Mathematical Notes

Mathematical Notes from the Editor

Why is it that Newsletters always seem to have the wrong number of pages? By the time it reached Ralph Allen Press, this edition, like the 25 that have preceded it, had 4n pages. (This time, n=9, not counting the booking forms which make up the centre pages.) But until a day or two before going to press, we were one page short, so this is my attempt to fill it up with a mathematical theme:

Timing Head Races

Crews of the 1970's and 1980's will remember taking part in what we used to call 'Internal Head Races'. These were rowed most weeks in the Lent Term by almost everyone in the club, so as many as 8 VIIIs took part - often on a Saturday afternoon. Publication of the results was always eagerly awaited. The original course was from the Quarter Mile Post to the Brook, and back again, though this was later moved 135 metres downstream to allow for easier boat manoeuvres between the 'up' and the 'down' courses. Many of the times are recorded in the Minute Book, and the graph is still safely preserved. The record is held by the 1982 crew (6.18); the 1975 crew managed 6:26, and the 2004 crew have the third ever fastest time (6:26.5). The slowest time ever recorded was in 2002, when some of the oarsmen who later rowed in the 1st VIII in 2004 were so slow that the time fell off the graph. It was approximately 7:50.

There was, of course, a stream running when the 2004 crew achieved a time of 6:26.5. The time 'up' was 7:04.4, and the time down was 5:54.8. The arithmetic mean of these is 6:28.6, so where does the 6:26.5 come from?

Allowance must be made for the stream. Suppose the length of the course is d, the speed of the boat in still water (assumed constant) is u and the speed of the stream is v. Let the time taken upstream by t1, and the time downstream be t2, and let the time the boat would have taken in still conditions be t.

The net speed of the boat upstream is u - v, and the speed downstream is u + v.

Apply distance=speed x time, to the upstream and downstream courses:

d = ( u - v ) x t1

d = ( u + v ) x t2

Also apply the formula to the course rowed in still water:

d = u x t

We wish to find t.

We eliminate d, u and v from these equations, and so find t in terms of t1 and t2. This is left as an exercise for the reader. Answer on page 32.

There are, of course, assumptions about boat speed, effects of wind, etc., implied here, but they are reasonable, and consistency of times by crews in various stream conditions indicates that the method is reliable.

Gift Aid

As Bluefriars is a charity, we can reclaim income tax on gifts from UK taxpayers.

Basic rate taxpayers

If you earn £100, you pay £22 in tax, and keep £78 to spend. You would have to earn £100 to give £78 away in presents or to good causes without using Gift Aid. (eg you would have to earn £100 to buy £78 of charity lottery tickets.)

If you give the £78 to a charity under Gift Aid, the charity can reclaim the £22 tax, making the total gift £100.

It is more efficient to use Gift Aid.

Higher rate taxpayers

If you earn £100, you pay £40 in tax, and keep £60 to spend.

If you give the £60 to a charity, the charity can reclaim £16.92 tax, making the total gift £76.92. This £76.92 does not count as part of your taxable income, so, if you have already paid 40% of £76.92 = £30.77 in tax, you can reclaim the £13.85 not already reclaimed by the charity.

Or : If a Higher Rate taxpayer earns £130, the 40% tax is £52, leaving £78 in the donor's pocket to spend. The donor would have to earn £130 to give £78 away in presents or to good causes without using Gift Aid. (eg you would have to earn £130 to buy £78 of charity lottery tickets.)

If you give £78 to a charity by Gift Aid, the charity can reclaim £22. This total of £100 is deducted from your taxable income before your own tax calculations are made, so you have to earn £100 (not £130) to give £78 (gift)+ £22 (tax reclaimed by the charity).

The conclusion is that Gift Aid is a good means to make donations to charities, especially for Higher Rate taxpayers.