An investment of $200,000 in an instrument that returns an annual rate

CI = \(P*[1+[m]r/2\)]^2[/m]
P = 200,000
r = rate of interest
2 = Compounded semi-annually.

220500 = 200000[1+r/2]^2
220500/200000 = [1+r/2]^2
441/400 = [1+r/2]^2
1+r/2 = 21/20
2+r = 2.1
r = 0.1

r = 10% interest.

C is the answer.

Solution:

We can use the compound interest formula A = P(1 + r/n)^(nt) to solve this problem. Here, A = 220,500, P = 200,000, n = 2, and t = 1, and we need to solve for r:

220,500 = 200,000(1 + r/2)^(2 x 1)

1.1025 = (1 + r/2)^2

√1.1025 = √[(1 + r/2)^2]

1.05 = 1 + r/2

0.05 = r/2

r = 0.10 = 10%

Alternate solution:

We see that the total interest earned for the year is 220,500 - 200,000 = $20,500. Since the interest is compounded semi-annually, we can make an educated guess that the interest earned for the first half of the year is $10,000 and that for the second half of the year is $10,500 (because of the effect of compounding interest).

Since 10,000/200,000 = 0.05 = 5%, we see that the interest rate for half the year is 5%; thus,the annual interest rate is doubled to 10%.

Answer: C
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(1st) we know that the interest taken in 1 year at r% compound interest is going to be > the interest taken in 1 year at the same r% but at SIMPLE Interest

We can find the simple interest earned in 1 year at this r% easily by determining the Interest earned and dividing by the initial principal

(20,500) / (200,000) = (205) / (2,000) = (41) / (400) = (10.25) / (100)

Thus, if the interest were taken at simple interest over 1 year, the interest rate would have been 10.25%

To achieve the same amount, we do not need as high of an interest rate if taken semi-annually at Compound Interest.

At this point, the only answer that makes sense is (C) 10%, but we can check rather quickly to see if this is the correct answer


End first 6 months: (5%) of ($200,000) ——-> would be HALF of $20,000 ——-> $10,000

: new principal is $210, 000

End of year: 5% of ($210,000) is ——-> HALF of $21,000 ——> $10,500

Final balance: $200,000 + $10,000 + $10,500 = $22,500

(C) 10% is the correct answer

Posted from my mobile device

On the GMAT, compounded interest is generally just a little more than simple interest.
If 20% simple interest yields X dollars, then 20% compounded interest will generally yield just a little more than X dollars.

Given an investment of $200,000, the interest earned here (20,500) is just a little more 10% simple interest (20,000).
Implication:
The value of r must be around 10%, implying that only answer choices C and D are viable.
Eliminate A, B, and E.

C: r=10
10% annual interest compounded semi-annually means that 5% interest is yielded every 6 months, as follows:
1st 6 months --> 200,000 + 5% of 200,000 = 200,000 + 10,000 = 210,000
2nd 6 months --> 210,000 + 5% of 210,000 = 210,000 + 10,500 = 220,500
Success!


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