The question you opened
20 percent more free, or 20 percent off
One shampoo bottle says 20 percent more free. The old bottle beside it is 20 percent off.
Which one gives you more shampoo per dollar?
The discounted bottle
Extra volume divides the price per ounce by 1.2, a cut of about 16.7 percent. Taking 20 percent off the price multiplies it by 0.8, a full fifth.
Price per ounce is the only number that settles it. Start with a 10 dollar bottle holding 10 ounces, which is 1 dollar an ounce. The bigger bottle holds 12 ounces for 10 dollars, so 10 divided by 12 is about 83.3 cents an ounce. The discounted bottle holds 10 ounces for 8 dollars, so 80 cents an ounce. The discount wins by about 3.3 cents on every ounce.
The arithmetic behind it is the difference between dividing and multiplying by the same number. Adding 20 percent more product divides the unit price by 1.2, and 1 divided by 1.2 is 0.833, a cut of 16.7 percent. Taking 20 percent off the price multiplies the unit price by 0.8, a cut of 20 percent. Division by 1 plus p always does less work than multiplication by 1 minus p.
The gap grows with the size of the offer. At 50 percent, extra product cuts the unit price by a third while a discount cuts it by half. At 100 percent, buy one get one free cuts the unit price by half while a 100 percent discount makes it free. This is why bonus packs are the cheaper promotion to run. There is a second reason shops prefer them.
Extra volume moves more product per customer and protects the shelf price, so the number people remember stays high. Discounts train shoppers to wait for the next sale. The habit that survives all of this is short. Convert both offers to price per unit before comparing anything, and treat the size printed on the front as part of the price rather than as a feature. Supermarket shelf labels usually carry the unit price in small type for exactly this reason.
Technique: Compare price per unit