Percentage

There are three modes here because "percent" answers three different questions and the formulas are not interchangeable. Finding 15% of 250 divides by 100. Finding what share 45 is of 150 divides by the whole. Finding the change from 80 to 100 divides by the old value — and that last denominator is where most percentage errors live.

One property worth knowing before you type anything: x% of y always equals y% of x, because both are xy ÷ 100. So 15% of 80 and 80% of 15 are both 12. That is not a curiosity, it is a shortcut: 4% of 75 is awkward to do in your head, while 75% of 4 is three.

The third mode reports a sign and a direction, because a change from 80 to 100 and a change from 100 to 80 are not the same size. The first is +25%, the second is −20%.

Three formulas, three different denominators

• Percent of a value: (Percent ÷ 100) × Value. 15% of 250 = 37.5.
• Part as a percent of a whole: (Part ÷ Whole) × 100. 45 out of 150 = 30%.
• Percent change: ((New − Old) ÷ Old) × 100. From 80 to 100 = +25%.

The denominator in the change formula is the starting value, always. Dividing by the new value instead gives 20 ÷ 100 = 20%, which is a real number answering a real question — "the increase is 20% of the end figure" — but it is not the percentage increase, and quoting it as one understates every rise and overstates every fall.

There is a fourth calculation people sometimes want and this tool does not do: percent difference between two values with no natural before-and-after, which divides by the average of the two rather than by either one. For 80 and 100 that is 20 ÷ 90 = 22.2%. Use it only when neither value is a baseline, such as comparing two lab measurements of the same quantity.

One of each, then the trap

1. 15% of $120. 0.15 × 120 = 18.00. Same arithmetic whether it is a tip, a discount or a commission.
2. 45 out of 150. 45 ÷ 150 × 100 = 30%. Useful for a score, a completion rate, or a budget line as a share of the whole.
3. From 80 to 100. (100 − 80) ÷ 80 × 100 = +25%.

Now the trap. A holding worth $100 falls 50% to $50, then rises 50% to $75. It is still down 25%, because the second 50% was taken on a smaller base. Percentage gains and losses of equal size do not cancel; they compound. Recovering a 50% loss needs a 100% gain, and recovering a 20% loss needs a 25% gain.

The same asymmetry runs the other way on prices. A $200 item marked up 20% to $240, then discounted 20%, sells for $192 — below where it started. Anywhere a percentage is applied twice, multiply the factors (1.2 × 0.8 = 0.96) rather than adding or subtracting the percentages.

What it takes to get back to even

The formula behind the column is gain = loss ÷ (100 − loss), and it explains why the last two rows look so brutal: after a 90% fall you hold ten cents on the dollar, and ten cents has to become a dollar again.

For quick estimates in the other direction: 10% is the decimal point moved one place left, 5% is half of that, 20% is double it, 15% is 10% plus half of 10%, and 1% is two places left. Chain them — 35% is 20% plus 10% plus 5%.

Discounts, markup, and margin — three places this is used wrong

Stacked discounts do not add. "20% off, plus an extra 10% at the register" is not 30% off. The second discount applies to the already-reduced price: 0.80 × 0.90 = 0.72, so it is 28% off. On a $200 coat that is $144, not $140.

Markup and margin are different numbers. An item costing $60 and selling for $100 carries a 66.7% markup (40 ÷ 60, computed on cost) and a 40% margin (40 ÷ 100, computed on price). Quoting the markup as the margin makes a business look far more profitable than it is; going the other way makes pricing come out too low.

Order matters for what you owe, not for the arithmetic. Multiplying by 0.8 and by 1.085 commutes, so a discount and sales tax give the same total in either order. What changes is the tax base: stores apply the discount first and charge tax on the reduced price, so the tax collected is lower. Enter the discounted figure when checking a receipt.

Percent, percentage point, and basis point

A rate moving from 4% to 5% has risen one percentage point and twenty-five percent. Both statements are correct and they describe the same move; using the wrong one in a headline changes the story by a factor of twenty-five. Finance splits it further with the basis point, one hundredth of a percentage point: a 25 basis point rate cut is 0.25 percentage points.

Percentages of different-sized groups cannot be averaged. A 50% pass rate among 10 students and a 10% pass rate among 1,000 is not a 30% overall rate — it is (5 + 100) ÷ 1010 = 10.4%. Recombine the raw counts, then take the percentage.

Two edge cases the change mode cannot resolve. If the old value is zero, percent change is undefined — any increase from nothing is infinite, so report the absolute figure instead. If the old value is negative, as with a company moving from a loss to a profit, the formula returns a number that is mathematically valid and practically meaningless; state the swing in currency.

Reading the output

Each of the three modes runs independently, so filling in one does not disturb the others and you can leave a previous answer on screen while working the next one. The three modes do not round alike: the first, which returns an amount rather than a percentage, shows two decimal places so cent-level money calculations stay honest, while the two modes that return a percentage show one.

The change mode adds a sign and a direction arrow: a positive result with an upward arrow for an increase, a negative result with a downward one for a decrease. Read the sign rather than the arrow when the numbers are small, because at one decimal place a change of −0.004% renders as −0.0% — a signed zero, still a decrease, and easy to read as nothing at all.

Frequently asked questions

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