ketkoag wrote:Deposit $1000 in an account, annual rate x percent (compound quarterly), no other activity for this account, is account balance over $1050 after one year?
1) (1+x/200)^2 > 1.05
2) (1+x/400)^2 >1.025
please explain by solving it completely.. coz i wanna know how to get the unique value of x.
We cannot get a unique value for x from either (or both) inequalities.
However, the question isn't asking for a unique value of x - the question is asking if the final balance is greater than $1050.
As always, when a common formula applies to a DS question, we should jot it down on our scratch paper. For compound interest:
Final value of the investment = P(1 + i)^t
P = amount of original investment
i = interest rate per compound period
t = number of compound periods
For this question, we know that:
Final value = 1000(1 + 1/4(x/100))^4
annual interest is x/100; since we're compounding quarterly, we multiply this by 1/4 to get a relevant interest rate of x/400. So:
FV = 1000 (1 + x/400)^4
and our question is:
is 1000(1 + x/400)^4 > 1050?
or
is (1 + x/400)^4 > 1.05?
(1) (1+x/200)^2 > 1.05
This statement tells us that if we were compounding semi-annually (x/200 and an exponent of 2), our multiplier would be greater than 1.05.
Well, we know that, with a fixed i, the more often we compound the more money we earn (e.g. at a 10% annual rate you earn more compounding twice a year than you do once a year).
So, if compounding semi-annually would put our FV over 1050, then compounding quarterly would put us even further over 1050: sufficient.
(2) (1+x/400)^2 >1.025
This statement tells us that for the first two quarters of the year our multiplier is greater than 1.025 (which is half of our target of 1.05 for the full year).
When we compound, our multiplier grows each time.
Let's illustrate with a simple example:
If you earn 10% interest per month (sign me up for that investment!) and invest $100:
start: 100
1 month: 110 (multiplier .1)
2 months: 121 (multiplier .21)
3 months: 133.10 (multiplier .331)
So, if we're already halfway to our target after 2 compound periods, we will definitely surpass our target: sufficient.
Each of (1) and (2) is sufficient alone: choose (D).
