to the square root of $X$. When $W=\sqrt8, X=2$. Find $W$ when $X=4$
Question 7
By forming an equation with a constant of proportionality, answer the following
a) $£5$ can be exchanged for $\$7$. Find how much $£20$ can buy in $\$$.
b) $5$ people can dig $8$ holes in an hour. They all work at the same rate. How many complete holes can $12$ people dig in an hour?
c) The $y$ coordinate on a graph is directly proportional to the square of the $x$ coordinate. The point $(20,2)$ lies on the graph. Find the equation of the graph and hence find the $y$ coordinate when $x=5$.
d) The amount of flour put into a bread mix is directly proportional to the cube root of the volume of the bread, once baked. When I put in $50$g of flour, I get a bread with volume $6859$cm$^3$. How much flour is needed to make a bread with volume $10,000$cm$^3$? Give you answer to the nearest gram.
e) Gravitational force is inversely proportional to the square of distance from the earth. At $6000$km from the centre of the earth (so on the earth surface) a person feels a gravitational force of $500$N (Newtons, which is a measure of force). How much gravitational force do they feel $90,000$km away?
Answers
Question 1
a) $y\propto x$, $y=kx$
b) $m\propto n$, $m=kn$
c) $a\propto b^2$, $a=kb^2$
d) $p\propto q^3$, $p=kq^3$
e) $u\propto \sqrt v$, $u=k\sqrt v$
f) $g\propto \sqrt[3] h$, $g=k\sqrt[3] h$
g) $c\propto d^5$, $c=kd^5$
h) $s\propto \sqrt[7] t$, $s=k\sqrt[7] t$
Question 2
a) $y=2x$
b) $a=5b$
c) $y=2x^2$
d) $f=5g^2$
e) $y=3x^3$
f) $n=\frac19n^3$
g) $y=\frac18x^2$
h) $y=5\sqrt x$
i) $h=\sqrt g$
j) $c=\frac12\sqrt[3] d$
k) $w=\frac32\sqrt[5] z$
Question 3
a) $y=30$
b) $y=15$
c) $b=\frac{40}{3}$ or $b=13\frac13$
d) $y=81$
e) $r=135$
f) $q=9$
g) $y=24$
h) $y=\frac43$ or $y=1\frac13$
Question 4
a) $y\propto \frac1x$, $y=\frac kx$
b) $P\propto \frac1Q$, $P=\frac kQ$
c) $P\propto \frac{1}{x^2}$, $P=\frac {k}{x^2}$
d) $a\propto \frac{1}{b^2}$, $a=\frac {k}{b^3}$
e) $h\propto \frac{1}{\sqrt g}$, $h=\frac {k}{\sqrt g}$
f) $E\propto \frac{1}{\sqrt[3] F}$, $E=\frac {k}{\sqrt[3] F}$
g) $y\propto \frac{1}{x^4}$, $y=\frac {k}{x^4}$
h) $s\propto \frac{1}{\sqrt[9] t}$, $s=\frac {k}{\sqrt[9] t}$
Question 5
a) $y=\frac2x$
b) $A=\frac5B$
c) $g=\frac{12}{h^2}$
d) $y=\frac{2}{x^2}$
e) $P=\frac{6}{Q^3}$
f) $S=\frac{20}{\sqrt T}$
g) $u=\frac{3}{\sqrt[3] w}$
h) $Y=\frac{\frac13}{\sqrt[6] X}$ or $Y=\frac{1}{3\sqrt[6]X}$
Question 6
a) $y=3$
b) $C=\frac27$
c) $y=12$
d) $a=\frac34$
e) $g=7$
f) $R=\frac23$
g) $g=\frac12$
h) $W=1$
Question 7
a) $\$28$
b) $19$ complete holes (rounded from $19.2$ holes)
c) $y=5x^2$ and $y=125$
d) $43$g to the nearest gram.
e) $G=2\frac29$N
Question 1
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is directly proportional to $x$
b) $m$ is directly proportional to $n$
c) $a$ is directly proportional to the square of $b$
d) $p$ is directly proportional to the cube of $q$
e) $u$ is directly proportional to the root of $v$
f) $g$ is directly proportional to the cube root of $h$
g) $c$ is directly proportional to the $5$th power of $d$
h) $s$ is directly proportional to the $7$th root of $t$
Question 2
By finding the constant of proportionality form an equation linking the variables
a) $y$ is directly proportional to $x$. When $y=10, x=5$.
b) $a$ is directly proportional to $b$. When $a=20, b=4$.
c) $y$ is directly proportional to the square of $x$. When $y=2, x=1$.
d) $f$ is directly proportional to the square of $g$. When $f=45, g=3$.
e) $y$ is directly proportional to the cube of $x$. When $y=24, x=2$.
f) $m$ is directly proportional to the cube of $n$. When $m=3, n=3$.
g) $y$ is directly proportional to the square of $x$. When $y=2, x=4$.
h) $y$ is directly proportional to the square root of $x$. When $y=10, x=4$.
i) $h$ is directly proportional to the square root of $g$. When $h=5, g=25$.
j) $c$ is directly proportional to the cube root of $d$. When $c=2, d=64$.
k) $w$ is directly proportional to the $5$th root of $z$. When $w=3, z=32$.
