A percentage is an obligation with the pound sign taken off

1

What the document says while the banner says three hundred

Two sentences describe the identical arrangement. The first says three hundred per cent. The second says that £1,200.00 must cross the reels before anything can leave. Both are accurate, neither contradicts the other, and they sit about as far apart as two sentences about one offer can sit.

The first is a ratio. Ratios have no unit, which is what lets them be large without committing anybody to anything, and three hundred is a satisfying size of ratio. The second is a sum of money in sterling, and a sum of money is a demand. Turning the first into the second needs two figures that the first does not carry: the payment, and the multiple written over whatever the payment produces.

On a payment of £10.00 the percentage produces £30.00 of credit and the account reads £40.00. A forty times written across the credited sum comes to £1,200.00 of turnover. Four hundredths of any turnover stays with the house where 96 pence comes back in the pound, so £48.00 of expected cost falls on £30.00 of credit and the close is minus £18.00.

Where each of those is settled is the point of the exercise. The ratio is settled on the banner, in type large enough to read across a room. The multiple that converts it into pounds is settled in the bonus terms, behind a line of small type. The turnover figure itself appears in neither, because no document has any reason to perform the multiplication for a reader.

Credit as the percentage states it £30.00 Expected close once £1,200.00 of turnover is demanded minus £18.00 Carried entirely by the clause giving the multiple £48.00

Forty-eight pounds is the subtraction this page performs. Nobody gets charged it, no statement records it, and the advertisement holds no sentence able to produce it. What it measures is the distance between a figure somebody would act on and the worth of the identical offer once its written condition has been applied.

2

Four pounds of obligation for every point advertised

Hold the payment at ten pounds and the relationship between the two sentences becomes a single constant. Each percentage point on the banner produces ten pence of credit. Each ten pence of credit carries forty times itself in turnover, which is four pounds. So every point added to the headline adds exactly £4.00 to the written obligation.

One point on the banner, four pounds in the document, all the way up the ladder

Turn it round and the asymmetry shows more plainly. Moving from two hundred per cent to three hundred alters the advertisement by one hundred, a quantity anybody can hold in mind. In the document the identical move adds four hundred pounds of turnover. One alteration, measured on two scales, with the bigger scale appearing nowhere in print.

The constant depends on the payment, and that is worth stating so the arithmetic can be carried elsewhere. At a payment of £5.00 the rate is two pounds a point. At £20.00 it is eight pounds. The rate is the payment multiplied by the multiple and divided by a hundred, so a reader wanting it for their own figures has one multiplication and one division to do.

What the rate establishes is that a percentage behaves like a tariff and not like a gift. Picking a larger headline picks a larger obligation, at a fixed rate of exchange, advertised in one currency and settled in another. Concealment is not the complaint. Both units simply never appear together, and setting them together is what parts three and four are for.

3

Both numbers at every rung, printed together

Here is the ladder with both sentences on it. The left column is what gets advertised. The column headed turnover is what the document asks for the same arrangement. Every row runs on a payment of ten pounds, a multiple of forty and a pound counting as a pound towards the requirement.

Column five wants a word of explanation, being the only one not lifted straight from the advertisement or the terms. A balance multiplied by twenty-five gives the turnover it funds before exhaustion becomes the expected outcome. The twenty-five falls out of the return and out of nothing else, four hundredths inverted, so that column is arithmetic rather than a description of the market.

Read the last column downwards and the ladder divides in two. Up to about one hundred and sixty-seven per cent the money is expected to outlast the requirement. Above that it is not, and the gap widens by the rung. At the top of the ladder the shortfall is £350.00, which means the balance is expected to be gone with more than a fifth of the turnover still outstanding.

The drawing makes one thing visible that the table states numerically. The advertised bar grows evenly up the ladder while the surviving portion of it never grows at all, because past the break-even multiple each extra pound of credit is worth sixty pence less than nothing. Scaling the headline scales the shortfall in exact proportion.

4

The one quantity that stays put while the headline climbs

Everything in the ladder moves except one figure. The payment is ten pounds at every rung, because the payment is what the reader chose and the percentage is what somebody else chose. That fixity is the structural fact this part is about, and it drives the threshold in part five.

