Kenny wants to invest $250 every three months at 5.2%/a compounded
quarterly. He would like to have at least $6500 at the end of his investment.How long will he need to make regular payments?
To determine how long Kenny will need to make regular payments, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($6500)
P = the amount Kenny invests each quarter ($250)
r = the annual interest rate (5.2% or 0.052)
n = the number of times interest is compounded per year (quarterly, so n = 4)
t = the number of years
Now we can rearrange the formula to solve for t:
(t) = ln(A/P) / (n * ln(1 + r/n))
Let's calculate it step by step:
1. Convert the annual interest rate to a quarterly rate:
For quarterly compounding, divide the annual interest rate by the number of compounding periods per year:
Quarterly interest rate (r) = 0.052 / 4 = 0.013
2. Substitute the given values into the formula:
(t) = ln(6500/250) / (4 * ln(1 + 0.013))
3. Calculate the natural logarithm (ln):
(t) = ln(26) / (4 * ln(1.013))
4. Use a calculator to find the natural logarithm (ln) of 26 and 1.013:
(t) = 3.258 / (4 * 0.012)
5. Multiply the result by 4 * 0.012:
(t) = 3.258 / 0.048
6. Divide the result by 0.048:
(t) ≈ 67.87
Therefore, Kenny will need to make regular payments for approximately 67.87 quarters or about 16.97 years to accumulate at least $6500. Keeping in mind that it's not possible to have a fraction of a quarter, we can round it up to 68 quarters or 17 years.