I'll be honest, I'm not entirely sure what's going on here, as none of what you've shown, neither your own workings, nor what you claim leads to the "correct answer," is actually correct (although, bizarrely, the actual answer is right, just the formula is wrong
). The
standard formula for compound interest is as follows:
\(\displaystyle A(t)=P \cdot \left( 1+\dfrac{r}{n} \right)^{nt}\)
Where
P is the initial principle,
r is the annual interest rate,
n is the number of compounding periods per year, and
t is the number of years. The formula that's supposedly the "correct answer" is thus:
\(\displaystyle A(t)=10000 \cdot \left( 1+\dfrac{0.03}{10} \right)^{10 \cdot 5}=10000 \cdot (1.003)^{50} \approx 10000 \cdot 1.1616 \approx 11615.73\)
As you can see, the actual numerical answer that formula spits out is not what you wrote. On top of that, the formula is incorrect for the problem you've posted. By comparing the formula to the one above, you'll note that n = 10. That means the formula would only be correct if you were to compound ten times a year. However, the problem text tells you that the interest is "compounded semiannually," or n = 2. Plugging in the correct variables gives us:
\(\displaystyle A(t)=10000 \cdot \left( 1+\dfrac{0.03}{2} \right)^{2 \cdot 5}=10000 \cdot (1.015)^{10} \approx 10000 \cdot 1.1605 \approx 11605.41\)
This fits with what you claim is the "correct answer." Now, getting down to troubleshooting your own work, what you've described leads me to believe that this is something like the formula you attempted to use:
\(\displaystyle A(t)=10000 \cdot \left(\dfrac{1+0.03}{10} \right)^{10 \cdot 5}=10000 \cdot (0.103)^{15} \approx 10000 \cdot 1.558 \cdot 10^{-15} \approx 1.5580 \cdot 10^{-11}\)
This is plagued by so so many problems, I scarcely know where to start. Right from the very beginning, you've mixed up the formula, by putting the 1 inside the numerator of the fraction instead of outside the fraction. Then you've again somehow decided that "compounded semiannually" means n = 10 or compounding 10 times per year. You computed 1 + 0.03 as equaling 1.03, and then you've also somehow computed 10 * 5 as being 15 instead of 50. And, after that... well, I don't even know what you did after that. You somehow got a totally different answer than anything even close to what the math suggests you should have.
I know I probably sound like I'm being really harsh and reaming you a new one, and I don't mean for it to come off that way. It's just seems as if you're just very confused, and you've been trying to follow the formula but you really don't even understand it. I posted a similar sentiment about formulas and rules in another topic that I'll share here too because I've seen it time and time again prove true:
Now, personally I find rules and formulas, in and of themselves, to be fairly unhelpful, because such a rule or formula often very quickly becomes some magical, mystical thing you memorize and it works but you'll be darned if you know how or why.
