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The last of the GRE practice questions I had trouble with involve interest and functions from introductory Algebra.

  1. Pat invested a total of 3,000 dollars. Part of the money was invested in a money market account that paid 10 percent simple annual interest, and the remainder of the money was invested in a fund that paid 8 percent simple annual interest. If the interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10 percent and how much at 8 percent?

    We are expected to be able to solve this in about 1 minute or less. I tried plugging in the interest formulas and still couldn't solve it, even after 20m. Is there some simple formula I'm missing or heuristic I can follow that makes this doable in the allotted time?

  2. I'm also having trouble finding the x-intercepts for these two functions:
    (a) f(x) = sqrt(x + 2)
    (b) f(x) = x + |x|

That's all! Thanks.

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up vote 2 down vote accepted

(1) If $x$ dollars were invested at $10$%, then $3000-x$ were invested at $8$%, and the total interest after one year was $0.1x+0.08(3000-x)=0.02x+240$. Since this is known to be $256$, and $256-240=16$, we must have $0.02x=16$ and $x=16/0.02=800$ dollars invested at $10$%; the remaining $2200$ dollars were invested at $8$%.

(2) In each case you need to solve $f(x)=0$. For (a) that’s $\sqrt{x+2}=0$, and squaring both sides should lead you to the solution almost immediately. For (b) it’s $x+|x|=0$, or $|x|=-x$. Recall the definition of $|x|$: $|x|=x$ when $x\ge 0$, and $|x|=-x$ when $x<0$. Since $|x|=-x$ is the equation that you need to solve, you’re done (and you have a whole ray of $x$-intercepts).

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