Percentage calculator

Three questions come up again and again with percentages, and none of them is the same operation. What is 15 percent of 240? What share of 180 is 45? How far has a price moved when it goes from 89.90 to 62.93? This page sets the three calculations side by side, each in its own card, with two fields and one result.

Nothing has to be submitted. A card updates as soon as both its fields contain a number, decimals are accepted with no step constraint, and the result appears to two decimals. The three cards are independent, so a figure can sit untouched in one while you work in another.

The difficulty with percentages is never the arithmetic. It is the base the percentage is applied to. A rise and a fall of the same size do not cancel out, two successive discounts do not add up, and a gap measured in points is not a percentage change. Those three traps are worked through below, with the full calculation each time.

How to use it

1. Find the card that asks your questionThree blocks follow one another down the page: what is X% of Y, X is what percentage of Y, and change from X to Y, whether that change is a rise or a discount.
2. First card, apply a rateType the rate into the X (%) field and the value into Y. The result is a bare number with no unit attached: whether it stands for pounds, grams or visitors is for you to keep track of.
3. Second card, express a shareX is the part, Y the whole, and the result carries a percent sign. A total of zero shows a dash rather than a result, since no share can be measured against nothing.
4. Third card, measure a changeX is the starting value, Y the finishing value. An increase is prefixed with a plus sign; a decrease carries its own minus sign. The order of the two fields is what fixes the direction of travel.
5. Correct the figures as you goBoth fields accept any decimal number, with no fixed step. Changing one digit recalculates that card immediately and leaves the other two exactly as they were.

The three formulas

Applying a rate: X% of Y = X × Y ÷ 100. Fifteen percent of 240 is 15 × 240 ÷ 100 = 36. Expressing a share: X ÷ Y × 100. Forty-five out of one hundred and eighty gives 45 ÷ 180 × 100 = 25 percent. Measuring a change: (Y - X) ÷ X × 100. From a price of 89.90 down to 62.93, that is (62.93 - 89.90) ÷ 89.90 × 100 = -30 percent.

The denominator is the only thing that moves from one formula to the next, and that is exactly where the mistakes live. A change is always measured against the starting value, never against the finishing value and never against the average of the two. Swapping the two fields of the third card therefore does not return the opposite number: 100 to 120 reads +20 percent, while 120 to 100 reads -16.67 percent.

A 20 percent rise followed by a 20 percent fall

The result is not where you started. Take 100: the 20 percent rise adds 20 and gives 120. The 20 percent fall now applies to 120, so it takes off 24 rather than 20 and leaves 96. Over the whole round trip, the third card reports (96 - 100) ÷ 100 × 100 = -4 percent.

The cause is the change of base: the first percentage is applied to 100, the second to 120. To come back to exactly 100 from 120 you have to remove 20 ÷ 120, that is 16.67 percent, which the third card confirms if you enter 120 then 100. A round trip at 50 percent is more striking still: a value cut in half has to double, in other words gain 100 percent, simply to return to its old level.

The same mechanism governs stacked discounts. Thirty percent off followed by twenty percent off is not fifty percent off: the price is multiplied by 0.70 and then by 0.80, which is 0.56, a real discount of 44 percent.

Percentage points, percentages and units

A rate moving from 4 percent to 6 percent gains two percentage points, while the third card announces +50 percent. Both statements are correct. Points measure the absolute gap between two rates; the percentage measures the relative move from one to the other. The confusion is routine in commentary on polls, market share and interest rates, and it disappears as soon as the sentence says which reading it means.

The cards carry no units of their own. The first returns a raw number, the other two a percentage. Feed in pounds and you read pounds; mix grams with litres and the page will produce a neatly formatted result with no meaning at all. Nothing warns you, because nothing here knows what the numbers stand for.

What these three cards do not do

None of them reverses a change. Recovering an original price from a sale price is a division: an item shown at 62.93 after 30 percent off was 62.93 ÷ 0.70 = 89.90 beforehand. The same mechanism governs the move from a tax-inclusive price to a tax-exclusive one, which the VAT calculator on this site handles directly, in both directions.

None of them chains two operations either. The result of one card has to be carried by hand into the next. None of them computes an average rate over several periods: that calls for a geometric mean, not the arithmetic mean of the individual changes.

The third card finally assumes a starting value other than zero. Start at zero and it shows a dash, since no relative change exists from nothing. Start at a negative value, a bank balance or a trading result, and the quotient is still computed, but the sign printed in front follows the order of the two numbers rather than the real direction of travel, so the display can end up putting a plus in front of a deterioration. On quantities that cross zero, work with the absolute gap instead.