is investing $4,000 at an interest rate of 5% (compounded annually) and $4,000 at an interest rate of 7% (compounded annually) always, sometimes, or never the same as investing $8,000 (the total of the two principals) at an interest rate of 6% (compounded annually)? why or why not? does it matter how long you leave it in the account?

explain using words and examples, and justify your answer.

Step-by-step explanation:

No, they are not always the same - the only time that they will be the same is after the first year:

$4000*1.05 + $4000*1.07 = $8480 = $8000*1.06

From there on, they will diverge. For example after the second year:

$4000*1.05^2 + $4000*1.07^2 = $8989.6

$8000*1.06^2 = $8988.8

After the third year:

$4000*1.05^3 + $4000*1.07^3 = $9530.672

$8000*1.06^3 = $9528.128

After the nth year:

The first option gives $4000*(1.05^n + 1.07^n)

The second option gives $8000*(1.06^n)

= $4000*(2)*(1.06^n)

= $4000*(1.06^n + 1.06^n)

Because 1.07^n - 1.06^n > 1.06^n - 1.05^n for n>1, the first option will be a better investment.

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