The standard bar of Imperial Leather soap is to be reduced from 125 grams to 100 grams – but sold for the same price. This must be true, because I read it in *The Times*.

But, surely, if it is *Imperial* Leather it should be reduced from 4.40924 ounces to 3.52739 ounces!

Or should it be rebranded as *Metric Leather?*

As it happens this provides a nice mathematical question! If the bar is reduced from 125 g to 100 g and sold at the same price, what is the resulting percentage increase in price?

The weight of the bar is reduced by 20%, because the reduction (25 g) is 20% of the original weight (125 g).

Is the price therefore increased by 20%? No! It’s increased by 25%! (Or, as we shall see, maybe more!)

A simple way to see this is to imagine that you buy 500 g of soap. That would be 4 of the 125 g bars, but 5 of the 100 g bars. So, you have to buy 5 instead of 4 to get the same amount of soap. The cost of 5 is 25 % greater than the cost of 4.

BUT … there’s always a ‘but’ when mathematics is applied to the real world. You don’t usually use the whole bar of soap, do you? You usually throw it away when it gets too small to be usable. With the smaller bars, the bit you throw away is going to be a larger proportion of the whole bar.

Say you throw the bar away when it has shrunk to 10 g. Then the 125-g bar and the 100-g bar actually provide 115 g and 90 g respectively of usable soap. If you pay the same for the 90 g as you did for the 115 g, that’s a price increase of 28%!

Derek, love it!

ReplyDeleteMetric units are a bit of a passion of mine, and in class I have been known to express the wish from afar that the US would go metric as soon as possible.

I hadn't thought about the bit of soap you throw away - good point. But if you stick the little bit left onto a new bar, you don't waste it :)

Lots of interesting math here - well posted.