Question 3
By finding the constant of proportionality, answer the following questions
a) $y$ is directly proportional to $x$. When $y=6, x=2$. Find $y$ when $x=10$
b) $y$ is directly proportional to $x$. When $y=2, x=20$. Find $y$ when $x=150$
c) $a$ is directly proportional to $b$. When $a=3, b=8$. Find $b$ when $a=5$
d) $y$ is directly proportional to the square of $x$. When $y=16, x=4$. Find $y$ when $x=9$
e) $r$ is directly proportional to the square of $s$. When $r=15, s=5$. Find $r$ when $s=15$
f) $p$ is directly proportional to the cube of $q$. When $p=9, q=3$. Find $q$ when $p=243$
g) $y$ is directly proportional to the square root $x$. When $y=18, x=36$. Find $y$ when $x=64$
h) $y$ is directly proportional to the cube root $x$. When $y=1, x=27$. Find $y$ when $x=64$
Question 4
Write a proportional statement with and without a constant of proportionality for the following
a) $y$ is inversely proportional to $x$
b) $P$ is inversely proportional to $Q$
c) $y$ is inversely proportional to the square of $x$
d) $a$ is inversely proportional to the cube of $b$
e) $h$ is inversely proportional to the square root of $g$
f) $E$ is inversely proportional to the cube root of $F$
g) $y$ is inversely proportional to the fourth power of $x$
h) $s$ is inversely proportional to the ninth root of $t$
Question 5
By finding the constant of proportionality form an equation linking the variables
a) $y$ is inversely proportional to $x$. When $y=\frac{2}{5}, x=5$.
b) $A$ is inversely proportional to $B$. When $A=10, B=\frac12$.
c) $g$ is inversely proportional to the square of $h$. When $g=3, h=2$.
d) $y$ is inversely proportional to the square of $x$. When $y=18, x=\frac13$.
e) $P$ is inversely proportional to the cube of $Q$. When $P=\frac34, Q=2$.
f) $S$ is inversely proportional to the square root of $T$. When $S=5, T=16$.
g) $u$ is inversely proportional to the cube root of $w$. When $u=1, w=27$.
h) $Y$ is inversely proportional to the sixth root of $X$. When $Y=\frac{1}{6}, X=64$.
Question 6
By finding the constant of proportionality, answer the following questions
a) $y$ is inversely proportional to $x$. When $y=4, x=3$. Find $y$ when $x=4$
b) $C$ is inversely proportional to $D$. When $C=\frac12, D=4$. Find $C$ when $D=7$
c) $y$ is inversely proportional to the square of $x$. When $y=3, x=4$. Find $y$ when $x=2$
d) $a$ is inversely proportional to the cube of $b$. When $a=6, b=1$. Find $a$ when $b=2$
e) $g$ is inversely proportional to the square root of $h$. When $g=4, h=49$. Find $g$ when $h=16$
f) $R$ is inversely proportional to the cube root of $S$. When $R=1, S=8$. Find $R$ when $S=27$
g) $f$ is inversely proportional to the cube of $g$. When $f=\frac12, g=2$. Find $g$ when $f=32$
h) $W$ is inversely proportional to the square root of $X$. When $W=\sqrt8, X=2$. Find $W$ when $X=4$
Question 7
By forming an equation with a constant of proportionality, answer the following
a) $£5$ can be exchanged for $\$7$. Find how much $£20$ can buy in $\$$.
b) $5$ people can dig $8$ holes in an hour. They all work at the same rate. How many complete holes can $12$ people dig in an hour?
c) The $y$ coordinate on a graph is directly proportional to the square of the $x$ coordinate. The point $(20,2)$ lies on the graph. Find the equation of the graph and hence find the $y$ coordinate when $x=5$.
d) The amount of flour put into a bread mix is directly proportional to the cube root of the volume of the bread, once baked. When I put in $50$g of flour, I get a bread with volume $6859$cm$^3$. How much flour is needed to make a bread with volume $10,000$cm$^3$? Give you answer to the nearest gram.
e) Gravitational force is inversely proportional to the square of distance from the earth. At $6000$km from the centre of the earth (so on the earth surface) a person feels a gravitational force of $500$N (Newtons, which is a measure of force). How much gravitational force do they feel $90,000$km away?
