tina has $1,000 per year she can invest to save money for her future. which option would allow the highest…

tina has $1,000 per year she can invest to save money for her future. which option would allow the highest growth for tinas investment? tina can start investing the whole amount this year at 5% interest. tina can start investing the whole amount this year at 7% interest. tina can start investing half of the amount two years from now at 5% interest. tina can start investing half of the amount two years from now at 7% interest.

tina has $1,000 per year she can invest to save money for her future. which option would allow the highest growth for tinas investment? tina can start investing the whole amount this year at 5% interest. tina can start investing the whole amount this year at 7% interest. tina can start investing half of the amount two years from now at 5% interest. tina can start investing half of the amount two years from now at 7% interest.

Answer

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1 + r)^n$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (as a decimal), and $n$ is the number of years.

Step2: Analyze Option 1

$P = 1000$, $r=0.05$, assume $n$ years. The amount after $n$ years is $A_1 = 1000(1 + 0.05)^n$.

Step3: Analyze Option 2

$P = 1000$, $r = 0.07$, assume $n$ years. The amount after $n$ years is $A_2=1000(1 + 0.07)^n$. Since $(1 + 0.07)^n>(1 + 0.05)^n$ for $n>0$, $A_2>A_1$.

Step4: Analyze Option 3

$P = 500$, $r = 0.05$, and the number of years of investment is $n-2$ (starting 2 years from now). The amount after $n$ years is $A_3 = 500(1 + 0.05)^{n - 2}$.

Step5: Analyze Option 4

$P = 500$, $r = 0.07$, and the number of years of investment is $n-2$ (starting 2 years from now). The amount after $n$ years is $A_4 = 500(1 + 0.07)^{n - 2}$. Comparing $A_2$ with $A_4$, $A_2=1000(1 + 0.07)^n=2\times500\times(1 + 0.07)^n$ and $A_4 = 500(1 + 0.07)^{n - 2}$. Since $A_2$ has a larger principal amount and more years of compounding, $A_2$ is the largest.

Answer:

Tina can start investing the whole amount this year at 7% interest.