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At the beginning of year 1

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At the beginning of year 1

by PGMAT » Sat Mar 31, 2012 10:46 am
At the beginning of year 1, an investor puts p dollars into an investment whose value increases at a variable rate of x_n% per year, where n is an integer ranging from 1 to 3 indicating the year. If 85 < x_n < 110 for all n between 1 and 3, inclusive, then at the end of 3 years, the value of the investment must be between:

(A) $p and $2p
(B) $2p and $5p
(C) $5p and $10p
(D) $10p and $25p
(E) $25p and $75p

Does any have a easy solution to this mgmat problem?

Thanks.
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by Bill@VeritasPrep » Sat Mar 31, 2012 3:19 pm
Since we've got fairly large ranges for answer choices, we can ballpark our calculations. I started with 100 and calculated the upper boundary (110% growth per year):

Start: $100
Year 1: 2.1(100) = $210
Year 2: 2.1(210) = (210 * 2 + .1*210) = $441
Year 3: 2.1(441) = (441*2 + .1*441) = $882 + $44 = $926.

Thus, our maximum value is between 9 and 10 times our starting point. The only range of values that fits is C.

This was made easier by the fact that we have ranges that don't overlap. Thus, all we had to do was find one possible value. On a percent problem like this where we have no concrete values, we can pick a starting point that's easy to work with (like 100!) and go from there.
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