prata wrote:How much more interest will Maria receive if she invests $1000 for one year at x percent annual interest, compounded semiannually, than if she invests $1000 for one year at x percent annual interest, compounded annually?
5x
10x
x^2/20
x^2/ 40
10x + x^2/10
[spoiler]OA: D[/spoiler]
We use the simple interest formula I = (P)(r)(t), where P is the principal, r is the interest rate, and t is the time. Note that using this simple interest formula for each of the two 6-month periods yields the same answer as using the compound interest formula for the entire year.
In the first scenario, Maria's principal of $1000 will earn interest at x percent, compounded semi-annually. This means that at the end of the first six months, her simple interest will be:
I = (1000)(x/100)(½) = 5x (We use t = ½ because the interest was earned for six months, or ½ year).
At the end of the first six months, her new principal will be 1000 + 5x and thus for the next six-month period, she earns interest on this new principal, so she earns:
I = (1000 + 5x)(x/100)(½) = (1000)(x/100)(½) + (5x)(x/100)(½) = 5x + (x^2)/40
The total amount of interest she earns for the entire year is
5x + 5x + (x^2)/40 = 10x + (x^2)/40
In the second scenario, Maria's principal earns interest only at the end of the year. So she earns:
I = (1000)(x/100)(1) = 10x
To answer the question, she earns 10x + (x^2)/40 - 10x = (x^2)/40 more interest in the first scenario.
Answer:
D
