If something rises 50% then falls 50%, where does it end up?

25% below where it started. 100 rises to 150, then falls by half of 150 — which is 75, not 100.

Why are percentage increases and decreases not symmetrical?

Because each percentage applies to whatever the value is at that moment, not the original. The fall is calculated on a bigger number than the rise was.

What percentage fall cancels a 50% rise?

A fall of 33.3%. To reverse any rise of r%, you need a fall of r ÷ (100 + r) × 100.

Why a 50% Rise Then a 50% Fall Does Not Get You Back

Start with 100. Add 50% and you have 150. Take away 50% and you have 75.

You are down a quarter, having applied equal and opposite percentages. This is not a trick — it is how percentages work, and it matters far more than it first appears.

Where the missing 25 goes

The rise and the fall are not the same size in absolute terms.

100 + 50% → the rise is 50 (half of 100) → 150
150 − 50% → the fall is 75 (half of 150) → 75

The increase added 50. The decrease removed 75. Percentages always apply to whatever the number is at that moment, and by the time you take the 50% off, the number is larger.

The order does not save you either. Take 50% off first and you get 50; add 50% back and you get 75. Same destination.

The recovery is always bigger than the fall

This is the part that catches people out in practice.

If an investment drops 50%, it needs to gain 100% to break even — not 50%. It fell from 100 to 50, and getting from 50 back to 100 means doubling.

The general rule: to reverse a fall of r%, you need a rise of

r ÷ (100 − r) × 100

Fall	Rise needed to recover
10%	11.1%
20%	25%
33%	50%
50%	100%
80%	400%

The gap widens fast. This is why large losses are so much worse than they look, and why “it only needs to bounce back by the same amount” is wrong every time.

It works the other way too

To cancel a rise of r%, you need a fall of

r ÷ (100 + r) × 100

A 50% rise is undone by a 33.3% fall. A 100% rise is undone by a 50% fall.

So if a supplier raises prices 25% and later offers “25% off”, you are not back to the old price — you are still paying about 6% more than before.

Where this bites in real life

Sequential discounts. 20% off, then a further 20% off, is not 40% off. It is 36%.

Averaging percentages. A stock up 10% one year and down 10% the next did not break even. It is down 1%. Averaging the two to zero is wrong, though it is done constantly.

Percentage recovery reporting. “Sales fell 40% then recovered 40%” sounds like a full recovery. Sales are actually still 16% below where they started.

The habit worth building

Never add or average percentage changes. Multiply them.

Convert each change to a multiplier — a 50% rise is ×1.5, a 50% fall is ×0.5 — and multiply through:

1.5 × 0.5 = 0.75 → 75% of the original

Multipliers compose correctly in any order and any quantity. Percentages do not. Once you are in the habit of converting first, this entire family of errors disappears.