Gap In NICs

Will somone explain exactly how the amount to make up a gap in National Insurance contributions (NICs) is calculated in general, please?

Having, finally, finished this, I realise it's become a huge message, but please try to read it!

I understand the basic principle that if one's income is below the lower earnings limit (LEL) for a suitable period then one must pay an amount equal to a number of weeks at the relevant Class-3 rate for that tax year, but I don't know exactly how either the amount or the number of weeks is derived, though obviously one can be deduced from the other.

I've spent a lot of time on the HMRC, Pension Service, CAB, DirectGov, AccountingWeb and MoneySavingExpert website, but I've seen few exact calculations, and none that seems to apply generally.

The NICs are based around weeks, and I don't understand how they are converted to be monthly or yearly in practice.

Consider the tax year from 2010 to 2011.  The LEL for that tax year is £97 weekly, £421 monthly and £5,044 yearly; the corresponding earnings threshold (ET), at which one pays contributions is £110 weekly, £476 monthly and £5,715 yearly; the Class-3 rate, £12.05.

For a first example, imagine a sole employement in that year for the month of February for which one is paid £420.  This is below the monthly LEL.  This seems to indicate that one would have a gap of a full year of 52 weeks in employment contributing to benefits, leading to an amount to be made up of 52 weeks times £12.05, which is £626.60.

However, February has 28 days, and as 4 times 7 is 28, the employment is for 4 weeks.  This means the weekly earnings are £420 divided by 4, which is £105 and is above the weekly LEL.  This seems to indicate that those 4 weeks do contribute to benefits, leaving a gap of the full 52-week year less 4 weeks, so 48 weeks, leading to an amount to be made up of 48 weeks times £12.05, which is £578.40.

For a second example, imagine a sole employment in that year for the months of July, August and September, for which one is paid £2,000 each month.

Think of the employment in terms of weeks.  The total earnings are 3 times £2,000, which is £6,000.  There are 31 days in each of July and August; and 30, in September.  This is a total of 92 days, and as 7 times 13 is 91, the employment is for a little over 13 weeks.  This means the weekly earnings are a little less than £6,000 divided by 13, which is £461.54 and is well above the weekly LEL.  This seems to indicate that those 13 weeks contribute to benefits, leaving a gap of the full 52-week less 13 weeks, so 39 weeks, leading to an amount to be made up of 39 weeks times £12.05 per week, which is £469.95.

Think now of the employment in terms of months.  The earnings each month are £2,000, which is well above the monthly LEL.  The employment is for a total of 3 months, which leaves a gap of 9 months.  If each month is 4 weeks, then this is a gap of 36 weeks, leading to an amount to be made up of 36 weeks times £12.05 per week, which is £433.80.  Although if the the 3 months employment is converted to weeks at 4 weeks a month, then there are 12 weeks contributing to benefits, leading to a gaph of the full 52-week year less 12 weeks, so 40 weeks, leading to an amount to be made up of 40 weeks times £12.05, which is £482.00.

Finally, think of the employment in terms of years.  The total earnings are £6000, which is well above the yearly LEL of £5,044, and NIDirect, at the link below, indicates that a qualifying year is one in which one pays more than the yearly LEL.  This seems to indicate that there is no gap at all, leading to an amount to be made up of £0.00.

http://www.nidirect.gov.uk/understanding-the-basic-state-pension

I want to understand how the amount to make up a gap is calculated, what would happen in the case of two overlapping employments and what effect paying some earnings into a work's contracted-out pension scheme would have.  Hence I'm interested in the following, preferably with links to webpages with confirmation.

1.  Explain how the amounts to make up gaps are calculated in general.

2.  Demonstrate how the rules apply in the first example above.

3.  Demonstrate how the rules apply in the second example above.

4.  Explain what effect making pension contributions, whilst sometimes being beneath the LEL, would have.

5.  Explain what would happen in the case of two overlapping employments, one paid weekly, one monthly, and both with pay that varies between being above and below the corresponding LELs.

If you've reached this far, I admire your diligence!

If you have an answer to any of the questions I am very impressed!