On January 1, 2010, Dave invests 70% of his retirement savings in Anta

MANHATTAN GMAT OFFICIAL SOLUTION:

The difficulty of this “percent change” problem is not so much conceptual as it is “executional”—you want to be able to solve it quickly, easily, and of course accurately. Let’s get to the workout!

You’ve got three investments (at various percent allocations) changing by various other percents over two time periods. The numbers don’t look too ugly, but you might suspect that the result will be hard to compute exactly, because the problem only asks for a range. Thus, you should be ready to switch to estimation at some point.

Pick a smart number for the total retirement savings Dave starts with—say, $10,000. (If you pick $100, you’ll wind up needing to track decimals, so give yourself a couple more zeros to start with.)

Here are the starting values:

L = $7,000
M = $2,000
S = $1,000

Apply the first year’s changes, so that you have these numbers on 1/1/2011:

Newer L = $7,000 + 5% = $7,350
Newer M = $2,000 + 10% = $2,200
Newer S = $1,000 + 15% = $1,150

Now apply the second year’s changes to L and M:

Newest L = $7,350 – 10% = $7,350 – $735 = $6,615
Newest M = $2,200 – 20% = $2,200 – $440 = $1,760

Add these to get $8,375. So the newest S must be $10,000 (the target final total of Dave’s retirement savings) minus $8,375, or $1,625.

The dollar change in S from 1/1/11 to 1/1/12 is $1,625 – $1,150 = $475. So the question is this: what percent change does $475 represent, from a starting point of $1,150? Since $1,150 is a nasty divisor, switch to benchmarks:

10% of $1,150 = $115.

So 20% is just double that, or $230.

And 40% is double that, or $460.

Since $475 is just slightly larger than $460 (but not enough to get you to 50%, which would be $460 + $115), x must be between 40 and 50.

Intuitively, it should make sense that you’d need a much bigger positive percent change in the smallest investment (S) to make up for even a moderate downturn in the larger investments (L and M), so if you were completely stuck for time and needed to guess in a hurry, you should favor C/D/E over A or B.

The correct answer is D.