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If $7,500 is invested in a savings account earning 5% interest compunded quarterly, how many years will pass until there is $20,000?
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To determine the number of years it will take for an investment to grow to $20,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of compounding periods per year
t = the number of years
In this case, we have:
P = $7,500
r = 5% = 0.05
n = 4 (compounded quarterly)
A = $20,000
Plugging in the given values and solving for t, we have:
$20,000 = $7,500(1 + 0.05/4)^(4t)
Divide both sides by $7,500:
20,000/7,500 = (1.0125)^(4t)
Simplify the left side:
2.6667 = (1.0125)^(4t)
To isolate the exponent on the right side, take the logarithm (base 1.0125) of both sides:
log₁.₀₁₂₅(2.6667) = log₁.₀₁₂₅[(1.0125)^(4t)]
Using logarithmic properties, we have:
(4t) = log₁.₀₁₂₅(2.6667) / log₁.₀₁₂₅(1.0125)
Now, divide both sides by 4:
t = [log₁.₀₁₂₅(2.6667) / log₁.₀₁₂₅(1.0125)] / 4
Calculating the right-hand side of the equation using a calculator or computer software, we find:
t ≈ 10.706
Therefore, it will take approximately 10.706 years for the investment to grow from $7,500 to $20,000 when earning 5% interest compounded quarterly.