Ten kinds of percentage question, answered in exact arithmetic rather than floating point, with the working shown. Includes percentage points, finding the whole, and tax both ways.
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The tip, the tax and everybody's share worked out in whole pennies, so the amounts printed add up to the total printed. Says who pays the odd penny, and what rounding does to the tip rate.
You need 15% of 23.7, and then you need to paste it somewhere.
Input
15% of 23.7
Result
3.555 on the screen, and 3.555 in the clipboard
Fifteen percent of 23.7 is exactly 3.555, and in floating point it is 3.5549999999999997, because 0.15 cannot be written exactly in binary. A display that rounds to six places hides that and a Copy button that copies the underlying value does not, so what you read and what you paste become different numbers. Here the value is held as a pair of whole numbers throughout, so both are 3.555. The same exactness makes the round trip work: 100 plus 15 percent, undone, is 100 rather than 100.00000000000001. Ten modes, including the one most calculators leave out, "X is P% of what number", and percentage points shown alongside relative change because 40% to 44% is four points AND ten percent.
Ten kinds of percentage question, answered without floating point. 15% of 23.7 is 3.555, not 3.5549999999999997, and what the Copy button copies is what the screen shows.
A percentage is a rational number, so every answer here is exact. That is not a detail: 15 percent of 23.7 is exactly 3.555 and a double gives 3.5549999999999997; 110 percent of 90.9 is exactly 99.99 and a double gives 99.99000000000001. A display that rounds to six places hides both, and a Copy button that copies the underlying value does not, so what you read and what you paste end up different numbers. Here they are the same number, always.
**Ten questions.** What is P% of X. X is what percent of Y. The change from X to Y. X increased or decreased by P%. What X was before a P% change, which is not the same as taking the percentage off again. X is P% of WHAT number, which is the question most percentage calculators leave out. From P% to Q%, which has two answers. P% of Q%. And a price with tax added or removed, which are different operations.
**Two definitions stated rather than assumed.** A percentage change from a NEGATIVE starting value: from -50 to -25 is a rise of 25 in a quantity whose size was 50, and dividing by the size gives +50 percent while dividing by the value gives -50 percent. This divides by the size, so a rise is positive whichever side of zero the numbers sit, and the page says so when it applies. And percentage POINTS: a rate going from 40% to 44% has risen four percentage points and ten percent, both true, and which one is meant changes the story. Both numbers are given together.
**The working is shown**, with the numbers in it rather than as a formula with letters, and each answer carries the extra lines that go with it: the amount added, the remainder, the multiplier, the check that reversing really returns to where you started. Inputs can be plain numbers, decimals, fractions like 3/4, mixed numbers like 1 2/3, percentages with a sign on them, or repeating decimals like 0.(3), and the tool shows the fraction it read. Anything it cannot read is refused by name, and a division by zero is explained rather than shown as Infinity.
Rounding, when you ask for it, is half away from zero, so 2.675 to two places is 2.68 rather than the 2.67 a double gives because it does not actually hold 2.675.
Take 20 percent off, and then work out what the price was before the discount, which is a division and not a second subtraction.
Add tax to a net figure, or find the net inside a gross one. 20 percent inside 120 is 20; 20 percent of 120 is 24.
A rate that went from 40% to 44% rose four points and ten percent. Both are given, so a headline can be checked.
37 out of 45 as a percentage, and how many more marks a target needs.
The percentage change between two figures, including when one or both are below zero.
Because 23.7 and 15 percent are held as exact fractions rather than as binary doubles. 0.15 cannot be written exactly in binary, so multiplying it is already slightly wrong before anything else happens.
Going from 40% to 44% is a rise of four percentage points, and also a rise of ten percent, because four is a tenth of forty. The Percentage points mode gives both at once.
Because the second percentage is of a smaller number. 100 less 20 percent is 80, and 80 plus 20 percent is 96. The "Undo a change" mode reverses it properly.
The change is divided by the SIZE of the starting value, so a rise is a positive percentage whichever side of zero the numbers are. The page says so whenever it applies.
Yes: 3/4, 1 2/3, 12% and 0.(3) are all read, and the tool shows the fraction it read them as.
Half away from zero, so 2.675 to two places is 2.68 and -0.5 to none is -1. The rounding is a display choice only; the answer itself is never rounded on the way.
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