Eleni and Sofia both invested $180,000 for one year at the same annual rate of interest, but in Eleni’s case the interest was compounded annually whereas in Sofia’s it was compounded semiannually. If Sofia received $162 more in interest than Eleni, how much interest did they receive in total?
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This one had me staring for a while. The $162 difference felt too small to work with until I realized you need to set up an equation with the two compounding formulas and solve for the rate. Definitely a 700+ level trap.
Same here. I tried plugging in the answer choices at first but it got messy fast. Setting up 180000[(1+r/2)^2 - (1+r)] = 162 is the way to go.
Yeah the key is realizing r is the same for both, that's what makes it solvable.
Wait, is the semiannual rate just r/2 for each period? I always second-guess myself on whether the annual rate gets divided like that.
Yes, nominal annual rate divided by number of compounding periods. So semiannually means r/2 per period, 2 periods.
Got r = 0.06 after expanding (1+r/2)^2 - (1+r) = r^2/4, so 180000 * r^2/4 = 162. Then just compute both interests and add. Took me way longer than it should have.
Nice, that expansion simplifies things a lot. I was trying to brute force it and wasted time.
Clever. I didn't see the r^2/4 shortcut, that's clean.
Solid question, the answer choices are spread out enough that once you find r you can pretty much lock in the total. Respect for the difficulty though, this is no joke under time pressure.
What it tests
Your ability to compute percentage change, percentage of a whole, and to work backwards from a final value.
Common trap
Treating successive percentage changes as additive — a 10% rise then a 10% fall is not back to the start.