An investor allocates a total principal of \10,000 between two investment options, Account… — GRE General Test

An investor allocates a total principal of $\$10,000$ between two investment options, Account X and Account Y. Account X earns simple interest at an annual rate of $8\%$ . Account Y earns interest at an annual rate of $8\%$ compounded semi-annually. Neither account receives additional deposits or withdrawals for a period of $2$ years. Which of the following statements MUST be true?

Select all such statements.

If the total principal is divided equally between Account X and Account Y, the total interest earned after $2$ years is greater than $\$1,640$ .Cevap
Allocating a greater portion of the total principal to Account Y strictly increases the total interest earned after $2$ years.Cevap
C
If the entire $\$10,000$ principal is placed in Account X, the total interest earned after $2$ years is $\$1,664$ .

Cevap

The correct statements are the statement regarding equal division yielding total interest greater than $1,640, and the statement that allocating a greater portion of principal to Account Y increases the total interest earned.

The statement regarding equal principal division is true because

5,000 placed in Account X yields

800 in simple interest and

5,000 placed in Account Y yields

849.29 under semi-annual compounding, giving a sum of

1,649.29 which exceeds

1,640. The statement regarding shifting allocation to Account Y is true because Account Y's effective two-year yield of 16.99% is higher than Account X's two-year simple yield of 16.00%.

Adım Adım Çözüm

Calculate the effective 2-year return for Account X under simple interest.

Simple interest over 2 years at an annual rate of 8% is

2 \times 8\% = 16\%

. Total return factor is

0.16

Simple interest accumulates linearly as

I = P \cdot r \cdot t

Calculate the effective 2-year return for Account Y under semi-annual compound interest.

Semi-annual rate is

\frac{8\%}{2} = 4\% = 0.04

. The number of compounding periods in 2 years is

2 \times 2 = 4

. Total compound multiplier is

(1.04)^4 = 1.16985856

. Thus, the total interest earned is

16.985856\%

of the principal.

Compound interest uses the period rate raised to the power of total compounding periods:

(1 + r_{period})^n

Evaluate the statement for equal division of $10,000.

With

5,000 in Account X: interest =

5,000 \times 0.16 =

800. With

5,000 in Account Y: interest =

5,000 \times 0.16985856 \approx

849.29. Total interest =

800 +

849.29 =

1,649.29. Since

1,649.29 > $1,640, this statement is TRUE.

Summing the interest from both accounts confirms the total exceeds $1,640.

Evaluate the statement comparing allocation shifts.

Since Account Y earns approximately

16.99\%

over 2 years and Account X earns

16.00\%

, the yield of Y exceeds X by approximately

0.99\%

. Thus, increasing the allocation to Account Y increases overall interest earned. This statement is TRUE.

Reallocating principal to an option with a higher effective yield strictly increases overall earnings.

Evaluate the statement for placing all funds in Account X.

If all

10,000 is in Account X, interest =

10,000 \times 0.16 =

1,600. The claim of

1,664 is FALSE.

1,664 results from calculating annual compound interest (

10,000 \times (1.08^2 - 1)$), which does not apply to simple interest.

Anahtar Kavram

Simple Interest vs. Compound Interest

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