Percentage change

Prices rise and fall, a jersey is on sale, the rent goes up, your savings grow. It always comes down to the same question: how much is left or added when something changes by a certain percentage? Whoever can compute this confidently will not be fooled at the checkout or in a contract.

Percent of the base value

The base value is the amount you start from, that is the full quantity or the 100 %. A percentage is a part of it. 10 % of 40 euros is $\frac{10}{100} \cdot 40 = 4$ euros. Remember: 1 % is always the hundredth part, and you build everything else from that.

Increase and decrease

For an increase the percentage is added to the base value, for a decrease it is subtracted. A book for 24 euros with 25 % off: the discount is 6 euros, you pay 18 euros. A jersey for 40 euros that becomes 25 % more expensive: 10 euros markup, so 50 euros. First find the part, then add or remove it.

The factor trick

It is faster with a single factor. Plus 20 % means times 1.2, because you pay 100 % plus 20 %. Minus 20 % means times 0.8, because 80 % are left. This way you compute in one step: 200 euros with a 25 % markup is 200 · 1.25 = 250 euros. Going backwards works the same way: if you only know the new price, divide it by the factor and you are back at the base value.

The base value can shift

Important: percentages always refer to the current base value. If a price first rises by 50 % and then falls by 50 %, you do not end up back at the start. 60 euros first become 90 euros, that is 30 euros more. The second 50 % now counts from those 90 euros, which is 45 euros off, leaving 45 euros. The cut of 45 euros is bigger than the markup of 30 euros because it is taken from a bigger base value.