# Compound Interest: $50000 Invested for 3 Years at 7.75% • MHB • kaye In summary: That is the same as 50000(1+ 0.019375)^12 which is the same as 50000(1.019375)^12 because 0.019375 is very small. kaye An alumnus of a local high school donated$50000 to the school. The amount was invested for 3 years at 7.75%, compounded quarterly. The school has agreed to use only the interest earned on the investment to buy sports equipment. How much money will be available for sports equipment at the end of the investments term?

$A = A_0 \left(1 + \dfrac{r}{n}\right)^{nt}$

$A$ = account balance after $t$ years
$A_0$ = initial account balance
$r$ = annual interest rate as a decimal
$n$ = number of compounding periods per year

I tend to prefer not blindly substituting into a formula.

If the interest rate is 7.75% p.a. compounded quarterly, then the quarterly rate is 1.9375%.

So every quarter, you increase by 1.9375%, thus you end up with 101.9375%.

Thus the multiplier is 1.019375

If you're investing for 3 years, then that's 12 quarters.

Thus $\displaystyle A = 50\,000 \times \left(1.019\,375 \right)^{12}$.

Even less "blindly": The interest rate is 7.75% annually so 7.75/4= 1.9375% per quarter as Prove It said. That means that after 3 months (one quarter of a year) interest of 50000(0.019375) will be added to the 50000 yielding 50000+ 50000(0.019375)= 50000(1.019375). Since the interest is compounded (neither principle nor interest is collected) quarterly, the next quarter the 1.9375% interest will be calculated on that new 50000(1.019375) so the interest at the end of the second quarter will be (50000(1.019375))(0.019375) and that will be added to the 50000(1.019375) so will be (50000(1.019375))(0.019375)+ 50000(1.019375)= 50000(1.019375)(0.019375+ 1)= 50000(1.0199375)^2.

The third quarter you do the same thing except this time you start with 50000(1.01375)^2 so the interest will be 50000(1.019375)^2(0.019375) added to 50000(1.019375)^2 to get 50000(1.019375)^2(0.019375)+ 50000(1.019375)^2= 50000(1.019375)^2(0.019375+ 1)= 50000(1.019375)^3.

In 3 years there are 3(4)= 12 quarters so you repeat this 12 times getting Prove It's 50000(1.019375)^12.

## What is compound interest?

Compound interest is the interest earned on both the principal amount and any accumulated interest from previous periods. It is calculated by adding the interest earned to the principal amount, and then calculating the interest on the new total.

## How is compound interest calculated?

Compound interest is calculated using the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

## What is the difference between simple and compound interest?

The main difference between simple and compound interest is that simple interest only calculates interest on the principal amount, while compound interest includes interest earned on both the principal and accumulated interest. This means that compound interest will result in a higher final amount compared to simple interest.

## How does the interest rate affect compound interest?

The interest rate has a significant impact on compound interest. A higher interest rate will result in a larger final amount, while a lower interest rate will result in a smaller final amount. This is because the interest is being compounded on a higher or lower amount each period.

## What factors can affect the final amount in a compound interest calculation?

The final amount in a compound interest calculation can be affected by the principal amount, interest rate, compounding period, and number of years. Additionally, any additional deposits or withdrawals made during the compounding period can also impact the final amount.

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