04. Direct and inverse proportion (using algebra)

Question 1

LEVEL 6

The force $F$ on a mass, $m$, is directly proportional to the acceleration, $a$, of the mass.

When $a = 350$, $F = 850$.

Find $F$ when $a = 140$.

Select the correct answer from the list below:

A: $360$

B: $340$

C: $280$

D: $300$

CORRECT ANSWER:   B: $340$

WORKED SOLUTION:

To start given that $F \propto a$, we need to find the constant of proportionality, $m$,  so that $F=ma$.

Hence we can find $m$ by substituting the values $a = 350$ and $F = 850$ into the equation,

 $850=m\times350$

Therefore $m$ is $\dfrac{17}{7}$ and $F=\dfrac{17}{7}\times a$

Next to find $F$ when $a = 140$, we can substitute this value into the equation so that,

$F=\dfrac{17}{7}\times140=340$

Question 2

LEVEL 6

The resistance $R$ (ohms) of a wire is directly proportional to the length $l$ (cm) of the wire.

When $l = 150$, $R = 750$.

Find $R$ when $l = 450$.

Select the correct answer from the list below:

A: $1250$

B: $1750$

C: $2250$

D: $2550$

CORRECT ANSWER: C: $2250$

WORKED SOLUTION:

To start given that $R \propto l$, we need to find the constant of proportionality, $k$,  so that $R=kl$.

Hence we can find $k$ by substituting the values $l = 150$ and $R = 750$ into the equation,

 $750=k\times150$

Therefore $k$ is $5$ and $R=5\times l$

Next to find $R$ when $l = 450$, we can substitute this value into the equation so that,

$R=5\times450=2250$

Question 3

LEVEL 6

The energy $E$ (Joules) of a gas is directly proportional to it’s temperature $T$ (Kelvin).

When $T = 280$, $E = 40$.

Calculate $T$ when $E = 9$.

Select the correct answer from the list below:

A: $30$

B: $95$

C: $63$

D: $76$

CORRECT ANSWER:  D: $63$

WORKED SOLUTION:

To start given that $E \propto T$, we need to find the constant of proportionality, $k$,  so that $E=kT$.

Hence we can find $k$ by substituting the values $T= 280$ and $E = 40$ into the equation,

 $40=k\times280$

Therefore $k$ is $\dfrac{1}{7}$ and $E=\dfrac{1}{7}T$

Next to find $T$ when $E = 9$, we can substitute this value into a rearrangement of the equation so that,

$T=7\times9=63$

Question 4

LEVEL 6

The energy $(E)$ released when matter is converted to energy is proportional to mass of that object $(m)$.

When $E = 1.0 × 10^{16}$ Joules, $m = 0.111$ kg.

Calculate the mass, in kg when $E$ is $1.8$ million Joules.

Select the correct answer from the list below:

A: $m = 2 × 10^{-11}$

B: $m = 2 × 10^{11}$

C: $m = 2 × 10^{-10}$

D: $m = 22 × 10^{-11}$

CORRECT ANSWER:   A: $m = 2 × 10^{-11}$

WORKED SOLUTION:

To start given that $E \propto m$, we need to find the constant of proportionality, $k$,  so that $E=km$.

Hence we can find $k$ by substituting the values $m= 0.111$ and $E =1.0 × 10^{16}$ into the equation,

 $1.0 × 10^{16}=k\times0.111$

Therefore $k$ is $9.01 × 10^{16}$ and $E=9.01 × 10^{16}\times m$

Next to find $m$ when $E = 1.8 × 10^{6}$, we can substitute this value into a rearrangement of the equation so that,

$m=1.8 × 10^{6}\div9.01 × 10^{16}=2 × 10^{-11}$

Question 5

LEVEL 6

$x$ is inversely proportional to $y$.

$x = 5$ when $y = 12$.

Work out the value of $y$ when $x = 4$.

Select the correct answer from the list below:

A: $15$

B: $18$

C: $14$

D: $10$

CORRECT ANSWER:  A: $15$

WORKED SOLUTION:

To start given that $x \propto \dfrac{1}{y}$, we need to find the constant of proportionality, $k$,  so that $x=\dfrac{k}{y}$.

