Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five

BTGmoderatorDC wrote: ↑

Mon Apr 20, 2020 6:03 pm

Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account?

A. 1.2%
B. 4.5%
C. 10%
D. 18%
E. 25.2%

OA B

Source: Magoosh

This question involves ugli numbers. If we go with the traditional method for solving, we would need a calculator. However, in the GMAT, a calculator is not allowed.

Anyway, let's understand how it would have been solved, applying the traditional method.

The formula for CI is as follows.

A = P(1 + r/100)^n;

A = Amount after 5 years = 76,662;
P = Principal = 61,293
r = Quarterly rate of interest. Thus, the answer would be 4r, the annual rate of interest
n = No. of periods = 4*5 = 20

76,662 = 61,293(1 + r/100)^20

(76,662/61,293)^(1/20) = 1 + r/100;

From this point, we are helpless.

Let's do this problem with another approach.

So, Diana earned an interest of 76,662 – 61,293 = ~15,000 earned in 5 years

=> 15,000/5 = 3,000 in a year

Simple rate of interest per annum = 3,000/60,000 = ~5%

However, 5% must be an incorrect answer as this is a CI question and the rate of interest must be less than 5%. The closest is 4.5%. Option A 1.2% is far too less.

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorDC wrote: ↑

Mon Apr 20, 2020 6:03 pm

Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account?

A. 1.2%
B. 4.5%
C. 10%
D. 18%
E. 25.2%


OA B

Source: Magoosh

Solution:

We can use the compound interest formula, where A is the final value of the investment, P is the original principal, r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the number of years:

A = P(1 + r/n)^(nt)

Here, we are given A = 76,662.25, P = 61,293, n = 4 and t = 5. We need to find r:

76,662.25 = 61,293(1 + r/4)^(4*5)

1.25075 = (1 + r/4)^20

^20√1.25075 = 1 + r/4

1.01125 = 1 + r/4

0.01125 = r/4

r = 0.045 = 4.5%

Alternate Solution:

The answer choices vary greatly, which is an indication that the interest rate can be approximated without using a calculator.

We see that 76,600 - 61,300 = $15,300 in interest was earned during the 5-year period. This is an average of a bit more than $3,000 per year in interest.


As a percentage of the original amount invested, we see that the yield is about 3,000/60,000, or 1/20, which is 5% per year. Of the answer choices, this is very close to 4.5% per year, and no other answer choice is a viable candidate.

Answer: B