document.write( "Question 543041: Your account with a 3.1% interest rate compounded 18 times a year has what effective interest rate? Please explain each step because I have more to work like this. \n" ); document.write( "

Algebra.Com's Answer #354740 by KMST(5328) $\"\"$ $\"About$

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If the interest did not compound, after a year $100 would turn into $100+$3.1=$103.10. They would be multiplying your balance times 0.031 $\"%283.1%2F100%29\"$ and adding that amount just once a year. You would end up with the initial balance times 1.031. They do add to your balance every day, but for the whole year they are always calculating the interest based on the original $100. If you asked for your balance after 1/18 of a year, you would see that the interest was calculated by multiplying your balance times $\"%283.1%2F100%29%2A%281%2F18%29=0.031%2A%281%2F18%29\"$
\n" ); document.write( "After the next 1/18 of a year, the same amount of interest would be added to the balance, always calculating 3.1% on the initial $100.
\n" ); document.write( "If interest compounds 18 times a year, it's as if they paid you the interest every 20 days or so (18 times a year), an you re-invested it. That way, they start paying you interest on the interest that you were paid before, and it is equivalent to a higher not-compounded interest rate. The balance would be the same after just 1/18 of a year, but 20 days later, they would be paying you interest on your new, higher balance, and you would have a tiny bit more money.
\n" ); document.write( "Every time, they multiply the current balance (not just the initial balance) times $\"0.031%2F18\"$ and add it to the balance. And the new balance that results is the old balance times $\"%281%2B0.031%2F18%29\"$ .
\n" ); document.write( "So after 1 year they have done that 18 times, and $100 has converted into
\n" ); document.write( "$ $\"100%2A%281%2B0.031%2F18%29%2A%281%2B0.031%2F18%29%2A%281%2B0.031%2F18%29\"$ ... $\"%281%2B0.031%2F18%29%2A%281%2B0.031%2F18%29\"$ =$ $\"100%2A%281%2B0.031%2F18%29%5E18\"$ =$ $\"100%2A1.031458\"$ =$103.15.
\n" ); document.write( "You get an extra five cents in this case.
\n" ); document.write( "The effective interest rate would be 3.1458%, the decimal part of the factor used to multiply the initial balance (0.031458). \n" ); document.write( "

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