There's definitely something confusing here.

1. "Collapsing the fractions" doesn't make particular sense. I can see there being two ways that the task may have been presented to you.

- If you were presented with these equations as they stand then a valid task would be "show that the following are true" or "demonstrate the validity of these equations" or something of that ilk.

- If you were only given the left-hand side, then it would have clear to say "work out" or "evaluate" or "simplify" the expressions.

2. A) is wrong. \(\displaystyle \frac{7}{16} - \frac{5}{16} - \frac{11}{12} \neq \frac{-21}{16}\). B) and C) are correct though.

Let's just say you have been given the stuff on the left hand side and you want to work out the answer. You are correct that to work out these questions you need to get a common denominator.

To get a common denominator, you need to work the lowest common multiple (LCM) of the values on the denominator of each fraction in your calculation. Then you use that number to work out the problem.

So, let's take C). We want to work out:

\(\displaystyle 1 \frac{1}{3} + \frac{5}{7} + \frac{8}{35}\)

Let's convert the first number to an improper fraction:

\(\displaystyle 1 \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{3+1}{3} = \frac{4}{3}\)

So now our question is

\(\displaystyle \frac{4}{3} + \frac{5}{7} + \frac{8}{35}\)

The denominators we have are 3, 7 and 35. What is the LCM of these?

**To get the LCM of three or more numbers, pick any two numbers, calculate the LCM for those two, then get the LCM of the number you just found with the next number... repeat until you have exhausted all your numbers.**

So... let's get the LCM of 3 and 7. They are both prime numbers, so the LCM is just \(\displaystyle 3 \times 7 = 21\).

Now we get the LCM of 21 and 35.

\(\displaystyle 21 = 3 \times 7\)

\(\displaystyle 35 = 5 \times 7\)

the highest common factor of the two numbers is 7. Therefore,

LCM = product of two numbers / highest common factor = \(\displaystyle \frac{3 \times 7 \times 5 \times 7}{7} = 3 \times 5 \times 7 = 105\)

So we want our denominator to be 105. The above method might look strange, but it actually works for *any* fraction problem, so if you learn it and practise it, you'll be able to do crazy hard questions without a calculator!

So, attempting to get the common denominator of 105, we multiply the top and bottom of each individual term by a number to get 105 on the bottom...

\(\displaystyle \frac{4}{3} + \frac{5}{7} + \frac{8}{35}\)

\(\displaystyle = \frac{4}{3}\times \frac{35}{35} + \frac{5}{7}\times \frac{15}{15} + \frac{8}{35}\times \frac{3}{3}\)

\(\displaystyle = \frac{140}{105} + \frac{75}{105} + \frac{24}{105}\)

\(\displaystyle = \frac{140+75+24}{105}=\frac{239}{105}\)

The multiplication step can be tricky. The trick is to use the LCM products you used earlier to help decide what to multiply by.

Let me know how you get on. If you find them really, really hard, it might be good to go back and practise some easier questions first before trying these ones again.