6a. The following problem is from the IXL website: \"Leah opened a savings accou

Question

6a. The following problem is from the IXL website: "Leah opened a savings account and deposited $1,000.00 as principal. The account earns 15% interest, compounded continuously. What is the balance after 1 year? Use the formula A Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms ( 2.71828), r is the interest rate expressed as a decimal, and t is the time in years. Round your answer to the nearest cent." Modify this problem so that it better engages students (starting small is fine) and so that students see the connection to the formula for accounts that are compounded n times per year. Then solve the problem yourself.

Explanation / Answer

Let modified problem can be given as , X opened a saving bank account and deposited 1$ as principal and account earns interest 100% compounded after every two month. So what is the balence in the account of X after one year?

let time unit is two month and interest rate is 100% annually so it is 100/6 percent for two month

hence at t=0 balence B=1$

at t=1(that is after two month), B=(1+1/6)$

at t=2, B=[(1+1/6)+(1+1/6)*(1/6)]=(1+1/6)(1+1/6)=(1+1/6)^2$

similarly

at t=3, B=(1+1/6)^3$

so on

at t=6(after the end of one year), B=(1+1/6)^6

so in general when you have a principal P which is compounded n times per year at a rate r yearly then balance after t years is P*[(1+r/n)^(nt)]

so when interest is compounded continously your n is very large because it is compounding at every instent of time that is n is tending to infinity.

now from advanced mathematics we know that as n tends to infinity limit of (1+r/n)^n is e^r

hence when interest is compounded continously balance is give as P*e^(rt) which is the limiting case when interest is compounding n times a year

now in the above problem balance B ia given as

B=(1+1/6)^6=2.52$ , when interest is compounding after two months and

B=pe^(rt)=e=2.72$, when interest is compounding continously

and values are very close.

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