Percentage calculator

The three kinds of percentage problem

Nearly every percentage question is one of three types. The fastest way to pick the right formula is to ask what you know and what you are looking for: the whole and a percent, a part and the whole, or two values at different points in time.

Always be clear about what the percentage is taken of. The quantity after "of" is the base, the 100%. If the base is wrong, the answer is wrong even when the arithmetic itself is perfect.

• Percent of a number: 15% of 240 is 0.15 × 240 = 36. For example, a 15% deposit on a 240 euro invoice.
• What percent: 18 correct answers out of 24 is 18 ÷ 24 × 100 = 75%. The base is the total number of questions.
• Percentage change: a salary rising from 1800 to 1908 euros is (1908 - 1800) ÷ 1800 × 100 = +6%.

Why +20% followed by -20% is not zero

This is the classic percentage trap. Say an item costs 100 €. The price goes up 20% to 120 €. Then it is discounted by 20%, but now 20% is taken of 120 euros, which is 24 €. The final price is 96 €, 4% below where it started.

The reason is that the second percentage uses a different base. Successive changes multiply rather than add: 1.20 × 0.80 = 0.96. Reversing the order gives the same result, because multiplication does not care about order.

The same logic explains asymmetric answers. A price falling from 250 to 200 euros is -20%. A price rising from 200 back to 250 euros is +25%. Both are correct: the base is 250 in the first case and 200 in the second.

Percent versus percentage point

When you compare two rates that are already percentages, there are two different ways to describe the change. A percentage point is the plain difference; a percentage change is the relative growth.

Take VAT: if the rate rises from 22% to 24%, that is an increase of 2 percentage points. As a percentage change it is (24 - 22) ÷ 22 × 100 = 9.09%. News stories and loan offers often blur the two, so the same change can sound very different.

The same applies to a loan interest rate going from 3% to 4%: it is 1 percentage point, yet the interest cost grows by roughly a third. If you enter the change 3 → 4 in the calculator, it shows that relative change, not points.

Working backwards: taking 24% VAT out of a price

An everyday example is extracting VAT from a price. The standard VAT rate in Estonia is 24%. If an item costs 49.90 € including VAT, a common mistake is to take 24% of that amount: 49.90 × 0.24 = 11.98 €. That overstates the tax, because the 24% was added to the price without VAT, not to the final price.

The correct way is to divide the final price by 1.24: 49.90 ÷ 1.24 = 40.24 € is the net price and the VAT is 49.90 - 40.24 = 9.66 €. Check: 24% of 40.24 is 9.66, exactly as it should be.

The same trick works for any percentage mark-up. If an item costs 69 € after a 15% price rise, the original price was 69 ÷ 1.15 = 60 €. Entering "15% of 69" instead gives 10.35 and a wrong original price of 58.65 €.

Checking your answer and common mistakes

A good habit is to estimate in your head before calculating exactly. You get 10% by moving the decimal point one place left: 10% of 347 is 34.7. From there 5% (half of it) and 20% (double) follow quickly.

It also helps to know that a% of b equals b% of a. For example, 8% of 50 is the same as 50% of 8, which is 4. The swapped version is often far easier to do mentally.

• When calculating a change, always divide by the old value, not the new one.
• If the starting value is 0, a percentage change cannot be calculated and the calculator shows 0.
• Results are rounded to two decimals, so with very small numbers the rounding can be noticeable.
• A rise of more than 100% is perfectly possible: from 40 to 100 is +150%, but a fall can never exceed -100%.