gmatt1476 wrote:
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?
(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.
DS29931.01
Solution:
Statement One Only:The account earned exactly $25 in interest at the 2.5% rate.
Since we don’t know the original amount deposited in the account, i.e., the principal, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if the principal is $970, then the interest earned at 2% will be $30, which is greater than the $25 interest earned at the 2.5% rate. However, if the principal is $980, then the interest earned at 2% will be $20, which is less than the $25 interest earned at the 2.5% rate.
Statement Two Only:The account was open for exactly three years.
Since we don’t know the length of time it takes the principal to reach the $1000 benchmark, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if it takes 2 years for the principal to reach $1000, then certainly the interest earned at 2% (during the first 2 years) will be greater than the $25 interest earned at the 2.5% rate (during the last year). However, if it takes 1 year for the principal to reach $1000, then certainly the interest earned at 2% (during the first year) will be less than the $25 interest earned at the 2.5% rate (during the last 2 years).
Statements One and Two Together:
Using statement one, we can calculate the time necessary to earn an interest of $25 from a principal of $1,000 at 2.5% compounded daily. Using statement two, we can subtract the time we calculated in the previous step from three years to determine the amount of time the principal acquired interest at the 2% rate.
Since we know the account was worth $1,000 at the end of this time, we can calculate the principal and the interest earned in this period. Finally, we can compare this interest to $25 and determine whether the account acquired more interest in the 2% period or the 2.5% period.
Recall that in a DS question, we only need to determine whether we have enough information to answer the question; we don’t have to perform the steps and find the actual answer. As we can see from above, statement one and statement two together provide sufficient information to answer the question.
Answer: CI used the same method as you did, and got the "right" answer. But I wasn't convinced because I (and you) am making an assumption here (the highlighted text above). How do we know that the starting balance for the 2nd period is 1000$ exactly? It could be 1000.01 or 1000.1 (you get my point). Because we are told that compounded happens daily and the daily could take it from just below 1000$ to just above 1000$. Of course, we are simplifying here, and in this case, it leads us to the right answer. But will it always? Why is it OK to ignore this rounding error?