Question

PLEASE HELP!!!

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.

Answer 1

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.

Answer 2

No, it is not the same because in option 1, the interest is calculated based on two different principal amounts and then added together. The principal amounts will be different every year because of the varying interest rates. Since one of the interest rates in option 1 is 7%, the principal will grow at a faster rate because the interest rate is applied to a greater and greater principal amount over time. In option 2, the interest is being calculated only on one principal