## November 16, 2009

### Nice Littler Earner

Smith and Jones were hired at the same time by Stainsbury’s Superdupermarket, with a starting salary of £10,000 per year. Every six months, Smith’s pay rose by £500 compared with that for the previous 6-month period. Every year, Jones’s pay rose by £1,600 compared with that for the previous 12-month period.

Three years later, who had earned more?

### 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

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