Hans invested 10,000 at an annual interest rate of x percent, compound

Question

Hans invested 10,000 at an annual interest rate of x percent, compounded annually. If the annual interest rate of x percent had been compounded semiannually, how much more interest, in dollars, would he have earned of his 10,000 investment for the first year in terms of x?

(A) $50x$

(B) $100x$

(C) $100x + \frac{x^2}{4}$

(D) $\frac{x^2}{2}$

(E) $\frac{x^2}{4}$

(A)  50x

(B)  100x

(C)  100x + \frac{x^2}{4}

(D)  \frac{x^2}{2}

(E)  \frac{x^2}{4}

Bunuel · Accepted Answer

In a year, $10,000 at an annual interest rate of x percent, compounded annually, earns an interest of  10,000*\frac{x}{100} = 100x .

When x percent is compounded semiannually, it implies applying x/2 percent twice a year. So, $10,000 would earn:

10,000*(1 + \frac{(\frac{x}{2})}{100})^2 - 10,000 =

= 10,000*(1 +\frac{ x}{200})^2 - 10,000 =

= 10,000*(1 + \frac{x}{100} + \frac{x^2}{40,000}) - 10,000 =

= 10,000 + 100x + \frac{x^2}{4} - 10,000 =

= 100x + \frac{x^2}{4} .

Therefore, the difference would be:

100x + \frac{x^2}{4} - 100x = \frac{x^2}{4} .

Answer: E.

KarishmaB · Answer

In case of semi annual compounding, the only extra component is interest on interest earned in the first 6 months. Interest earned in first 6 months will be x/2% of 10,000. Interest earned on this interest in the next 6 months will be x/2% of x/2% of 10,000 which is \frac{x}{200} * \frac{x}{200} * 10,000 = \frac{x^2}{4} Answer (E) Here is a discussion on difference between simple and compound interest and how to handle the differential efficiently: https://youtu.be/2h_luc7buWE

gmatophobia · Answer

One way is to solve this algebraic, and another way is to assume values to solve this question.

Let x = 4%

...Hans invested 10,000 at an annual interest rate of x percent, compounded annually...

Interest Earned =  10000 * \frac{4}{100 }= 400

...If the annual interest rate of x percent had been compounded semiannually...

Amount =  10000 * (1+\frac{4}{200})^2 = 10000 * (\frac{51}{50})^2

=  (51)^2 * 4 = (50+1)^2*4 = 10404

Interest Earned = $ 404

...how much more interest, in dollars, would he have earned of his 10,000 investment for the first year....

Difference in Interest Earned =  404 - 400 = 4

Answer Choice Elimination

(A)   50x   ⇒  50 * 4 
eq 4 ⇒  Eliminate

(B)   100x   ⇒  100 * 4 
eq 4 ⇒  Eliminate

(C)   100x + \frac{x^2}{4}   ⇒  100x + \frac{4^2}{4} 
eq 4 ⇒  Eliminate

(D)   \frac{x^2}{2}   ⇒  \frac{4^2}{2} 
eq 4 ⇒  Eliminate

(E)   \frac{x^2}{4}   ⇒  \frac{4^2}{4} = 4 ⇒  Answer

Option E

Kinshook · Answer

XimeSol

Given: Hans invested 10,000 at an annual interest rate of x percent, compounded annually.

Asked: If the annual interest rate of x percent had been compounded semiannually, how much more interest, in dollars, would he have earned of his 10,000 investment for the first year in terms of x?

Compounded annually: - 
Principal P = 10000
Rate of interest = r = x/100
Period of compounding n = 1
A = P (1 + r)^n = 10000 (1 + x/100)^1 
Interest = A - P = 10000 (1 + x/100 - 1) = 100x

Compounded semiannually: -
Principal P = 10000
Rate of interest = r = x/200
Period of compounding n = 2
A = P (1 + r)^n = 10000 (1 + x/200)^2
Interest = A - P = 10000 (1 + x/200)^2 - 10000 = 10000 (1 + x^2/40000 + x/100 - 1) = 10000 (x/1000 + x^2/40000)

More interest = 10000 (x/100 + x^2/40000) - 100x = x^2/4

IMO E

korteraz · Answer

10,000(1 + x/200)^2 - 10,000(1 + x/100) =
10,000((1 + x/200)^2 - (1 + x/100)) =
10,000(1 + x/100 + (x^2)/(200^2) -1 - x/100) =
10,000(x^2/(4*100^2) =
x^2 / 4

sarthak1701 · Answer

PRO TIP: If you are short on time, you can easily reject the first 3 choices because the difference in compounded and simple interest for 1 year cannot be this high.

Goldenfuture · Answer

let the interest be 10% so 1000 is what we make in 1 year and semi-annual compounding is 500+500+ 25 
Difference is25 = x^2/4
In 1 min

Hans invested 10,000 at an annual interest rate of x percent, compound

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Hans invested 10,000 at an annual interest rate of x percent, compound

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards