To find the required annual interest rate to the nearest tenth of a percent for $1400 to grow to $1700, when interest is compounded quarterly for 7 years, we can carry out these steps:
1. We establish our initial balance, which is $1400.
2. We establish our final balance, which is $1700.
3. We establish the number of compounding periods per year, which, given that it's compounded quarterly, is 4.
4. We establish the number of years the money is invested for, which is 7 years.
5. We can then use the formula to calculate the future value of an investment:
FV = P*(1 + r/n)^(nt)
where:
- FV is the future value of the investment
- P is the principal amount (initial balance)
- r is the annual interest rate
- n is the number of compounding periods per year
- t is the number of years the money is invested for
6. We can rearrange this formula to solve for r:
r = ((FV/P)^(1/(nt)) - 1) * n
7. Substituting the given values into the formula, we get:
r = ((1700/1400)^(1/(7*4)) - 1) * 4
8. Now, we can solve the equation. Doing so gives us r, which then needs to be converted to a percentage (by multiplying by 100) to find the annual interest rate.
9. The final step is to round the annual interest rate to the nearest tenth.
Hence, using these steps and performing the calculations, we find that the required annual interest rate to the nearest tenth of a percent for $1400 to grow to $1700 if interest is compounded quarterly for 7 years is approximately 2.8%.