, 2022

With 1 kg of muffin batter, exactly 24 mini muffins and 2 large
muffins can be made. Thus with 2 kg of muffin batter, exactly $2\times24=48$ mini muffins and $2\times2=4$ large muffins can be made, and
so $n=4$.
Solution 1

With 2 kg of muffin batter, exactly 36 mini muffins and 6 large
muffins can be made. With 2 kg of muffin batter, exactly 48 mini muffins
and 4 large muffins can be made. Adding these, we get that with $2 $\textrm{}$ kg}+2 $\textrm{}$ kg}= 4 $\textrm{}$
kg}$of muffin batter, exactly$36+48=84$mini muffins and$6+4=10$ large muffins can be made, and so
$x=4$.

Solution 2

Let $m$ be the number of kilograms
of muffin batter needed to make 1 mini muffin. Let $\ell$ be the number of kilograms of muffin
batter needed to make 1 large muffin. Since 1 kg of muffin batter makes
exactly 24 mini muffins and 2 large muffins, then $1=24m+2\ell$.

Since 2 kg of muffin batter makes exactly 36 mini muffins and 6 large
muffins, then $2=36m+6\ell$.

Subtracting the second equation from 3 times the first equation, we get
$3\times1-2=3\times(24m+2\ell)-(36m+6\ell)$
or $1=36m$, and so $m=\dfrac{1}{36}$.

Substituting $m=\dfrac{1}{36}$
into the second equation, we get $2=36\left(\dfrac{1}{36}\right)+6\ell$ or
$1=6\ell$, and so $\ell=\dfrac16$.

Since $\dfrac{1}{36}$ kg of muffin
batter is needed to make 1 mini muffin, then $84\cdot\dfrac{1}{36}=\dfrac73$ kg of
muffin batter is needed to make 84 mini muffins.

Since $\dfrac{1}{6}$ kg of muffin
batter is needed to make 1 large muffin, then $10\cdot\dfrac{1}{6}=\dfrac53$ kg of muffin
batter is needed to make 10 large muffins.

Thus, exactly 84 mini muffins and 10 large muffins can be made with
$\dfrac73+\dfrac53=\dfrac{12}{3}=4$
kg of muffin batter, and so $x=4$.
In part (b) Solution 2, it was determined that $\frac{1}{36}$ kg of muffin batter is
needed to make 1 mini muffin, and $\frac{1}{6}$ kg of muffin batter is needed
to make 1 large muffin.

Therefore, the number of kilograms of muffin batter needed to make 1
large muffin is 6 times the number of kilograms of batter needed to make
1 mini muffin $\left(6\times\frac{1}{36}=\frac16\right)$.

In other words, the batter needed to make 1 large muffin can make 6 mini
muffins.

Then, using the amount of batter that makes 7 large muffins, $6\times7=42$ mini muffins can be
made.