One of the GRE’s favorite tricks involves successive percents. For example:
If a ticket increased in price by 20%, and then increased again by 5%, by what percent did the ticket price increase in total?
Although it may seem counterintuitive, the answer is not 25%.
To understand why, consider a concrete example. Say that the ticket initially cost $100. After increasing by 20%, the ticket price went up to $120 ($20 is 20% of $100).
Here is where it gets tricky. The ticket price goes up again by 5%. However, it increases by 5% of the new price of $120 (not 5% of the original $100 price). Thus, 5% of $120 is 0.05(120), or $6. Therefore, the final price of the ticket is $120 + $6, or $126, not $125.
You can now see that two successive percent increases, the first of 20% and the second of 5%, do not result in a combined 25% increase. In fact, they result in a combined 26% increase (because the ticket price increased from $100 to $126).
Successive percents cannot simply be added together. This holds for successive increases, successive decreases, and for combinations of increases and decreases. If a ticket goes up in price by 30% and then goes down by 10%, the price has not in fact gone up a net of 20%. Likewise, if an index increases by 15% and then falls by 15%, it does not return to its original value! (Try it—you will see that the index is actually down 2.25% overall.)
A great way to solve successive percent problems is to choose real numbers and see what happens. The preceding example used the real value of $100 for the initial price of the ticket, making it easy to see exactly what happened to the ticket price with each increase. Usually, 100 will be the easiest real number to choose for percent problems. This will be explored in greater detail in the next section.
You could also solve by converting to decimals. Increasing a price by 20% is the same as multiplying the price by 1.20.
Increasing the new price by 5% is the same as multiplying that new price by 1.05.
Thus, you can also write the relationship this way:
Original × (1.20) × (1.05) = final price
When you multiply 1.20 by 1.05, you get 1.26, indicating that the price increased by 26% overall.
This approach works well for problems that involve many successive steps (e.g., compound interest, which will be addressed later). However, in the end, it is still usually best to pick $100 for the original price and solve using concrete numbers.
If your stock portfolio increased by 25% and then decreased by 20%, what percent of the original value would your new stock portfolio have?