A newly couple has $15,000 toward the purchase of a house. For the type and size of house the couple is interested in buying, an estimated down payment of $20,000 will be necessary. How long will the money have to be invested at 10% compounded quarterly to grow to $20,000?
This is what I have so far
A=P(1+r/n)^(n*t)
20,000 = 15,000 (1+0.1/4)^(4*t)
divide both sides by 15,000
4/3 = (1 + 0.1/4)^(4*t)
4/3 = 1.025^(4*t)
Now I'm stuck and don't know if should take the natural logs
log 4/3 = log [1.025^(4*t)]
log 4/3 = 4*t log 1.025
Divide both sides by log 1.025
solve for t
P = Po(1+r)^n = 20,000.
r = 0.1/yr * 0.25yr. = 0.025 = Quarterly % rate expressed as a decimal.
n = 4Comp/yr * t yrs. = 4t.
15,000(1.025)^4t = 20,000.
(1.025)^4t = 20,000/15,000 = 1.333,
4t*Log1.025 = Log1.333.
4t = Log1.333/Log1.025 = 11.7,
t = 2.91 Yrs.
Taking the natural logarithm (ln) of both sides of the equation can definitely help you solve for the unknown variable, which in this case is 't'.
Let's start by taking the natural logarithm of both sides:
ln(4/3) = ln(1.025^(4*t))
Now, since the ln and the exponentiation (raised to the power of) are connected by the property of logarithms, we can move the exponent down:
ln(4/3) = 4*t * ln(1.025)
Next, divide both sides by 4 * ln(1.025):
t = ln(4/3) / (4 * ln(1.025))
Now you have the equation for 't', which represents the number of quarters needed for the investment to grow to $20,000. Simply calculate the value on the right-hand side of the equation to determine the value of 't'.
Using a calculator, you can find the solution:
t ≈ 20.098
Since the time (t) represents the number of quarters, based on this calculation the money will need to be invested for approximately 21 quarters (rounded up) to grow to $20,000 at an interest rate of 10% compounded quarterly.