Question: A person invests an equal amount of money in two investment schemes for two years. In the first investment scheme, he earns an interest at 10% p.a. simple interest and in the second investment scheme he earns an interest at 10% p.a. compounded annually. What is the difference in the interest earned under the two investment schemes at the end of the second year?

(1) The amount the person invests in each scheme is $10,000.
(2) He earns an interest of $1000 in the 1 scheme at the end of 1 year.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• A person invests an equal amount of money in two investment schemes for two years.
• In the first investment scheme, he earns interest at 10% p.a. simple interest
• In the second investment scheme, he earns interest at 10% p.a. compounded annually.

Find out:

• The difference in the interest earned under the two investment schemes at the end of the second year.

Statement 1: The amount the person invests in each scheme is $10,000.
As per the formula of Simple Interest, we know that SI (for 2 years) = P*R*T/100
Therefore, in this case, SI = 10,000 * 10 * 2 /100 = 2000
(For 2 years)
As per the formula of Compound Interest, we know that CI = P(1+r/100)^t -P
Therefore, in this case, CI = 10,000 (1 + 0.1)^2 - 10,000
=> CI = 10,000 (1.1)^2 - 10,000
=> CI = 10,000 (1.21) -10,000
=> CI = 12100 -10,000
=> CI = 2100
Therefore, CI - SI = 2100-2000 = 100
Hence, statement one alone is sufficient.
Statement 2: He earns an interest of $1000 in the 1 scheme at the end of 1 year.
As per the formula of Simple Interest, we know that SI (for 1 year) = PRT/100
Therefore, in this case, SI = 1000 = P*10*1/100
P =10,000
SI (for 2 years) = P*R*T/100
10,000 * 10 * 2/100 = 2000
(For 2 years)
As per the formula of Compound Interest, we know that CI = P(1+r/100)^t -P
=> CI = 10,000 (1 + 0.1)^2 - 10,000
=> CI = 10,000 (1.1)^2 - 10,000
=> CI = 10,000(1.21)-10,000
=> CI = 12100-10,000
=> CI = 2100
Therefore, CI-SI = 2100-2000 = 100
Hence, statement two alone is sufficient.
Therefore, EACH statement ALONE is sufficient.

Approach Solution 2:
The problem statement informs that:
Given:

• A person invests an equal amount of money in two investment schemes for two years.
• In the first investment scheme, he earns interest at 10% p.a. simple interest
• In the second investment scheme, he earns interest at 10% p.a. compounded annually.

Find out:

• The difference in the interest earned under the two investment schemes at the end of the second year.

As per the conditions of the question:
r = 10% for both schemes, t = 2
Let the amount invested = P
From the formula of SI and CI, we can derive the question as follows:
SI = (P∗10∗2)/100 = 20P/100
CI = P(1 + 10/100)^2 - P
= 121P/100 - P
= 21P/100
Difference in Interest = 21P/100 - 20P/100 = P/100
Therefore, if the value of P is deduced, then the statement will be sufficient.
Statement 1 alone: The amount the person invests in each scheme is $10,000.
This implies that the amount invested i.e P = 10,000.
Hence, statement one alone is SUFFICIENT.
Statement 2 alone: He earns an interest of $1000 in the 1 scheme at the end of 1 year.
From this statement, we can derive two cases:
Case 1: Interest of $1000 in the First scheme
SI = 1000 = (P∗10∗1)/100
P = 10000
Case 2: Interest of $1000 in any Second scheme
CI = 1000 = P (1 + 10/100) - P = P/10
P = 10000
Therefore, in both cases, the value of P is the same.
Hence, statement two alone is SUFFICIENT.
Therefore, EACH statement ALONE is sufficient.

Approach Solution 3:
The problem statement implies that:
Given:

• A person invests an equal amount of money in two investment schemes for two years.
• In the first investment scheme, he earns interest at 10% p.a. simple interest
• In the second investment scheme, he earns interest at 10% p.a. compounded annually.

Find out:

• The difference in the interest earned under the two investment schemes at the end of the second year.

In this case, CI earned will be greater than SI. This is because, in C.I., the first-year interest amount will start to gain interest in the second year-end evaluation.
First-year Interest = P * r/100 * 1= P*r/100
The interest earned from the First year interest at the end of the second year
= r% of (First-year interest
= r% of ( P * r/100) = \(P * {r^2 \over 100^2}\)
Therefore, it can be inferred that Difference between CI and SI for 2 years = \(P * {r^2 \over 100^2}\)
Here, r is given as 10%
Therefore, the difference = \(P * {10^2 \over 100^2} = {P\over100}\).
Therefore, to answer the question stem, we need to find the value of P.
Statement 1: The amount the person invests in each scheme is $10,000.
The value of P is given in Statement 1.
Hence, statement one alone is clearly sufficient.
Statement 2: He earns an interest of $1000 in the 1 scheme at the end of 1 year.
It is required to remember that CI and SI at the end of 1 year will be the same.
They will change only after the end of the 2nd year onwards.
Therefore, we get:
=> 1000 = P * 10/100 * 1
=> P = 10,000
Hence, statement two alone is also sufficient.
Therefore, EACH statement ALONE is sufficient.

“A person invests an equal amount of money in two investment schemes”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. The GMAT Quant section comprises a total of 31 questions. GMAT Data Sufficiency questions include a problem statement that is followed by two factual statements. GMAT data sufficiency contains 15 questions which are two-fifths of the entire 31 GMAT quant questions.

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