# Shortcuts for annual percentage change calculations?

Given the number of apples was 2,100,000 3 years ago, is 500,000 this year and has declined at a constant rate during the stated period. Which is the closest estimate of the annual percentage drop between 3 years ago and this year?

The choices are: (1) 20% (2) 25% (3) 40% (4) 55%

What would be the quickest way to solve it in ~3 minutes without a calculator? I would be very thankful for any help!

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## 1 Answer

Think of it this way, you started off with x amount of apples and in 3 years you are left with about .25x, which is 25% of the apples you had 3 years ago. Once you make this approximation, you can start playing with a simple example. Say you start with 1. If you reduce this quantity by 40% you get .6. If you reduce it by 40% again, you get about .36, and if you reduce it yet again by 40% you get about .25, which is 25% of what you started with. These kinds of problems are meant to get you to approximate quantities, and even though working with large numbers is a little tough, once you realize you can work with smaller numbers, its just about playing with the problem with the given choices.

On the exam, you might have thought this through like this. suppose the quantity declines by 50% yearly(you choose 50% because it is easy to work with). then $1 \rightarrow .5 \rightarrow .25 \rightarrow .125$ so a 50% decline is too much. This eliminates choice (4).

Lets try 25%. $1 \rightarrow .75 \rightarrow$ (something greater than .5) $\rightarrow$ (something greater than .375) This tells us that 25% is too low. So the only answer must be 40%

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Thank you very much! This will definitely help. – Vadym Jun 20 '11 at 17:16