4 comments by 2 or more people

1. Iain

   Jones? Or is this one of those ones where I miss something entirely elementary and go off in the wrong direction?

   Smith = 13600
   Jones = 14800

   16 Nov 2009, 16:11

2. Steve Rumsby

   I think the question is about overall earnings over the period, rather than just the final salary. Does that help?

   16 Nov 2009, 16:39

3. Simon Whitehouse

   Smith has earnt more.

   Jones has had an average pay of £11,600 and so been paid a total of £34,800 over three years. Smith has earnt an average of £12,500 and so been paid a total of £37,500 over three years.

   The question says that “Every six months, Smith’s pay rose by £500 compared with that for the previous 6-month period”.

   An increase of £500 compared with the 6-month period equates to a rise of £1000 in his salary. And he gets that every 6 months. So, he’s getting a £2k rise every year and he’s seeing some of it before Jones does.

   16 Nov 2009, 17:06

4. Eleanor Lovell

   Sorry Iain – you missed something…Simon got it though. This is Ian Stewart’s explanation:

   Surprisingly, Smith earned more – even though £1,600 per year is greater than Smith’s accumulated £500 + £1,000 over a year.

   To see why, tabulate their earnings for each six-month period:

   	Smith	Jones
   Year 1 first half	£5,000	£5,000
   Year 1 second half	£5,500	£5,000
   Year 2 first half	£6,000	£5,800
   Year 2 second half	£6,500	£5,800
   Year 3 first half	£7,000	£6,600
   Year 3 second half	£7,500	£6,600

   Note that Jones’s £1,600 splits into two amount of £800 for each half-year, so his half-yearly figures increase by £800 every year. Smith’s half-year figures increase by £500 every half-year. Despite that, Smith is ahead in every period after the first, and gets ever further ahead as time passes.

   In fact, at the end of year n, Smith has earned a total of 10,000n + 5,00n(2n-1) pounds, while Jones has earned a total of 10,000n + 800n(n-1) pounds.

   So Smith – Jones = 200n² + 300n, which is positive and grows with n.

   17 Nov 2009, 16:16