The Quarter That Was a Fifth

Dear friend,

The riddle first, as the tradition demands. Back in the old town there were two brushmakers, one in a yard off the Shambles and one by the bridge, and both of them would have given you the very same account of how they set their prices.

Two brushmakers in the town I have never named, the same grey bristle and the same beech stocks.
Both bored, drew and pitched a brush that would outlast two of the cheap sort.
Both said, and fully believed, that they took a quarter on everything they sold.
One added his quarter to what the bristle, the wood and the firing had cost him.
One set his price so that a quarter of every shilling taken stayed in the shop.
At Michaelmas one of them paid for a new boring machine. Which one, and why could the other not?

Hold your answer. Story time.

The Story

Two brushmakers, taken on as boys in the same trade in the same decade, each with a shop that smelled of hot pitch and wet beech. Both bought grey bristle from the same dealer, sorted into lengths on a baize cloth. Both bored their stocks, drew the knots through with wire, pitched and trimmed them, and sent out a brush that a careful householder would still be using when two cheap ones had gone bald and been thrown on the fire. On the quality of the work there was nothing to choose between them, and the parish knew it.

The first brushmaker had a rule, and he kept to it honestly for twenty years. Whatever a dozen brushes cost him in bristle, stocks, wire, pitch and coal, he added a quarter, and that was his price. Eight shillings of materials and firing became ten shillings asked. Nor was he a man who could be talked down. When a customer pushed on the figure he would say that he took a quarter and no more, and that a quarter was little enough for a trade that had kept him up to his elbows in pitch since he was eleven.

The second brushmaker went at the sum from the other end. He did not begin by asking what he should add. He began by asking what should stay. If a quarter of the money was to remain in the shop, then the goods and the firing had to be covered by the other three quarters, and so eight shillings of costs required a price of ten shillings and eightpence. That was the figure he chalked up, and he could not have told you what fraction he had added to anything, because adding was not the sum he was doing.

In the town this difference amounted to three farthings on a tenpenny brush, and nobody ever mentioned it. Both men were busy, both were thought reasonable, and neither was ever the cheap brushmaker. They each sold something near seven hundred dozen in the year. Eightpence a dozen, a figure neither of them would have crossed the road for, came to a little over three and twenty pounds by the following Michaelmas. The second man paid twenty of it for a treadle boring machine from a maker in Birmingham, which took the hardest hour out of every day of his week and let a boy do what had needed a man. The first went on boring his stocks by hand, and said that trade was flat and that people would not pay for good work any more.

Be fair to him. He was not idle, not careless, and not vague about money, and he refused to undercut himself for two decades. His whole error was one word in the middle of his own rule, and nothing in the ordinary run of a working year was going to point at it.

The Lesson, Since You've Earned It

What you add to your costs and what stays with you are two different fractions of two different numbers, and only one of those numbers ever passes through your hand. Add a quarter to eight shillings and you have a price of ten. The two shillings you kept is a quarter of your dealer's figure and a fifth of your own. You announced a quarter, you believed a quarter, and a fifth is what went in the drawer.

A fifth instead of a quarter sounds like a quibble for a clerk. It is a fifth of your whole keeping gone, every week and on every line, and the keeping is the only part of the money that was ever yours. That is the real damage, because almost everything rests on it. Whether you can knock something off for a builder who buys steadily, which looks quite different once you know what a reduction actually takes out of your year. Whether a second pair of hands pays for itself. Whether you can put twenty pounds towards a machine. Every one of those is answered with the kept fraction, and a man working from the added fraction answers all of them a quarter too generously. He does not find out as a loss, either, since there is no loss to find. He finds out as a thing he did not buy.

The error survives because it feels like arithmetic has been done. A man who adds a quarter is not guessing. He has a rule, he applies it to every job, and that confidence is precisely what stops him checking which number he applied it to. The remedy is not more sums but a different starting point, and it is small enough to write on the first page of your day book. To keep a fifth of what comes in, add a quarter to your costs. To keep a quarter, add a third. To keep a third, add a half. To keep two fifths, add two thirds. To keep a half, double them. The figure on the right is always the larger of the two, and that sentence is the whole of this letter.

Two cautions. The first is that the fraction you keep after the goods is not profit, and must not be spent as though it were. It is what remains when the materials are paid for and before anything else at all is, so it must still carry the rent, the cart, the quiet weeks and the days nobody pays you for. The second is that the share must not be chosen by appetite. Work backwards from what your year actually costs and how much work you can honestly do in it, arrive at the share you need, and only then convert it into something to add. A figure reached by wishing will be refused by the first customer who hears it.

Three questions, for the week in front of you:

1. Out of every shilling that came over my counter last week, how many pence stayed once the goods were paid for? Not from the rule, from the receipts. One ordinary week, the money taken set against the materials bought for the work that earned it. Most men find the answer two pence under the one they would have sworn to.
2. Which of my prices were made by adding to a cost, and when was that cost last looked at? Every price built that way is suspect twice over, once for the fraction and once for the figure underneath it, since the dealer has raised his prices and your rule has no way of noticing.
3. What share do I actually need, and what must I therefore add to get it? Settle the share first, convert it with the list above, and write what is to be added beside each line in your book, where whoever prices a job in your absence will find it.

The Answer to the Riddle

At Michaelmas one of them paid for a new boring machine. Which one, and why could the other not?

The second man, the one by the bridge, who had never added a quarter to anything in his life. He had decided what must stay and then worked out what to ask, which put eightpence on every dozen brushes that left his shop and three and twenty pounds in his hand by the end of the year. The man in the yard off the Shambles had taken his quarter from his dealer's figure instead of his own, and had kept a fifth for twenty years in perfectly good faith, with a rule he was proud of and would defend to your face. The same brush, the same bristle, the same honest intention about the worth of the work. One of them reckoned on the money coming in rather than the money going out.

The verse:

A quarter on your costs, my friend,
comes back a fifth to you.
So reckon on the money in,
as all your creditors do.

Until next time. Go and take one week of real receipts and work out how many pence in the shilling stayed with you once the goods were paid for.

P.S. The same error does its worst the moment you give anything away. If a fifth of every shilling stays with you, a tenth off your price takes half of that fifth, and a fifth off takes the whole of it and leaves you working for the exercise. The man who believes he keeps a quarter will agree to reductions he thinks he can bear twice over, in perfectly good faith, with a sum in his head that was wrong before the customer opened his mouth.

Also in the archive: The Four Days Out of the Shop, Half an Inch a Year and Sevenpence a Gallon.