Keep the whole, add the extra
A bus fare goes up by 20%. The old fare is the whole, 100%. After the rise you pay all of the old fare and 20% of it on top, so the new fare is 100% + 20% = 120% of the old one.
The top bar is the whole old amount, 100%. The bar under it is the same whole cut into tenths, and the 2 colored tenths are the 20% added on.
On a number
Increase 40 by 20%. 10% of 40 is 4, so 20% of 40 is 2 × 4 = 8. Keep the whole 40 and add the 8 on: 40 + 8 = 48.
Each part is 10% of 40, which is 4. The 40 is 10 parts; with 2 more parts added it is 12 parts, 48.
One multiplication
48 is 120% of 40. As a fraction, 120% is , and . So 120% of 40 is 40 × 1.2 = 48, found in one step without working out the 8 first. The number 1.2 is called the multiplier.
Every rise has a multiplier: 100% plus the rise, written as a decimal. A 10% rise is × 1.1, a 50% rise is × 1.5, and a 5% rise is , which is × 1.05.
0%: × 1, the amount is unchanged, 40 × 1 = 40
Set +20% and read the one multiplier: 40 × ?
The dial opens at 0%, where the multiplier is × 1 and 40 stays 40. Turn it up to +20%: the multiplier becomes × 1.2, and the lower bar grows to 48.
Falls work the same way
A fall of 20% keeps 100% − 20% = 80% of the amount, and 80% is . So a 20% fall is one multiplication by 0.8: 40 × 0.8 = 32.
A rise keeps the whole and adds, so its multiplier is more than 1. A fall keeps only part of the whole, so its multiplier is less than 1.
After a 20% fall, 8 of the 10 tenths are left: 80%, which is × 0.8.
Three slips
Increasing 40 by 20% is not 40 + 20 = 60. The rise is 20% of 40, which is 8, not the number 20 itself.
The answer is not 8 either. 8 is only the extra, and an increase keeps the whole 40 as well.
And a 20% fall is not × 0.2. Multiplying by 0.2 keeps only the 20% that fell away; the amount keeps the other 80%, so the multiplier is 0.8.