Marcus deposited $8,000 to open a new savings account that earned …five percent annual interest, compounded semi-annually. If there were no other transactions in the account, what the amount of money in Marcus’s account one year after the account was opened?
(A) $8,200
(B) $8,205
(C) $8,400
(D) $8,405
(E) $8,500

There is indeed a formula to calculate final balance for compounded interest (check here: math-number-theory-percents-91708.html) though there are at least two shorter ways to solve this problem.

Approach #1:
5 percent annual interest compounded semi-annually --> 2.5% in 6 moths.

For the first 6 moths interest was 2.5% of $8,000, so $200;
For the next 6 moths interest was 2.5% of $8,000, plus 2.5% earned on previous interest of $200, so $200+$5=$205;

Total interest for one year was $200+$205=$405, hence balance after one year was $8,000+ $405=$8,405.

Answer: D.

Approach #2:
If the interest were compounded annually instead of semi-annually then in one year the interest would be 5% of $8,000, so $400. Now, since the interest is compounded semi-annually then there would be interest earned on interest (very small amount) thus the actual interest should be a little bit more than $400, only answer choice D fits.

Answer: D.

Hope it's clear.
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This is a much simpler problem. I have never used exponents in calculating interest...

Compounding period is half a year so R = 0.05 * 1/2 = 0.025

8000 * (1+ 0.025) = 8200 = total at 1/2 year

8200 * (1+ 0.025) = 8405 = total at year-end

Shortcut = figure total at year end for annually compounding interest... 8000 * (1 + .05) = 8400. More frequent compounding results in ever-so-slightly increased totals so 8405 fits.

8000 ( 1 + 5/200 )^(2*1)

Bumping for review and further discussion.
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Yes: 5 percent annual interest compounded semi-annually --> 2.5% in 6 moths.
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The basic formula for this question is compounded interest-

Payment(1 + r)^nt where

r= rate of interest- that is...how much percent?
n= number of times per year - this is key- this is what sets compounded interest apart from simple interest
t= year- or really this variable is just equal to 1- well for the sake of this formula on the GMAT- advanced finance is different

But the theory behind this formula is actually quite simple within the context of other GMAT percentage formulas...here's why. When we want to express a percentage increase such as a 20% increase of $100.00 on a stock then we would actually multiply the payment amount $100.00 by 1.2- if we wanted to express a 100% percent increase of that $100.00 we would multiply by "2" and if we wanted to express a 120 increase of $100.00 we would multiply by "2.2" - NOTE: a 150% increase is not the same as 150% of a number. The latter is warning is basically the idea of finance- you have a payment which is represented by P in the both the compounded and simple interested formula. The parentheses is quite simple if you understand the previously mentioned technique for percentage increase- the parentheses 1 + "r" or rate of interest is basically just that... we had 1 just like we did with a 20 percent increase of 1 ($100.00 x (1.2). The exponent in the formula simply expresses...well how many times are we doing that? So if you put down a $100.00 dollar stock on Tesla and it increases 20 percent every year then the amount we would have in two[u][/u] years is just $100(1.20)^2 which is really just $100(1 +20/100)^2 or in even in another form $100(1 +.20)^2

Anyways so what we have here is

$8,000.00(1 + .025)^2 ... this is the same as $8,000.00(1 + 25/100)^2 or even $8,000(1.025)^2

Thus
"D"

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