The two percentage mistakes
A percentage is a fraction with a hidden denominator, and everything that goes wrong with percentages goes wrong because the denominator changed while nobody was looking.
There are really only two mistakes. The base moves between the rise and the fall, and percentages of different bases get added together as if they were the same thing.
A rise and the matching fall are different numbers
Go from 80 to 100 and that is a 25% rise — 20 more, on a base of 80. Go from 100 back to 80 and that is a 20% fall — 20 less, on a base of 100. Same twenty, different denominators, different percentages.
This is not a curiosity. A share that falls 50% needs to rise 100% to recover. A price cut by 20% has to rise by 25% to get back to where it was, so "we put the 20% back on" leaves you below the original price.
The general rule: to undo a change of x per cent you need the opposite change measured against the new base, which is always a different figure. Anything that promises to reverse a percentage with the same percentage is wrong.
80 -> 100 (100-80)/80 = +25%
100 -> 80 (80-100)/100 = -20%
Down 50% then up 50% -> 75% of where you started
Work it out: Percentage Calculator →
Percentages of different things do not add
Thirty per cent off followed by twenty per cent off is not fifty per cent off, because the second one applies to what is left after the first. It works out at forty-four per cent.
The same applies going up. Two successive 10% rises are a 21% rise, not 20%, because the second is taken on the larger figure. Over many periods that gap is the whole of compound growth.
The reliable technique is to multiply what remains rather than subtract what goes. Twenty per cent off means paying eighty per cent, so multiply by 0.8. Chain them: 0.7 times 0.8 is 0.56, so you pay 56% and save 44%. It works in either direction and it never gets the base wrong, because you are never subtracting.
Which average, and why it matters
The mean is the total divided by the count. The median is the middle value once they are sorted. The mode is the most common one. They answer different questions and they can be very far apart.
The mean is the one that gets used by default and the one most easily distorted. A single very large value drags it away from anything typical, which is why incomes are reported as medians and why an average that sounds surprising is usually a mean with an outlier in it.
The rule of thumb: if the data is roughly symmetrical, the mean is fine. If it has a long tail — incomes, house prices, response times, anything that can be very large but not very small — the median is the honest answer.
Work it out: Average Calculator →
Fractions are exact and decimals are not
A third has no decimal representation. Three thirds is exactly one; three lots of 0.333 is 0.999. That is why anything working in thirds, sixths or sevenths should stay in fractions until the very end.
Adding and subtracting fractions needs a common denominator, and the lowest common multiple is what supplies it — that is what the LCM is actually for. The greatest common factor is the other half of the pair: it is what you divide by to put a fraction into lowest terms.
A useful check on both: the GCF times the LCM always equals the product of the two original numbers. For 48 and 60 the GCF is 12 and the LCM is 240, and 12 times 240 is 2,880, which is 48 times 60.
One more that is worth knowing rather than deriving each time: dividing by a fraction means multiplying by its reciprocal. Turn the second fraction upside down and multiply. That is why dividing by a half doubles a number, which otherwise looks like a mistake.
1/2 + 1/3 -> common denominator 6 -> 3/6 + 2/6 = 5/6
GCF(48,60) = 12, LCM(48,60) = 240
12 x 240 = 2,880 = 48 x 60
Work it out: Fraction Calculator →
Questions
Why is a 20% fall not the opposite of a 20% rise?
Because the base changes. 80 to 100 is a 25% rise; 100 back to 80 is a 20% fall. The same absolute change is a different percentage of a different starting figure.
If something drops 50%, how much must it rise to recover?
100%. Halving and then doubling gets you back — a 50% rise on the reduced figure only reaches 75% of where you started.
Do successive discounts add up?
No. 30% then 20% is 44% off, not 50%, because the second discount applies to the already reduced price. Multiply the remainders: 0.7 x 0.8 = 0.56.
What is the quickest way to work out a percentage?
Multiply by what remains rather than subtracting what goes. 20% off is a multiplication by 0.8; a 15% rise is a multiplication by 1.15. It chains correctly and never loses track of the base.
Should I use the mean or the median?
The mean for roughly symmetrical data, the median where there is a long tail — incomes, prices, waiting times. A single large value moves a mean a long way and a median hardly at all.
Why do I need the lowest common multiple?
To add or subtract fractions, which needs a shared denominator. The greatest common factor does the opposite job: it is what you divide by to simplify a fraction.