Answers
Question 1
a) $y\propto x$, $y=kx$
b) $m\propto n$, $m=kn$
c) $a\propto b^2$, $a=kb^2$
d) $p\propto q^3$, $p=kq^3$
e) $u\propto \sqrt v$, $u=k\sqrt v$
f) $g\propto \sqrt[3] h$, $g=k\sqrt[3] h$
g) $c\propto d^5$, $c=kd^5$
h) $s\propto \sqrt[7] t$, $s=k\sqrt[7] t$
Question 2
a) $y=2x$
b) $a=5b$
c) $y=2x^2$
d) $f=5g^2$
e) $y=3x^3$
f) $n=\frac19n^3$
g) $y=\frac18x^2$
h) $y=5\sqrt x$
i) $h=\sqrt g$
j) $c=\frac12\sqrt[3] d$
k) $w=\frac32\sqrt[5] z$
Question 3
a) $y=30$
b) $y=15$
c) $b=\frac{40}{3}$ or $b=13\frac13$
d) $y=81$
e) $r=135$
f) $q=9$
g) $y=24$
h) $y=\frac43$ or $y=1\frac13$
Question 4
a) $y\propto \frac1x$, $y=\frac kx$
b) $P\propto \frac1Q$, $P=\frac kQ$
c) $P\propto \frac{1}{x^2}$, $P=\frac {k}{x^2}$
d) $a\propto \frac{1}{b^2}$, $a=\frac {k}{b^3}$
e) $h\propto \frac{1}{\sqrt g}$, $h=\frac {k}{\sqrt g}$
f) $E\propto \frac{1}{\sqrt[3] F}$, $E=\frac {k}{\sqrt[3] F}$
g) $y\propto \frac{1}{x^4}$, $y=\frac {k}{x^4}$
h) $s\propto \frac{1}{\sqrt[9] t}$, $s=\frac {k}{\sqrt[9] t}$
Question 5
a) $y=\frac2x$
b) $A=\frac5B$
c) $g=\frac{12}{h^2}$
d) $y=\frac{2}{x^2}$
e) $P=\frac{6}{Q^3}$
f) $S=\frac{20}{\sqrt T}$
g) $u=\frac{3}{\sqrt[3] w}$
h) $Y=\frac{\frac13}{\sqrt[6] X}$ or $Y=\frac{1}{3\sqrt[6]X}$
Question 6
a) $y=3$
b) $C=\frac27$
c) $y=12$
d) $a=\frac34$
e) $g=7$
f) $R=\frac23$
g) $g=\frac12$
h) $W=1$
Question 7
a) $\$28$
b) $19$ complete holes (rounded from $19.2$ holes)
c) $y=5x^2$ and $y=125$
d) $43$g to the nearest gram.
e) $G=2\frac29$N
Solutions
Question 1
a)
b)
c)
d)
e)
f)
g)
h)
Question 2
a) $\begin{align}y&\propto x\\y&=kx\\\text{when }y=10,\,x=5\,\,\,\,\,\,\,10&=k\times5\\\frac{10}{5}&=k\\k&=2\\\text{therefore }y&=2x\end{align}$
b) $\begin{align}a&\propto b\\a&=kb\\\text{when }a=20,\,b=4\,\,\,\,\,\,\,20&=k\times4\\\frac{20}{4}&=k\\k&=5\\\text{therefore }a&=5b\end{align}$
c) $\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=1\,\,\,\,\,\,\,2&=k\times1^2\\k&=2\\\text{therefore }y&=2x^2\end{align}$
d) $\begin{align}f&\propto g^2\\f&=kg^2\\\text{when }f=45,\,g=3\,\,\,\,\,\,\,45&=k\times3^2\\45&=k\times9\\\frac{45}{9}&=k\\k&=5\\\text{therefore }f&=5g^2\end{align}$
e)
f) $\begin{align}m&\propto n^3\\m&=kn^3\\\text{when }m=3,\,n=3\,\,\,\,\,\,\,3&=k\times3^3\\3&=k\times27\\\frac{3}{27}&=k\\k&=\frac19\\\text{therefore }m&=\frac19n^3\end{align}$
g) $\begin{align}y&\propto x^2\\y&=kx^2\\\text{when }y=2,\,x=4\,\,\,\,\,\,\,2&=k\times4^2\\2&=k\times16\\\frac{2}{16}&=k\\k&=\frac18\\\text{therefore }y&=\frac18x^2\end{align}$
h) $\begin{align}y&\propto\sqrt x\\y&=\sqrt x\\\text{when }y=10,\,x=4\,\,\,\,\,\,\,10&=k\times\sqrt4\\10&=k\times2\\\frac{10}{2}&=k\\k&=5\\\text{therefore }y&=5\sqrt x\end{align}$
i) $\begin{align}h&\propto\sqrt g\\h&=k\sqrt
