One of our colleagues was in the butcher's shop near to Quiz Master Shop Towers, pricing up lamb joints for Easter.

He remarked to the butcher that his prices were quite high, and the butcher agreed. "The wholesale prices are high at the moment, as importing lamb is quite difficult, but I'm hoping the prices will drop soon. If the wholesaler would give me five more legs for £100, I could drop my price by £1.50 and still maintain the same profit margin."

He returned the next week, and true to his word the butcher had reduced the price of a leg of lamb by £1.50. However, our colleague still felt this to be a bit pricey and asked if prices would be dropping further. "Well" replied the butcher, "I think that they might be coming down over the next week. Now if I could get another five legs for my £100 I could drop my price by a further £1 and still retain the same margin."

So the puzzle is to find how much is the butcher charging for a leg of lamb, and what is his profit margin?

Last week the butcher bought x - 5 legs of lamb for £100 and sold them at y% profit. This week he has bought x legs of lamb for £100 and retained his profit margin. And he hopes that next week he will buy x + 5 legs of lamb for £100, still selling at y% profit.

Last week each leg of lamb cost

$$ \frac{100}{x - 5} $$

and sold for

$$ \frac{100 + y}{x - 5} $$

This week each leg of lamb cost

$$ \frac{100}{x} $$

and sold for

$$ \frac{100 + y}{x} $$

And next week each leg of lamb should cost

$$ \frac{100}{x + 5} $$

and sell for

$$ \frac{100 + y}{x + 5} $$

If we subtract this week's selling price from last week's selling price the difference is £1.50 (or 3/2), and if we subtract next week's selling price from this week's selling price the difference is £1. So that:

$$ (a) \frac{100 + y}{x - 5} - \frac{100 + y}{x} = \frac{3}{2} $$

$$ (b) \frac{100 + y}{x} - \frac{100 + y}{x + 5} = 1 $$

From (b)

$$ (100 + y)(\frac{1}{x} - \frac{1}{x + 5}) = 1 $$

$$ 100 + y = \frac{x (x + 5)}{5} $$

And from (a)

$$ \frac{x (x + 5)}{5 (x - 5)} - \frac{x (x + 5)}{5x} = \frac{3}{2} $$

$$ 10x ^ 2 + 50x = 15x ^ 3 - 75x $$

So x = 25

and then

$$ 100 + y = \frac{25 (25 + 5)}{5} $$

So y = 50

Last week the butcher bought 20 legs of lamb for £100 (£5 each) and sold them for £7.50. This week he bought 25 legs of lamb for £100 (£4 each) and sold them for £6. And he hopes to buy 30 legs of lamb for £100 next week (£3.33 each) and sell them for £5.