A price rises 20% and then falls 20%. Most people expect to be back where they started. Starting at 100, you rise to 120, then fall by 20% of 120, which is 24, landing at 96.
That 4% is not rounding. It is the single most common numerical mistake in everyday work, and the same misunderstanding produces wrong tax figures and wrong loan comparisons.
Percentage change is directional
The trap is that the percentage is always of something, and the something changes.
| Question | Formula |
|---|---|
| What is X% of Y | Y × X / 100 |
| X is what percent of Y | X / Y × 100 |
| Change from A to B | (B − A) / A × 100 |
| Reverse: B is A after X% increase | A = B / (1 + X/100) |
The last row is the one people get wrong. If a price is 118 after an 18% increase, the original is 118 / 1.18 = 100, not 118 − 18 = 100 by coincidence of the numbers, and definitely not 118 × 0.82 = 96.76.
Also worth separating: a percentage point and a percent are different units. A rate going from 4% to 6% has risen two percentage points, or fifty percent. Both statements are true and they describe the same change, which is why the ambiguity is so useful to people arguing a case.
A percentage calculator is worth using for the reverse cases specifically, because those are the ones where the wrong method produces a plausible-looking number.
GST and the inclusive-price problem
Tax calculations are the same reverse-percentage problem with money attached.
Adding GST to a base price is simple: base × (1 + rate). At 18%, a base of 1,000 becomes 1,180.
Removing GST from an inclusive price is division: inclusive / (1 + rate). From 1,180 at 18%, the base is 1180 / 1.18 = 1,000 and the tax is 180.
For Indian GST there is a further split that matters on the invoice rather than in the total. An intra-state supply divides the rate into CGST and SGST, half each, so 18% appears as 9% plus 9%. An inter-state supply uses a single IGST line at the full 18%. The amount payable is identical; the presentation is not, and getting it wrong is a compliance issue rather than an arithmetic one. A GST calculator that shows the split is doing that part.
What an EMI actually is
An equated monthly instalment is a fixed payment covering interest on the outstanding balance plus enough principal that the loan clears exactly at the end of the term.
EMI = P × r × (1 + r)^n / ((1 + r)^n − 1)
P is the principal, n the number of months, and r the monthly rate, which is the annual rate divided by 12 and then by 100. Using the annual rate here is the usual error and produces a wildly wrong figure. An EMI calculator does this correctly and, more usefully, shows the amortisation schedule underneath; the exponentiation is awkward enough by hand that a scientific calculator, such as the one on any phone, is the minimum if you want to check it yourself.
The payment is constant. Its composition is not, and that is the part worth understanding:
| Stage | Interest | Principal |
|---|---|---|
| First instalment | Most of it | Very little |
| Middle of term | Roughly even | Roughly even |
| Final instalment | Almost nothing | Almost all |
Because interest is charged on the outstanding balance, and the balance starts high, early payments are mostly interest. This has a direct consequence: a lump sum repaid early removes far more total interest than the same sum repaid late, because it removes principal that would otherwise have accrued interest for the whole remaining term.
It also explains why a small rate difference matters so much over a long term. On a 25-year loan, half a percentage point can change the total interest by more than a year's payments, because the effect compounds across 300 months. Comparing two offers by their EMI alone hides this; compare total interest paid.
Units, and the one that is different
Most unit conversion is multiplication by a constant. A metre is 3.28084 feet, a kilogram is 2.20462 pounds, and the ratio holds at every value.
Temperature does not work that way, because the scales have different zero points:
°F = °C × 9/5 + 32
There is an offset, so the ratio between two temperatures is not preserved. 20°C is not twice as warm as 10°C in Fahrenheit, and "a 5 degree rise" means different amounts in the two scales. This is the one conversion where intuition from length and weight actively misleads, and it is why a unit converter handles temperature as a special case.
The small conversions
Two that come up more than you would expect:
Number to words exists because cheques, contracts and invoices require the amount spelled out. The reason is fraud prevention: digits can be altered with a pen, words cannot be extended convincingly. Note that the Indian system groups in lakhs and crores rather than thousands and millions, so the same figure is written differently depending on the convention.
Roman numerals follow a subtractive rule where a smaller value before a larger one is subtracted: IV is 4, IX is 9, XL is 40. Only powers of ten are subtracted, and only from the next two values up, which is why IC is not a valid way to write 99. It is XCIX.
Every calculation here is arithmetic on numbers you typed, so none of it needs a server. That is worth having for the loan and tax cases in particular, where the figures being entered are salary, outstanding debt and turnover.