g\\5&=k\sqrt{25}\\k&=1\\h&=\sqrt g\end{align}$
j)
k)
Question 3
a) $\begin{align}y&\propto x\\y&=kx\\\text{when }y=6,\,x=2\,\,\,\,\,\,\,6&=k\times2\\\frac{6}{2}&=k\\k&=3\\\text{therefore }y&=3x\\x=10\,\,\,\,\,\,\,y&=3\times10\\y&=30\end{align}$
b)
c) $\begin{align}a&\propto b\\a&=kb\\\text{when }a=3,\,b=8\,\,\,\,\,\,\,3&=k\times8\\k&=\frac38\\\text{therefore }a&=\frac38b\\a=5\,\,\,\,\,\,\,5&=\frac38\times b\\5\div\frac38&=b\\5\times\frac83&=b\\b&=\frac{40}{3}\end{align}$
d)
e)
f) $\begin{align}p&\propto q^3\\p&=kq^3\\\text{when }p=9,\,q=3\,\,\,\,\,\,\,9&=k\times3^3\\9&=k\times27\\\frac{9}{27}&=k\\k&=\frac13\\\text{therefore }p&=\frac13q^3\\p=243\,\,\,\,\,\,\,243&=\frac13\times q^3\\243\times3&=q^3\\729&=q^3\\\sqrt[3]{729}&=q\\q&=9\end{align}$
g)
h)
Question 4
a)
b)
c)
d)
e)
f)
g)
h)
Question 5
a) $\begin{align}y&\propto \frac{1}{x}\\y&=\frac{k}{x}\\\text{when }y=\frac25,\,x=5\,\,\,\,\,\,\,\frac25&=\frac k5\\\frac{2}{5}\times5&=k\\k&=2\\\text{therefore }y&=\frac2x\end{align}$
b) $\begin{align}A&\propto \frac{1}{B}\\A&=\frac{k}{B}\\\text{when }A=10,\,B=\frac12\,\,\,\,\,\,\,10&=\frac {k}{\frac12}\\10\times\frac12&=k\\k&=5\\\text{therefore }A&=\frac5B\end{align}$
c) $\begin{align}g&\propto \frac{1}{h^2}\\g&=\frac{k}{h^2}\\\text{when }g=3,\,h=2\,\,\,\,\,\,\,3&=\frac {k}{2^2}\\3\times2^2&=k\\3\times4&=k\\k&=12\\\text{therefore }g&=\frac{12}{h}\end{align}$
d)
e)
f)
g)
h)
Question 6
a) $\begin{align}y&\propto \frac1x\\y&=\frac kx\\\text{when }y=4,\,x=3\,\,\,\,\,\,\,4&=\frac k3\\4\times3&=k\\k&=12\\\text{therefore }y&=\frac{12}{x}\\x=4\,\,\,\,\,\,\,y&=\frac{12}{4}\\y&=3\end{align}$
b) $\begin{align}C&\propto \frac1D\\C&=\frac kD\\\text{when }C=\frac12,\,D=4\,\,\,\,\,\,\,\frac12&=\frac k4\\\frac12\times4&=k\\k&=2\\\text{therefore }C&=\frac{2}{D}\\D=7,\,\,\,\,\,\,\,C&=\frac{2}{7}\end{align}$
c) $\begin{align}y&\propto \frac{1}{x^2}\\y&=\frac{k}{x^2}\\\text{when }y=3,\,x=4\,\,\,\,\,\,\,3&=\frac {k}{4^2}\\3\times4^2&=k\\3\times16&=k\\k&=48\\\text{therefore }y&=\frac{48}{x^2}\\x=2,\,\,\,\,\,\,\,y&=\frac{48}{2^2}\\y&=\frac{48}{4}\\y&=12\end{align}$
d)
e)
f)
g) $\begin{align}f&\propto \frac{1}{g^3}\\f&=\frac{k}{g^3}\\\text{when }f=\frac12,\,g=2\,\,\,\,\,\,\,\frac12&=\frac {k}{2^3}\\\frac12\times2^3&=k\\\frac12\times8&=k\\k&=4\\\text{therefore }f&=\frac{4}{g^3}\\f=32,\,\,\,\,\,\,\,32&=\frac{4}{g^3}\\32\times g^3&=4\\g^3&=\frac{4}{32}\\g^3&=\frac18\\g&=\sqrt[3]{\frac18}\\g&=\frac12\end{align}$
h)
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 $\$28$.
b)
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 $x=5$.
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.
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
