Reverse Percentages

Sometimes you know an amount after a percentage change and want the original amount it came from. The trick is to divide by the multiplier — not to subtract the percentage.

If a price already includes 20% tax, the original price was the price divided by 1.20:

\text{original} = \text{price} \div 1.20

And if something is reduced by 30% to \pounds x, then only 70% remains, so the original was

\text{original} = x \div 0.70

It is tempting to add back 30% of the new price — but 30% of the smaller number is the wrong size. Always undo a multiply with a divide.

The rule

To find the original amount before a percentage change:

identify the multiplier — a 20% rise gives 1.20, a 25% fall gives 0.75;
the amount you know is \text{original} \times \text{multiplier};
so reverse it by dividing: \text{original} = \text{known} \div \text{multiplier}.

The common mistake is to add or subtract that percentage of the new amount — that uses the wrong base. Always divide.

See it on a bar

We pay \pounds 40 after a 20% discount, so that is 80% of the original. Find what one 20% slice is worth, then build back up to 100%.

In one step: 40 \div 0.80 = 50. The original price was \pounds 50.