Follow the consequences in order. Credit rises in proportion to the headline. Turnover rises in proportion to the headline, forty times faster in absolute terms. The balance rises too, but it rises from a base of ten that does not move, so it rises more slowly than either of the other two in relative terms.

Everything on this page follows from that sentence. Twenty pounds of balance at the foot of the ladder reaches fifty at its top, a growth of two and a half times. Across the same span turnover runs from £400.00 to £1,600.00, a growth of four. What is being asked for outgrows what is there to meet it, and the cause is nothing more exotic than a fixed component sitting inside one of the two.

Nothing a reader does moves the mismatch. Paying in more raises both sides of the comparison in the same proportion, since the credit and the balance both scale with the payment, so a payment of £100.00 at three hundred per cent arrives at the identical verdict as a payment of ten. The threshold in part five is a property of the percentage and the multiple, and the size of the payment drops out of the algebra entirely.

That is an unusual result on a subject where the usual advice is to think about how much to put in. On this particular question the amount makes no difference whatever. What decides whether a matched offer can be worked to the end is the pair of numbers chosen by the operator, and the reader’s contribution to the arithmetic is to supply a scale factor that cancels.

5

Solving for the percentage nobody is expected to finish

Set the two sides of the comparison against each other and solve. Turnover demanded is the multiple times the credit, and credit is the headline times the payment. Turnover funded is twenty-five times the balance, and the balance is the payment plus the credit. Write the demand as no larger than the funding and the payment cancels off both sides.

What survives is a statement about two numbers only. The headline, expressed as a multiplier, times the quantity got by taking twenty-five away from the multiple, must come to no more than twenty-five. At a multiple of forty the bracket is fifteen, so the headline cannot exceed twenty-five fifteenths, which is one and two thirds, or 166.67 per cent.

Check it against the ladder and the rows agree. At one hundred and fifty per cent the balance funds £625.00 against a demand of £600.00. At one hundred and sixty-six and two thirds the two sides meet at £666.67 exactly. At two hundred the demand of £800.00 overtakes the funding of £750.00, and everything above that rung is further behind.

Everything turns on that bracket, which is why it repays a second look. Where a clause writes twenty-five, the credit under it is expected to break even exactly, that figure being what 96 pence back in the pound produces. Should the terms write twenty-five or less, the bracket comes out at nought or below and no headline of any size can outrun the money. Write more than twenty-five and the bracket turns positive, at which point a finite threshold begins to exist.

Two readings of that table are worth separating. The first is the obvious one: a larger multiple pulls the threshold down, so the room a headline has to grow in shrinks as the terms get harder. The second is sharper. The threshold does not merely move as the multiple rises past twenty-five. It comes into existence there, having been absent at every multiple below.

That is the same twenty-five doing two different jobs, and it is not a coincidence. A balance funds twenty-five times itself in turnover, and a credit pays for itself at a multiple of twenty-five. Both figures are one divided by the four hundredths a 96 per cent return keeps, so the point where a credit stops being worth anything is also the point where a headline becomes capable of outrunning the money behind it.

6

The same question put to a seven-day window

Money is one constraint and time is the other, and they give different answers. A requirement that the balance could in principle fund may still be unreachable inside the period the terms allow, and the period is the quietest clause on a matched offer because it changes no figure in the arithmetic.

The window posited below is seven days carrying an hour of pressing apiece, and both halves of it are this page’s own assumption rather than anything published. At the ten pence this site works to, a licensed build keeping 2.5 seconds between cycles allows at most 1,440 of them inside an hour, so the week yields 10,080 presses and £1,008.00 of turnover.

Put the ladder through it. Four hundred pounds at one hundred per cent fits easily. Eight hundred at two hundred per cent fits. Twelve hundred at three hundred per cent does not, and the headline where the window runs out is the one producing £1,008.00 of demand, which is 252 per cent.

Three hundred per cent therefore sits past both thresholds at once, and the two are independent of each other. The money runs out at about a hundred and sixty-seven per cent and the clock runs out at two hundred and fifty-two, so a reader at the worked rung is short of time and short of balance for separate reasons.

