Percentage change — the new value, and the rise needed to get back

The two cards on the left are the before and after values. The three on the right show, from the top, the change, the percentage change, and the after value as a percentage of the before value. Changing either input updates all three.

Percentage change divides the change by the before value

A percentage change says how much a value went up or down, as a percentage of where it started:

percentage change = (after − before) ÷ before × 100

Here, after minus before is 800 − 1,000 = −200. Divided by the before value, 1,000, and multiplied by 100, that is −20%. A minus sign means a decrease; a plus means an increase.

The divisor is always the before value, because the question is "what share of the starting value was gained or lost?" The same formula works for prices, sales, visitor numbers or anything else you count or measure.

The bottom-right card shows the after value as a percentage of the before value: 800 ÷ 1,000 × 100 = 80%. The percentage change is that figure minus 100 (80 − 100 = −20). Reports sometimes give one and sometimes the other — "sales were 80% of last year's" and "sales fell 20% on last year" describe the same change.

Swapping before and after changes the percentage

With before set to 800 and after to 1,000 on Board 1, the percentage change becomes 25%. The change is the same 200, but it is now divided by 800 instead of 1,000. Going from 1,000 to 800 is a 20% decrease; going from 800 to 1,000 is a 25% increase. That difference is what sits behind the "rise needed to get back" further down.

The new value is the old value times (1 + the change)

Working the other way, a starting value and a percentage change give the new value. A 20% fall leaves 80%, so the value is multiplied by 0.8. A 20% rise makes it 120%, so the value is multiplied by 1.2.

new value = old value × (1 + percentage change ÷ 100)

For a fall, the percentage change goes in as a negative number. Board 2 applies this twice in a row.

Top left is the starting value, with the first percentage change below it. The second percentage change sits at the bottom in the middle. Both are sliders running from −90% to 100%. The value after each change and the overall change from the start update with them.

Down 20%, then up 20%, does not return to the start

Board 2 starts with 1,000 falling 20% and then rising 20%. After the first change it is 800; after the second it is 960, and the overall change is −4%. It is the same sum as cutting the price of a 1,000 item by 20% to 800, then raising it by 20% again and ending up at 960.

The two 20%s are taken from different numbers:

• The fall — 20% of 1,000, so it loses 200
• The rise — 20% of 800, so it gains 160

Down 200, up 160: 1,000 − 200 + 160 = 960. The rise is taken from the smaller number left after the fall, so it comes up short by 20% of the 200 that was lost — 40, exactly the gap to the start.

As multiplication, 0.8 × 1.2 = 0.96: two changes leave 96% of the start. Doing it the other way round — up 20% first, then down 20% — passes through 1,200 and still ends at 960, since the order of a multiplication does not change the answer.

The same happens with sales that fall 20% one month and rise 20% the next: they end at 96% of where they were two months earlier.

The rise needed to get back

Board 3 works out the rise that takes 800 back to 1,000.

Top left is the starting value, with the fall below it. In the middle is the value after the fall, and the two cards on the right give the amount needed to get back and the rise that amount represents. The card at the bottom right draws the rise needed as the fall changes.

To go from 800 back to 1,000 takes 200 more. Divided by the current value, 800, that is 200 ÷ 800 × 100 = 25%.

The amount lost and the amount to win back are both 200. Only the divisor differs: the fall is measured against the original 1,000, the recovery against the 800 left after it. 800 is smaller, so the same 200 is a bigger percentage of it. It is the same sum as swapping before and after on Board 1.

The starting value makes no difference. With it set to 250 on Board 3, the value after the fall is 200, the amount to get back is 50, and the rise needed is still 25%. Because only the size of the fall matters, the rule can be written as:

rise needed to get back = 1 ÷ (1 − fall) − 1

For a 20% fall (0.2) that is 1 ÷ 0.8 − 1 = 1.25 − 1 = 0.25, or 25%. 1.25 is the factor that turns 800 into 1,000; subtracting the original 1 leaves 0.25, the increase.

The bigger the fall, the faster the climb back grows

The Draw button on the bottom-right card plots the rise needed as the fall goes from 0% to 90%. On the left the line stays close to the fall itself; towards the right it rises steeply. Once drawn, moving the fall slider updates the figures at the top of the card too.

• A 10% fall — a rise of 11.1% to get back
• A 20% fall — 25%
• A 30% fall — 42.9%
• A 50% fall — 100% (halved, so it has to double)
• A 75% fall — 300% (down to a quarter, so it has to grow fourfold)
• An 80% fall — 400% (down to a fifth, so fivefold)
• A 90% fall — 900% (down to a tenth, so tenfold)

For small falls, the rise needed is only a little bigger than the fall. Past 50%, the gap opens up quickly. A 100% fall leaves 0, and no percentage of 0 is anything but 0, so there is no way back; the formula becomes 1 ÷ 0, which has no answer.

Other numbers

These are Board 2 results with different first and second changes. The starting value stays at 1,000.

Going down and then up by the same percentage always ends below the start. The shortfall is the percentage multiplied by itself: 0.1 × 0.1 = 0.01, or 1%, for 10%; 0.3 × 0.3 = 0.09, or 9%, for 30%; 0.5 × 0.5 = 0.25, or 25%, for 50%. The bigger the percentage, the further from the start the result ends up.

Two 10% rises add up to 21%, not 20%, because the second 10% is taken from 1,100. Two 10% falls add up to 19%, not 20%. A 20% fall followed by a 25% rise lands exactly on 1,000 again, an overall change of 0% — matching the 25% from Board 3.

Board 1 also works for a change in a rate. If a rate goes from 10% to 12%, 10 as before and 12 as after give a percentage change of 20%. The plain difference, 2, is called "2 percentage points". "Up 2 percentage points" and "up 20%" describe the same change, one as a difference and one as a percentage change.

Where this calculation stops working

A percentage change divides by the before value, so it cannot be worked out when the before value is 0. With before set to 0 on Board 1, the percentage-change and ratio cards show that they cannot calculate. A change from 0 to 100 can only be described by the amount, 100.

A negative before value does not give a meaningful percentage either. If a profit goes from −100 to 50, Board 1 shows a percentage change of −150%, although the value went up. For a move from a loss to a profit, the change itself (150 here) is the clearer figure.

Read next — Margin and markup — how they differ and how to calculate each It calculates margin and markup from a cost and a price: the same profit divided by the selling price or by the cost, which gives two different percentages such as 20% and 25%.

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