Direct and Inverse Proportion

First, find the scale factor:

$210div525=0.4$

Next, multiply the scale factor by the amounts of each ingredient.

Sugar: $475times0.4=190text{ g}$

Butter: $225times0.4=90text{ g}$

Eggs: $15times0.4=6$ eggs

Set up equation:

$sqrt{y}=kx$

Substitute values to find $k$:

$sqrt{36}=9k$

$9k=6$

$k=dfrac{2}{3}$

Find $y$ using the new value of $x$:

$sqrt{y}=dfrac{2}{3}x$

$sqrt{y}=dfrac{2}{3}times21$

$sqrt{y}=14$

$y=196$

$F$ is inversely proportional to the square of $r$ (i.e. $r^2$), so

$F propto dfrac{1}{r^2}$

We can rewrite this equation using $k$ to represent the constant of proportionality:

$F = dfrac{k}{r^2}$

We know that when $F = 50, r = 3$, so we can work out $k$ by substituting these known values into the equation:

$50 = dfrac{k}{3^2}=dfrac{k}{9}$

Since we want to work out the value of $k$, we need to rearrange the formula to make $k$ the subject which we can do by multiplying each side by $9$:

$k = 50 times 9 = 450$

We can now rewrite the original equation with a value for the constant $k$:

$F = dfrac{450}{r^2} = dfrac{450}{12^2} = 3.125$

Since $y$ is inversely proportional to $x$, we can write this as an equation, as follows:

$y propto dfrac{1}{x}$

We can rewrite this equation using $k$ to represent the constant of proportionality:

$y = dfrac{k}{x}$

 

From the table, we know that when $x = 4, y = 7.5$. If we substitute these values into our equation, we can work out the value of the constant $k$.

Since

$y = dfrac{k}{x}$

then

$7.5 = dfrac{k}{4}$

 

Since we want to work out the value of $k$, we need to rearrange the formula to make $k$ the subject which we can do by multiplying each side by $4$:

$begin{aligned} k &= 7.5 times 4 k &= 30 end{aligned}$

 

We can now rewrite the original equation with a value for the constant $k$:

$y = dfrac{30}{x}$

 

Now that we have calculated the value of the constant $k$ and rewritten the original proportionality equation, we can work out the value of $y$ when  $x = 60$:

When $x = 60, y= dfrac{30}{60} = 0.5$

 

We can also work out the value of $x$ when  $y = 12$:

When $y = 12$, $12 = dfrac{30}{x}$

If we multiply both sides by $x$ and divide both sides by $12$, we have rearranged the equation making $x$ is the subject:

$x = dfrac{30}{12} = 2.5$

Therefore when $y = 12, x = 2.5$

 

So, the completed table should look like this:

 

a) Since $M$ is directly proportional to $h$, we can write this as an equation, as follows:

$M propto h$

We can rewrite this equation using $k$ to represent the constant of proportionality:

$M = kh$

 

We know that when $M = pounds155.80, h = 9.5$ hours, so we can work out $k$ by substituting these known values into the equation:

$155.80 = k times 9.5$

 

Since we want to work out the value of $k$, we need to rearrange the formula to make $k$ the subject which we can do by dividing each side by $9.5$:

$k = dfrac{155.80}{9.5} = 16.4$

 

We can now rewrite the original equation with a value for the constant $k$:

$M = 16.4h$

 

 

b) Now that we have calculated the value of the constant $k$ and rewritten the original proportionality equation, we can work out the value of $h$ when  $M = pounds688.80$.

 

Since

$M = 16.4h$

then

$pounds688.80 = 16.4h$

If

$16.4h = pounds688.80$

then

$h = pounds688.80 div16.4 = 42$

 

So it will take Sasha $42$ hours to earn $£688.80$.

a)  $x$ is directly proportional to $y$, so

$x propto y$

We can therefore write this as an equation where $k$ is a constant:

$x = ky$

 

We now need to work out the value of the constant $k$ which we can do by substituting in the known values for $x$ and $y$:

Since

$x = ky$

then

$2 = k times 8$

 

If we make $k$ the subject of the formula by dividing both sides by $8$, we can work out the value of $k$:

$dfrac{2}{8} = k$

 

Therefore $k$ has a value of $frac{2}{8}$, which can be simplified to $frac{1}{4}$.

 

Now that we know the value of the constant $k$, we can write the equation connecting $x$ and $y$:

Since

$x = ky$

then

$x = frac{1}{4}y$

or

$x = frac{y}{4}$

 

 

b)  We now have an equation connecting $x$ and $y$, so we can work out the value of $x$ when $y$ is $32$.

 

Since

$x = frac{y}{4}$

then

$x = frac{32}{4}$

 

$32 div4 = 8$

So

$x = 8$

 

 

c)  We now have an equation connecting $x$ and $y$, so we can work out the value of $y$ when $x$ is $50$.

 

Since

$x = frac{y}{4}$

then

$y = x times 4$

 

$50 times 4 = 200$

So

$y = 200$

a)  Since the time taken ($t$) is inversely proportional to the square of the number of staff on duty ($s$), we can write a basic equation as follows:

$t propto dfrac{1}{s^2}$

This formula can be rewritten using $k$ as a constant that connects $t$ and $s$:

$t = dfrac{k}{s^2}$

 

We can now work out the value of $k$ if we substitute in the known values for $t$ and $s$:

Since

$t = dfrac{k}{s^2}$

then

$20 = dfrac{k}{4^2}$

so

$20 = dfrac{k}{16}$

 

To work out the value of $k$, we need to rearrange the formula, making $k$ the subject.  We can do by multiplying both sides by $16$:

$20 times 16 = k$

So

$k = 320$

 

Since we now know the exact value of $k$, we can rewrite the proportionality formula:

Since

$t = dfrac{k}{s^2}$

then

$t = dfrac{320}{s^2}$

 

 

b)  It would be too easy to assume that if you double the staff, then the time taken would be halved!  Sadly, this is not the case.

 

If the number of staff is doubled, then there would be $8$ staff members on duty instead of $4$.  We know from part a) that $k = 320$ and we know that $s = 8$.  If we substitute these values into our formula, it should look like this:

Since

$t = dfrac{k}{s^2}$

then

$t = dfrac{320}{8^2}$

so

$t = dfrac{320}{64}$

 

$320 div 64 = 5$ minutes

 

Therefore, if you double the number of staff the orders take $5$ minutes to be received instead of $20$ minutes, so the orders are received $4$ times faster.