Interest problem - GMAT Prep Exam pack 2

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]

Pick an easy number. Say x = 20. If her initial $1000 is compounded semi-annually, she'll get 10% every six months. After six months, she'll receive 1000*.10 = 100. Now she's up to $1100. Another six months pass, she she earns another 10%: 1100*.10 = 110. So now she's got 1100+110 = 1210. If she started with 1000, she's earned 210 in interest.

If, however, the money were only compounded annually, she'll earn 20% on her 1000 after 1 year, or 1000*.20 = 200.

The difference is 210 - 200 = 10.

Now plug x = 20 into the answer choices and see what spits out 10. Only D works.

Investment: 1000

Rate: x

Compounding: Semi-Annual

1000 * (1 + (x/2)/100)² = 1000 * ( 1 + x/100 + x²/40,000) = 1000 + 10x + x²/40

Compounding: Annual

1000 * (1 + x/100) = 1000 + 10x

Difference: x²/40

The correct answer is D.

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