An investor opened a money market account with a single deposit of $60

Interest earned for the first three quarters = $125 + $130 + $145 = $400

GOAL: The total interest earned on the account for the year to be 10 percent of the initial deposit?
10% of $6000 = $600

$600 - $400 = $200
So, we want the investor to earn $200 of interest during the 4th quarter.

After the first 3 quarters, the amount in the account = $6000 + $400 = $6400

What percent of $6400 equals $200?
To find out if we need to write $200/$6400 as a percent.

200/6400 = 2/64 = 1/31 = 3/93 ≈ 3.1%

So, to earn $200 interest in the 4th quarter, we need a QUARTERLY interest rate of about ≈ 3.1%

Since the question asks for the ANNUAL interest rate, we must multiply 3.1% by 4 to get 12.4%

Answer: E

Amount = principal + interest...(formula)

10 percent of the initial deposit 6000 (principal) is 600. Therefore, the 4th quarter's amount must be 6000 + 600 = 6600.

It's given that the interest earned in the first 3 quarters = 125 + 130 + 145 = 400. Therefore, the amount at the end of the quarter must be 6000 + 400 = 6400. Please note that this amount 6400 will be the principal for the 4th quarter.
Interest needed = 6600 - 6400 = 200.
(200 / 6400) × 100 = 3.125%
Therefore, A is the best answer.

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Given: An investor opened a money market account with a single deposit of $6000 on Dec. 31, 2001. The interest earned on the account was calculated and reinvested quarterly. The compound interest for the first 3 quarters of 2002 was $125, $130, and $145, respectively.

Asked: If the investor made no deposits or withdrawals during the year, approximately what annual rate of interest must the account earn for the 4th quarter in order for the total interest earned on the account for the year to be 10 percent of the initial deposit?

First quarter: -
Principle = $6000
Interest = $125
Interest rate = $125/6000 * 12/3 * 100% = 25/3 % = 6 1/3%

Second quarter: -
Principle = $6125
Interest = $130
Interest rate = $130/6125 * 12/3 * 100% =

Third quarter: -
Principle = $6255
Interest = $145
Interest rate = $145/6255 * 12/3 * 100% =

Fourth quarter: -
Principle = $6400
Interest = $6000*10% - $400 = $200
Interest rate = $200/6400 * 12/3 * 100% = 12.5%

IMO E

Here is a detailed solution to understand the basics - 

1st Quarter

• P = 6000
• CI for the 1st quarter = $125
• CI formula = P(1+r/400)-P
• CI = 6000(1+r/400)-6000
• So r = 8.33% for the 1st Quarter

2nd Quarter

• P = 6125 (as we reinvested the $125 CI from the 1st quarter)
• CI for the 2nd quarter = $130
• CI formula = P(1+r/400)-P
• CI = 6125(1+r/400)-6125
• So r = 8.48% for the 2nd  Quarter

3rd Quarter

• P = 6125+130 = 6255 (as we reinvested $130 again)
• CI for the 3rd quarter = $145
• CI formula = P(1+r/400)-P
• CI = 62551+r/400)-6255
• So r = 9.27% for the 3rd  Quarter

4th Quarter

• P = 6255+145 = 6400
• CI for the 4th quarter = 10% of overall 12 months (600) - 400 (for 9 months) = 200
• CI formula = P(1+r/400)-P
• CI = 6400(1+r/400)-6400
• So r = 12.5% for the 4th Quarter

A = 12.5%

"approximately what annual rate of interest must the account earn for the 4th quarter"

Krunaal Had the question STEM instead been "approximately what rate of interest must the account earn for the 4th quarter", we wouldn't multiply the final rate of interest (3.125) by 4 right?

Responding to a pm:

We need the total interest earned to be 10%. Since initial deposit is $6000, we need interest to be $600.

In 3 quarters, interest of $120 + 130 + 145 = $400 has been earned. So we need to earn $200 from the last quarter.

The total amount for the last quarter is 6000 + 400 = $6400

Rate of interest = \(\frac{200}{6400} * 100 = 3.1\) approx

This is the rate of interest for the quarter. So annual rate will simply be 4 * 3.1 = 12.5%

Answer (E)

Here is a discussion on simple and compound interest: https://anaprep.com/arithmetic-solving- ... -interest/