Percent of a Percent

Mental Math · Axiom Academy

Understanding successive percentage changes and why "20% off, then 10% off" is NOT the same as "30% off"—exploring the mathematics of compound discounts. It's tempting to think that successive discounts simply add together. After all, 20% + 10% = 30%, right? But this logic ignores a crucial fact: the second discount applies to the already-reduced price, not the original price. 2. Applying the First Discount Let's work with a concrete example: a 100 item. When we apply a 20% discount, we're really multiplying by 0.80 (which represents keeping 80% of the price). New price after first discount: 80 3. Applying the Second Discount Here's where the compounding happens. The 10% off doesn't apply to the original 100—it applies to the already-discounted 80. So we're calculating 10% of 80, not 10% of 100. Price after first discount: 80 Final price after both discounts: 72 Instead of adding 20% + 10%, we multiply (1 - 0.20) × (1 - 0.10). This gives us the compound effect. Let's see this visually and algebraically. First discount: keep 80% → multiply by 0.80 Second discount: keep 90% → multiply by 0.90 Combined effect: 0.80 × 0.90 = 0.72 Let's directly compare the incorrect and correct methods to see the difference clearly. 20% + 10% = 30% off 100 × 0.70 = 70 Too much savings! 0.80 × 0.90 = 0.72 100 × 0.72 = 72 Actually 28% off 6. General Formula for Any Successive Changes

This is the written version of the interactive lesson above. See the full Mental Math course.