Question

If $450 is invested at 6% compounded A (annually), B (quarterly), C (monthly), what is the amount after 7 years? How much interest is earned?

Answer 1

Answer:

Step-by-step explanation:

Here's the gameplan for this.  First of all we need a general formula, then we will define the variables for each.

The general formula for all of these is the same:

$A(t)=P(1+\frac{r}{n})^{nt}$

where A(t) is the amount after the compounding, P is the initial investment, n is the number of compoundings per year, r is the interest rate in decimal form, and t is time in years.  

Then after we find the amount after the compounding, we will subtract the initial amount from that, because the amount at the end of the compounding is greater than the initial amount.  It's greater because it represents the initial amount PLUS the interest earned.  The difference between the initial amount and the amount at the end is the interest earned.

For A:

A(t) = ?

P = 450

n = 1

r = .06

t = 7

$A(t)=450(1+\frac{.06}{1})^{(1)(7)}$

Simplifying gives us

$A(t)=450(1.06)^7$

Raise 1.06 to the 7th power and then multiply in the 450 to get that

A(t) = 676.63 and

I = 676.63 - 450

I = 226.63

For B:

A(t) = ?

P = 450

n = 4 (there are 4 quarters in a year)

r = .06

t = 7

$A(t)=450(1+\frac{.06}{4})^{(4)(7)}$

Simplifying inside the parenthesis and multiplying the exponents together gives us

$A(t)=450(1.015)^{28}$

Raising 1.015 to the 28th power and then multiplying in the 450 gives us that

A(t) = 682.45

I = 682.45 - 450

I = 232.75

For C:

A(t) = ?

P = 450

n = 12 (there are 12 months in a year)

r = .06

t = 7

$A(t)=450(1+\frac{.06}{12})^{(12)(7)}$

Simplifying the parenthesis and the exponents:

$A(t)=450(1+.005)^{84}$

Adding inside the parenthesis and raising to the 84th power and multiplying in 450 gives you that

A(t) = 684.17

I = 684.17 - 450

I = 234.17