Find the number of years for $4800 to grow to $9400 at 6% compounded quarterly. Round to the nearest tenth of a year.
9400/4800 = (1.015)^4y = 1.9583
where y is the number of tears.
Solve for y. I recommend using logs
4y = log1.9583/log1.015
y = 11.3 years
To find the number of years for an initial investment of $4800 to grow to $9400 at a 6% interest rate compounded quarterly, 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 (initial amount)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case,
A = $9400
P = $4800
r = 6% = 0.06 (decimal form)
n = 4 (compounded quarterly)
Let's plug in these values and solve for t:
$9400 = $4800(1 + 0.06/4)^(4t)
Dividing both sides by $4800:
1.9583333333 = (1.015)^(4t)
Now, we can take the natural logarithm of both sides:
ln(1.9583333333) = ln(1.015)^(4t)
Using the logarithmic property ln(a^b) = b * ln(a):
ln(1.9583333333) = 4t * ln(1.015)
Dividing both sides by ln(1.015):
t = ln(1.9583333333) / (4 * ln(1.015))
Using a calculator to evaluate the expression:
t ≈ 9.5 years
Therefore, it would take approximately 9.5 years for $4800 to grow to $9400 at a 6% interest rate compounded quarterly.
To find the number of years it takes for an amount to grow at a given interest rate compounded quarterly, 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 (initial amount)
r = the annual 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 = $4800
A = $9400
r = 6% = 0.06 (as a decimal)
n = 4 (compounded quarterly)
Let's substitute these values into the formula and solve for t:
$9400 = $4800(1 + 0.06/4)^(4t)
Divide both sides of the equation by $4800:
$9400/$4800 = (1 + 0.06/4)^(4t)
Simplify the right side of the equation:
1.9583333... = (1.015)^(4t)
Now, take the natural logarithm (ln) of both sides of the equation to isolate the exponent:
ln(1.9583333...) = ln((1.015)^(4t))
Using logarithm properties, bring down the exponent:
ln(1.9583333...) = 4t * ln(1.015)
Divide both sides of the equation by ln(1.015):
ln(1.9583333...) / ln(1.015) = 4t
Now, solve for t by dividing by 4 and rounding to the nearest tenth of a year:
t = ln(1.9583333...) / (4 * ln(1.015))
t ≈ 5.9 years
Therefore, it takes approximately 5.9 years for $4800 to grow to $9400 at a 6% interest rate compounded quarterly.