Find the time required for an investment of 5000 dollars to grow to 7900 dollars at an interest rate of 7.5% per year, compounded quarterly
6.155 years.
solve for t years in
5000(1+.075/4)^(4t) = 7900
To find the time required for an investment to grow, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial investment amount
r = the interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we have:
P = $5000
A = $7900
r = 7.5% = 0.075
n = 4 (quarterly compounding)
Let's substitute these values into the formula and solve for t:
$7900 = $5000(1 + 0.075/4)^(4t)
Divide both sides of the equation by $5000:
$7900/$5000 = (1.01875)^(4t)
1.58 = 1.01875^(4t)
Now, take the logarithm (base 1.01875) of both sides to isolate the exponent:
log₁.₀₁₈₇₅ (1.58) = log₁.₀₁₈₇₅ (1.01875^(4t))
t = log₁.₀₁₈₇₅ (1.58) / 4
Using the logarithm function on a calculator or spreadsheet application, we can find the value of t:
t ≈ log₁.₀₁₈₇₅ (1.58) / 4 ≈ 0.0988 / 4 ≈ 0.0247 years
To convert this to months, multiply by 12:
t ≈ 0.0247 * 12 ≈ 0.2964 months
Therefore, it would take approximately 0.2964 months for an investment of $5000 to grow to $7900 at an interest rate of 7.5% per year, compounded quarterly.
To find the time required for an investment to grow, we can use the formula for compound interest:
A = P * (1 + r/n)^(nt).
Where:
A = the ending balance or future value of the investment
P = the principal or initial investment amount
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, the principal (P) is $5000, the future value (A) is $7900, the annual interest rate (r) is 7.5% or 0.075 (expressed as a decimal), and the interest is compounded quarterly (n = 4). We need to find the time (t).
Plugging in the known values, the formula becomes:
7900 = 5000 * (1 + 0.075/4)^(4t).
To solve for t, we need to rearrange the equation and solve for t using logarithms:
(1 + 0.075/4)^(4t) = 7900/5000.
Taking the natural logarithm (ln) of both sides:
ln((1 + 0.075/4)^(4t)) = ln(7900/5000).
Using the logarithmic identity (ln(a^b) = b * ln(a)):
4t * ln(1 + 0.075/4) = ln(7900/5000).
Now, we divide both sides of the equation by 4 * ln(1 + 0.075/4) to isolate t:
t = ln(7900/5000) / (4 * ln(1 + 0.075/4)).
Using a calculator, we can find t.
Calculating:
t ≈ ln(7900/5000) / (4 * ln(1 + 0.075/4))
≈ ln(1.58) / (4 * ln(1.01875))
≈ 0.471 / (4 * 0.0186)
≈ 0.471 / 0.0744
≈ 6.33.
Therefore, the time required for the investment to grow from $5000 to $7900 at an interest rate of 7.5% per year, compounded quarterly, is approximately 6.33 years.