Percentage Calculator

Almost every percentage problem is one of five shapes. Naming the shape first is what makes the arithmetic easy — the formulas themselves are trivial once you know which one you need.

This is the single most common percentage error in reporting. If an interest rate moves from 4% to 5%, that is a rise of one percentage point — but a 25% relative increase. Both statements are true and they describe the same change.

Use percentage points when comparing two percentages directly (market share, rates, poll numbers). Use percent change when describing how much a quantity grew relative to where it started. Mixing them up can make a modest change sound dramatic, or the reverse.

Take 100, add 20% to get 120, then subtract 20% — you land on 96, not 100. The two percentages are applied to different bases: the increase used 100, the decrease used 120. Percentage moves compound rather than add.

The same asymmetry explains why a loss hurts more than an equal-sized gain helps. A 50% drop needs a 100% gain to recover. A 20% drop needs a 25% gain. To undo a P% decrease you need an increase of P ÷ (100 − P) × 100 percent.

A frequent mistake is subtracting a percentage from a final figure to recover the original. If a price of 120 already includes a 20% markup, the original is not 120 − 24. It is 120 ÷ 1.20 = 100. You divide by the factor that was applied, you do not subtract the percentage of the wrong base.

• Remove an added P%: divide by (1 + P ÷ 100).
• Remove a discount of P%: divide by (1 − P ÷ 100).
• Find a whole from a known part: divide the part by its share, then multiply by 100.

Percentages are commutative: P% of N equals N% of P. That makes some awkward problems trivial. 4% of 75 looks unpleasant; 75% of 4 is obviously 3.

Build the rest from 10% and 1%. Move the decimal one place for 10%, two places for 1%, then combine: 35% is three tens plus a half-ten. Halving and doubling handles 5%, 20%, and 25% without any written work.