Question

$3100.00 is invested in an

account with a 3.2% interest rate
that is compounded quarterly.

How much money is in the
account at the end of one year?

Answer 1

To determine the final amount in an account with quarterly compounded interest after one year, we use the compound interest formula A = P(1 + r/n)^(nt). With a principal of $3100, an annual rate of 3.2%, quarterly compounding, and a time span of 1 year, the amount is approximately $3199.42.

Step-by-step explanation:

To calculate the amount of money in an account after one year with quarterly compounded interest, we can use the formula for compound interest, which is A = P(1 + r/n)^(nt), where:

A is the amount of money accumulated after n years, including interest.

P is the principal amount (the initial amount of money).

r is the annual interest rate (decimal).

n is the number of times that interest is compounded per year.

t is the time the money is invested for, in years.

In this case:

P = $3100

r = 3.2% or 0.032

n = 4 (since the interest is compounded quarterly)

t = 1 year

Now substituting the values into the formula, we get:

A = 3100(1 + 0.032/4)^(4*1)

First, we calculate 0.032/4 which is 0.008 and then add 1 to it, getting 1.008. Next, we raise this sum to the power of 4 because the interest is compounded quarterly for 1 year and then multiply it by the principal amount:

A = 3100(1.008)^4

Finally, calculating the power and then the product:

A = 3100 * 1.032328 (approximately)

A ≈ $3199.42

So, after one year, the money in the account, with interest compounded quarterly, is approximately $3199.42.

Answer 2

Answer: $3516.26

Step-by-step explanation:
For this problem, let’s assume that the money was put into the account on the first day of the year. With this in mind, we know that there are four quarters in a year. So, let’s set up a series of expressions to find the annually compounded interest.

Quarter 1: $3100 + ($3100 * .032) = $3199.2

Quarter 2: $3199.2 + ($3199.2 * .032) = $3301.5744

Quarter 3: $3301.5744 + ($3301.5744 * .032) = $3407.22478

Quarter 4: $3407.22477 + ($3407.22477 * .032) = $3516.25596

So, the exact amount would be $3516.25596. The rounded amount would be $3516.26.

We can extract from this series of expressions an equation to solve this faster.

Consider our base principle, P, and the interest, I. And let Q be the periods of interest, in our case quarters. We can say the following:

Returns = P (1 + I)^Q

With this we can calculate the same value as before:

P (1 + I)^Q
= 3100 (1+0.032)^4
= 3100 (1.13427612)
= 3516.25597
= $3516.26

Cheers.