Work out a percentage

How it works

Three different questions share the word percent, and mixing them up is one of the commonest arithmetic errors in ordinary paperwork. Fifteen percent of eighty is a share of a quantity. Twenty-five as a percent of two hundred is a ratio between two quantities. A move from four percent to five percent is a change in a rate. Every one of them has its own formula and gives a different number, and the wrong answer sits in a spreadsheet cell looking perfectly plausible until somebody checks it against the source figures.

Percent change runs on the old number

Percentage change divides the movement by where you started rather than where you ended up. A rise from 100 to 150 counts as 50 percent, because the 50 of movement is measured against the original 100. That same 50 measured against 150 reads 33 percent instead, and 33 percent is what you get travelling in the other direction. The asymmetry catches people constantly. Take 100 up by 50 percent to reach 150, then take 50 percent off that result, and you land on 75 rather than back where you began.

The everyday version of the same trap is a discount stacked on a discount. Twenty percent off followed by another twenty percent off does not come to forty percent off. The multipliers compound: 0.8 times 0.8 is 0.64, so the real reduction is 36 percent. Shop signage often relies on the reader adding the two figures together.

Removing a percentage means dividing

A gross figure already contains the percentage, so recovering the net figure is a division. A total of 120 that includes 20 percent tax came from a net of 100, because 100 times 1.2 is 120. Subtracting 20 percent of 120 gives 96, wrong by four units and wrong in the same direction every time. The reason is that the tax was charged on the smaller number, so a percentage taken off the gross total gets computed from the wrong base. Divide by 1.2 and the net comes back exactly.

Markup and margin diverge for the same reason. A cost of 80 marked up by 25 percent sells at 100, and the margin on that sale is 20 percent, because margin is measured against the selling price while markup is measured against the cost.

Points and percent are different units

A rate that moves from 4 percent to 5 percent has risen by one percentage point and by 25 percent. Both readings are true and both describe the same event, so any report quoting one without naming the unit is ambiguous. The relative figure sounds more dramatic and gets quoted more often for that reason. One point added to a rate of 4 percent is a quarter more. The same point added to a rate of 40 percent is 2.5 percent more. The base decides how large a move of a single point appears.

Where the odd trailing digits come from

Binary floating point cannot hold a tenth exactly, and percentages are made of tenths. Taking twenty percent off 100 twice comes out as 64.00000000000001 in a double. Adding 0.1 to 0.2 gives 0.30000000000000004. Multiplying 4.35 by 100 gives 434.99999999999994. Rounding inherits the fault rather than curing it, because 1.005 stored as a double sits a hair under the halfway mark, so the usual rounding trick returns 1.00 where 1.01 is correct.

Everything here runs on exact integers scaled by powers of ten, and the single rounding step happens at the very end, half away from zero. A recovered net of 100 therefore reads 100.00, two stacked twenty percent discounts on 100 read 64.00, and raising the decimal places control reveals arithmetic rather than artefacts.