Question
solve.
James invests $25,000 into an account paying 8% annual interest. How many years will it take for his account to reach $43,000?
James invests $25,000 into an account paying 8% annual interest. How many years will it take for his account to reach $43,000?
Answers
GPT 3.5
To solve this problem, we can use the formula for calculating compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest
P = the principal amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, James invested $25,000 at an interest rate of 8% annually, so
P = $25,000
r = 0.08
We want to find out how many years it will take for the account to reach $43,000, so
A = $43,000
Substitute these values into the formula and solve for t:
$43,000 = $25,000(1 + 0.08/1)^(1t)
$43,000 = $25,000(1.08)^t
Divide both sides by $25,000:
1.72 = 1.08^t
Take the natural logarithm (ln) of both sides:
ln(1.72) = ln(1.08)^t
ln(1.72) = t * ln(1.08)
Now divide by ln(1.08) to solve for t:
t = ln(1.72) / ln(1.08)
t ≈ 7.16
Therefore, it will take approximately 7.16 years for James' account to reach $43,000.
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest
P = the principal amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, James invested $25,000 at an interest rate of 8% annually, so
P = $25,000
r = 0.08
We want to find out how many years it will take for the account to reach $43,000, so
A = $43,000
Substitute these values into the formula and solve for t:
$43,000 = $25,000(1 + 0.08/1)^(1t)
$43,000 = $25,000(1.08)^t
Divide both sides by $25,000:
1.72 = 1.08^t
Take the natural logarithm (ln) of both sides:
ln(1.72) = ln(1.08)^t
ln(1.72) = t * ln(1.08)
Now divide by ln(1.08) to solve for t:
t = ln(1.72) / ln(1.08)
t ≈ 7.16
Therefore, it will take approximately 7.16 years for James' account to reach $43,000.