a)

When faced with dividing fractions, it much easier to switch one of the fractions around and multiply them together instead of divide them.

$$\left( \frac{\var{f_coprime}}{\var{g_coprime}}\div\frac{\var{h_coprime}}{\var{j_coprime}} \right) \equiv \left( \frac{\var{f_coprime}}{\var{g_coprime}}\times\frac{\var{j_coprime}}{\var{h_coprime}} \right) = \frac{\var{fj}}{\var{gh}}$$

Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\var{gcd1}$. 

This gives a final answer of $\displaystyle\simplify{{fj}/{gh}}$.

b)

$$\frac{\var{f1_coprime}}{\var{g1_coprime}}\div\frac{\var{h1_coprime}}{\var{j1_coprime}} \equiv \left( \frac{\var{f1_coprime}}{\var{g1_coprime}}\times\frac{\var{j1_coprime}}{\var{h1_coprime}} \right)=\frac{\var{f1j1}}{\var{g1h1}}$$

Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\var{gcd2}$.

This gives a final answer of $\displaystyle\simplify{{f1j1}/{g1h1}}$.

c)

$${\var{f3}\frac{\var{g3_coprime}}{\var{h3_coprime}}}\div{\var{f4}\frac{\var{g4_coprime}}{\var{h4_coprime}}}$$

The first thing to do is to change the mixed numbers into improper fractions.

An improper fraction is a fraction where the numerator is greater than the denominator. To change a mixed fraction to an improper fraction, multiply the integer part of the mixed number by the denominator, and add it to the existing numerator to make the new numerator of the improper fraction. The denominator will stay the same. 

$${\var{f3}\frac{\var{g3_coprime}}{\var{h3_coprime}}}\equiv\frac{(\var{f3}\times\var{h3_coprime})+\var{g3_coprime}}{\var{h3_coprime}}=\frac{\var{f3h3}+\var{g3_coprime}}{\var{h3_coprime}}=\frac{\var{f3h3+g3_coprime}}{\var{h3_coprime}}$$

$${\var{f4}\frac{\var{g4_coprime}}{\var{h4_coprime}}}\equiv\frac{(\var{f4}\times\var{h4_coprime})+\var{g4_coprime}}{\var{h4_coprime}}=\frac{\var{f4h4}+\var{g4_coprime}}{\var{h4_coprime}}=\frac{\var{f4h4+g4_coprime}}{\var{h4_coprime}}$$

We now have our mixed numbers as improper fractions.

$$\frac{\var{f3h3+g3_coprime}}{\var{h3_coprime}}\div\frac{\var{f4h4+g4_coprime}}{\var{h4_coprime}}$$

Now, use the same method as in parts a) and b) to divide by switching around one fraction and changing the division symbol to multiplication.

$$\frac{\var{f3h3+g3_coprime}}{\var{h3_coprime}}\div\frac{\var{f4h4+g4_coprime}}{\var{h4_coprime}}\equiv\frac{\var{f3h3+g3_coprime}}{\var{h3_coprime}}\times\frac{\var{h4_coprime}}{\var{f4h4+g4_coprime}}=\frac{\var{num}}{\var{denom}}$$

Finally, the last thing to do is to simplify your answer down by finding the highest common divisor in the numerator and denominator, which in this case is $\var{gcd3}$.

By doing this, you will get a final answer of

$$\simplify{{num}/{denom}}$$

d)

$$\frac{\var{a}}{(\simplify[all,!collectNumbers]{{b}-{c}/{d}})}$$

Consider the denominator first, as following the rules of BODMAS, you should address brackets first.

You need to get a common denominator for both terms on the denominator, like this:

$$\var{b}\times\frac{\var{d}}{\var{d}} = \frac{\var{bd}}{\var{d}}$$

This now allows you to complete the addition or subtraction as both terms have a common denominator. 

$${\simplify[all,!collectNumbers]{{bd}/{d}-{c}/{d}}} = \frac{\var{bd_c}}{\var{d}}$$

This means that the expression is now:

$$\frac{\var{a}}{\frac{\var{bd_c}}{\var{d}}}$$

Dealing with this requires a bit of manipulation. You have to change the divisor of the denominator to be a mulitplier of the numerator. The denominator, ${\var{bd_c}}$ was being divided by ${\var{d}}$ but by flipping it around, the numerator, ${\var{a}}$ will be mulitplied by ${\var{d}}$. The value of the expression remains the same.

$$\frac{\var{a}}{\frac{\var{bd_c}}{\var{d}}}\equiv \frac{(\var{a})\times(\var{d})}{\var{bd_c}}= \frac{\var{ad}}{\var{bd_c}}$$

From this, you can try to cancel the expression down by finding the highest common factor of the numerator and denominator, to give a final answer of

$$\simplify{{ad}/{bd_c}}$$