Percentage Calculator Guide: Percentages, Increases and Discounts

What a percentage actually means

The word percentage comes from the Latin phrase per centum, meaning by the hundred, and that origin is the whole definition. Fifty percent means fifty out of every hundred. Twenty-five percent means twenty-five out of every hundred. A percentage is therefore not a separate kind of number but a convenient way of writing a fraction whose denominator is one hundred, and every percentage calculation is a fraction calculation wearing a percent sign.

That framing explains why percentages are so useful for comparison. Saying that forty-five students out of a class of three hundred passed an exam is accurate but hard to compare with another class of different size. Saying that fifteen percent passed makes the comparison immediate, because both figures are now expressed against the same base of one hundred. The Percentage Calculator performs that standardisation instantly, but the idea behind it is worth holding onto: a percentage only means something when you know what whole it refers to.

Percentages appear in more places than most people notice. Exam marks, interest rates, tax rates, sale discounts, tips, pay rises, survey results, battery levels and statistical claims all use the same notation. Learning the three basic percentage calculations therefore pays off across school, work and everyday money decisions, and it protects you against the most common numerical trick in advertising and politics, which is quoting a percentage without naming its base.

The three calculations you need

The first calculation finds a percentage of a number: what is twenty percent of two hundred and fifty? The method is to multiply the number by the percentage and divide by one hundred, so two hundred and fifty multiplied by twenty, divided by one hundred, gives fifty. The second calculation finds what percentage one number is of another: forty-five as a percentage of three hundred is forty-five divided by three hundred, multiplied by one hundred, which gives fifteen percent. The third calculation finds a percentage change: the difference between the new and old values, divided by the old value, multiplied by one hundred.

The third calculation is the one that causes the most trouble, because the base is the starting value, not the finishing value and not an average of the two. A price that rises from eighty to one hundred has increased by twenty, and twenty divided by eighty is nought point two five, so the increase is twenty-five percent. The same twenty-unit move in the opposite direction, from one hundred down to eighty, is a twenty percent decrease, because twenty divided by one hundred is nought point two. The same absolute change gives two different percentages because the bases differ, and anyone who quotes a percentage change without naming the starting value is leaving out the most important part of the claim.

A related trap is adding percentages that have different bases. A price that rises by ten percent and then falls by ten percent does not return to its starting value. Start at one hundred: a ten percent rise gives one hundred and ten, and a ten percent fall from one hundred and ten removes eleven, leaving ninety-nine. Percentage changes compound against a moving base, which is why successive discounts and successive rises must be applied step by step rather than added together.

A worked example, step by step

Consider a student who scored forty-five marks out of a possible three hundred across three papers, and wants to know the overall percentage. The whole is three hundred and the part is forty-five. Divide forty-five by three hundred, which gives nought point one five, and multiply by one hundred, which gives fifteen percent. That is a low score, but at least it is an honest one, and the student now knows exactly how far the mark sits from any pass threshold the course sets.

Now consider a shop price. A jacket is priced at two hundred and fifty, and a sale offers twenty percent off. Twenty percent of two hundred and fifty is two hundred and fifty multiplied by twenty, divided by one hundred, which is fifty, so the sale price is two hundred and fifty minus fifty, which is two hundred. If the shop then raises that sale price by twenty-five percent, the new price is two hundred plus twenty-five percent of two hundred, which is two hundred plus fifty, which is two hundred and fifty again. The two percentages look like they should cancel, but they do not cancel by addition; they cancel here only because the second percentage is applied to the smaller, discounted base, and the arithmetic happens to line up.

Finally, consider a pay rise. A monthly salary of eighty thousand rises to one hundred thousand. The increase is twenty thousand, and twenty thousand divided by the starting salary of eighty thousand is nought point two five, so the rise is twenty-five percent. Quoting the rise as twenty percent would mean dividing by the new salary instead of the old one, which is a different calculation answering a different question. Whenever a percentage change is quoted to you, ask which value was used as the base, and recompute it if the answer matters.

Mistakes that even careful people make

The first mistake is confusing percentage points with percent. If a tax rate rises from twenty percent to twenty-five percent, that is a rise of five percentage points, but it is a twenty-five percent increase in the rate itself, because five divided by twenty is nought point two five. News reports and advertisements blur this distinction constantly, and the only defence is to recompute from the underlying figures whenever both are available.

The second mistake is treating a percentage as if it had no base. A claim that something grew by fifty percent is meaningless until you know fifty percent of what, over what period, and compared with which starting point. A small business that grows from two customers to three has grown by fifty percent, and a large business that grows from two thousand customers to two thousand and one has grown by almost nothing, yet the first claim sounds far more impressive. Always ask for the underlying numbers.

The third mistake is rounding percentages before finishing a calculation. A percentage that is rounded to a whole number at an intermediate step can shift a final money figure by more than the rounding seems to justify, especially on large amounts. Keep the full precision until the final step, then round the answer to a sensible number of decimal places, usually two for money and one or none for reporting.

Where percentages meet real decisions

In money, percentages set the terms of almost every deal: interest on savings and loans, tax on purchases, commission on sales, and discounts in shops are all percentages applied to a base amount. The Discount Calculator specialises in sale prices, the Sales Tax Calculator handles tax-inclusive pricing, and the guide to the Average Calculator shows how to summarise percentage scores, such as a set of exam results, into a single representative figure. Each of these tools is the same percentage arithmetic applied to a particular context, and learning the general method means you can check any of them by hand.

In study and work, percentages grade performance, track progress and set targets. A student who knows that forty-five out of three hundred is fifteen percent also knows how many more marks are needed to reach any threshold, because the gap in marks is the gap in percentage points multiplied by the total and divided by one hundred. A manager tracking a project that is sixty percent complete knows that forty percent of the work remains, provided the percentage refers to the same measure of work throughout, which is a condition worth verifying before reporting it.

Percentages are simple, but they are not trivial, and the difference between a correct and an incorrect percentage is often a single choice: which number is the whole. Get that choice right, keep your precision until the end, and the arithmetic takes care of itself.