Problem Solving

Hi prata,

This question requires knowledge of the formulas for Simple Interest and Compound Interest:

Simple Interest = Principal x (1 + RT)
Compound Interest = Principal x (1 + R)^T

Where R is the yearly interest rate and T is the number of years. When compounding MORE than once a year, you must divide R by the number of periods and multiply T by the same number of periods.

We're asked for the DIFFERENCE in the amount of interest that is generated under two different situations. We can TEST VALUES to answer this question. IF... X = 20...

$1,000 at 20% compounded semi-annually (meaning twice a year) =

$1,000 x (1.1)^2 =
$1,000 x (1.21) = $1210

$1,000 at 10% compounded annually =

$1,000 x (1.2)^1 =
$1200

The difference is $10, so we're looking for an answer that equals 10 when we plug X=20 into it. There's only one answer that matches....

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

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