Run the clock the other way and the size of the task turns concrete. Turnover of £1,200.00 at ten pence is 12,000 presses. Against the same gap between cycles those occupy 30,000 seconds, being eight hours and twenty minutes of pressing, which across seven days falls out at one hour and eleven minutes a day with not one of them missed.

A longer window moves the second threshold and leaves the first exactly where it is. At thirty days of the same hour the week supplies £4,320.00 and no rung on the ladder is out of time, while the money still runs out at a hundred and sixty-seven per cent. The clock is a clause an operator chooses. The other threshold is arithmetic and cannot be drafted around.

7

What a ceiling on the multiple does to that threshold

Everything above takes the multiple as given. Suppose instead that something holds it down. A British licence does exactly that, and because the whole of this page turns on the interaction between a headline and a multiple, the effect is larger here than the figures alone suggest.

Ten times the credited sum is where that ceiling sits, settled in a public rulebook. Its home is the code provision covering incentives, heading a short list of prohibitions, and the date it took hold is 19 January 2026. Two kinds of operating licence stand outside it, one for technical equipment and one for software. A multiple is all the ceiling governs. About a percentage it has nothing whatever to say.

That distinction decides how the ceiling behaves up the ladder, and the published wording supports working it out rather than asserting it. A headline of three hundred still produces £30.00 of credit, because nothing caps a percentage. Ten over that credit is £300.00 of turnover rather than £1,200.00. The ratio of turnover to credit is ten to one at every rung, so the ladder keeps its shape and loses three quarters of its size.

Expected close where the terms write forty times minus £18.00 Expected close where a ceiling holds the multiple to ten £18.00 What the ceiling by itself is worth at this rung £36.00

Thirty-six pounds, and the sign changes with it. Ten lots of £30.00 send £300.00 over the reels, feed £12.00 to the margin and leave eighteen pounds standing, against a credited sum that under the written multiple was expected to take eighteen pounds of the reader’s own money with it when it went.

The deeper effect is on part five rather than on the price block. Run the threshold algebra at a multiple of ten and the bracket is fifteen below nought, so the inequality holds for every headline however large. Under a ten times ceiling there is no percentage at which the balance stops being able to fund the requirement, at any size, on any payment. The ceiling does not move the threshold. It abolishes it.

For scale, the practice the ceiling was drafted against is published by the regulator as multiples reaching up to sixty times. Run the same algebra there and the threshold falls to 71.43 per cent, which is below every rung on the ladder in part three. Three hundred per cent at sixty times demands £1,800.00 against a balance funding £1,000.00, and the credit closes at minus £42.00.

8

Converting a ratio into the sum it commits you to

Three multiplications and one comparison hand a reader both sentences in place of one. Payment times headline yields the credit. That credit times the multiple yields the turnover demanded. The balance times twenty-five yields what the balance will fund. Set the final pair of figures against each other and stop.

Each of those steps runs on a figure the advertisement supplies, apart from the middle one, which is where all the work happens. Without the multiple a reader holds a ratio and nothing besides, and a ratio will not compare with anything, for the plain reason that it is not denominated in the unit the comparison takes place in.

What comes back from the comparison is a verdict and not a sum, which suits a decision taken in under a minute. Should the funded figure be the larger, the money is expected to see the requirement out, and the remaining question is what survives. Should the demanded figure be the larger, the requirement is not expected to be met at all, and a figure for what survives has nothing left to describe.

Two cautions keep the method honest. Where a build counts at less than a pound in the pound, that rate divides into the turnover and pushes every demand upward, and such a rate sits nowhere in writing on a great deal of what gets advertised. A window tighter than the one posited in part six drags the second threshold down, and the periods used above are illustrations rather than observations.

Carried across the family the method changes nothing but its inputs, and the family is where it pays off. A par match asks £400.00, two hundred per cent asks £800.00, this rung asks £1,200.00 and four hundred asks £1,600.00, all on the same unchanged ten pounds. A reader comparing headlines is comparing ratios. A reader comparing obligations is comparing the thing that decides it.