Hence we can find m by substituting the values $x = 5$ and $y = 12$ into the equation,

 $5=\dfrac{k}{12}$

Therefore $k$ is $5\times12=60$ and $x=\dfrac{60}{y}\times a$

Next to find $y$ when $x = 4$, we can substitute this value into the equation so that,

$y=\dfrac{60}{4}=15$

Question 6

LEVEL 6

The gravitational force $F$ (Newtons) between two masses is inversely proportional to the square of the distance $d$ between them.

When $d = 8$, $F = 10$.

Calculate $F$ when $d = 10$.

Select the correct answer from the list below:

A: $10.8$

B: $9.2$

C: $6.4$

D: $5.2$

CORRECT ANSWER:  C: $6.4$

WORKED SOLUTION:

To start given that $F \propto \dfrac{1}{d^{2}}$, we need to find the constant of proportionality, $k$,  so that $F=\dfrac{k}{d^{2}}$.

Hence we can find $k$ by substituting the values $F = 10$ and $d = 8$ into the equation,

 $10=\dfrac{k}{64}$

Therefore $k$ is $10\times64=640$ and $F=\dfrac{640}{d^{2}}\times a$

Next to find $F$ when $d = 10$, we can substitute this value into the equation so that,

$F=\dfrac{640}{100}=6.4$

Question 7

LEVEL 6

The time in minutes $(T)$ for meals to be served at a busy restaurant is inversely proportional to the square of the number of waiters $(W)$ working at that time.

It takes $20$ minutes for meals to be served when $12$ waiters are working.

Find an equation connecting $T$ and $W$.

Select the correct answer from the list below:

A: $T = \dfrac{2880}{W^{2}}$

B: $T = \dfrac{W^{2}}{2880}$

C: $T = 2880\times W^{2}$

D: $T = 2880\times W$

CORRECT ANSWER:  A: $T = \dfrac{2880}{W^{2}}$

WORKED SOLUTION:

To start given that $T \propto \dfrac{1}{W^{2}}$, we need to find the constant of proportionality, $k$,  so that $T=\dfrac{k}{W^{2}}$.

Hence we can find $k$ by substituting the values $T = 20$ and $W = 12$ into the equation,

 $20=\dfrac{k}{144}$

Therefore $k$ is $120\times144=2880$ and $T=\dfrac{2880}{W^{2}}$

Question 8

LEVEL 6

The mass $m$ of a liquid in a cylindrical container is proportional to the square of the radius $r$.

When $r = 7$, $m = 16$.

Find $r$ as a fraction in its simplest form, when $m = 9$.

Select the correct answer from the list below:

A: $r = \dfrac{21}{4}$

B: $r = \dfrac{28}{3}$

C: $r = \dfrac{16}{7}$

D: $r = \dfrac{44}{3}$

CORRECT ANSWER:   A: $r = \dfrac{21}{4}$

WORKED SOLUTION:

To start given that $m \propto r^{2}$, we need to find the constant of proportionality, $k$,  so that $m=kr^{2}$.

Hence we can find $k$ by substituting the values $m = 16$ and $r = 7$ into the equation,

 $16=49k$

Therefore $k$ is $16\div49=\dfrac{16}{49}$ and $m=\dfrac{16}{49}r^{2}$

Next to find $r$ when $m = 9$, we can substitute this value into the equation so that,

$r^{2}=9\div\dfrac{16}{49}=\dfrac{441}{16}$

Question 9

LEVEL 6

The mass of a solid sphere $(m g)$ is proportional to its radius $(r)$ cubed.

When $r = 6$, $m = 7560$.

Find the value of $m$ when $r = 5$.

Select the correct answer from the list below:

A: $4375$

B: $6575$

C: $4350$

D: $5225$

CORRECT ANSWER:    A: $4375$

WORKED SOLUTION:

To start given that $m \propto r^{3}$, we need to find the constant of proportionality, $k$,  so that $m=kr^{3}$.

Hence we can find $k$ by substituting the values $m = 7560$ and $r = 6$ into the equation,

 $7560=216k$

Therefore $k$ is $7560\div216=35$ and $m=35r^{3}$

Next to find when $r = 5$, we can substitute this value into the equation so that,

$m=35\times 5^{3}=4375$