Mail on Sunday

The Mail on Sunday “Style” editor requested an equation for whether or not to buy expensive designer clothes. After some hesitation, I decided that there was something sensible that I could say and which would get at least a few readers thinking about mathematics, and with the approval of Rob Eastaway (former president of the U.K.’s Mathematical Association) I went ahead.

When we are deciding whether to buy something, we often start by weighing up all the factors, but usually give up in despair at the difficulty of the task. One way to bypass this problem is to look at what else we might spend the money on, and to spend it on whatever we feel is likely to give us the most satisfaction. That’s the best solution, and one that we quite often use intuitively.

Just for interest, though, let’s assume that we can assess the factors involved in terms of money, and that those factors are actually additive. If you are buying a fantastic coat for a special occasion, you might say that it’s worth I pounds to you because it will impress others, an additional pounds because it keeps you warm, pounds because itenhances the rest of your outfit, and a further pounds because it just makes you feelgreat. Add them all together (and subtract any negatives, such as the cost of dry cleaning C) to get a total value V. If this comes to the price or more, then you should buy the coat. In other words:

            If (+ W + E + G  C) ≥ P, then buy!

But wait! You are probably going to wear the coat more than once. Maybe you should multiply the value by the number of times you are likely to wear the coat. On the other hand, it may not have such a high value (V2) to you the second time around, and even less the third time. Perhaps you should work out all the different values V1V2, V3 etc and add them up.

It would help if we knew what the relationship was between V1V2, V3 etc. It could depend on how long you have had the coat, how fast fashions are changing, or any number of other factors. The easiest one to work out is if the coat’s value to us diminishes by some factor k each time we wear it.

The total value VTOTAL of the coat over its lifetime would then be:

            VTOTAL  = V1+ k V1+ k² V1+ k³ V1+ … …
= V1( 1+ k + k² + k³ + … … )

If its value halved each time we wore it, for example, then k would be 0.5, and
VTOTAL  V1 ( 1 + ½ + ¼ + 1/8 + ….. )

The sum inside the brackets gets closer and closer to 2 the more terms you add, so if you wear the coat a lot, it will be worth twice as much to you as if you wore it only once.
The mathematician’s shorthand for ( 1+ k + k² + k³ + … … ) is ∑ k(n-1) ( ∑ just means “the sum of”, and n assumes the successive values 1, 2, 3, etc. until you get up to the number of times you will wear the coat ). So if

             V1 ∑k(n-1) ≥ P, then buy!

If you happen to have a high school mathematics student in the family, they might recognize why I have picked this particular example. It is because ∑ k(n-1) can be worked out easily for a large number of terms! It is called a geometric series, and the sum gets closer and closer to {1 / (1- k)} the more terms you add.

You could, of course, avoid all this maths by just using your intuition. After all, you used it to work out V1 in the first place and, in any case, who wants to be doing maths when they’re buying a coat?

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