Solve the problem.
Suppose you start saving today for a $20,000 down payment that you plan to make on a house in 10 years. Assume that you make no deposits into the account after your initial deposit. The account has quarterly compounding and an APR of 3%. How much would you need to deposit now to reach your $20,000 goal in 10 years?
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment 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
In this case, we want to find the initial deposit (P), so we rearrange the formula as:
P = A / (1 + r/n)^(nt)
Given:
A = $20,000
r = 3% or 0.03 (as a decimal)
n = 4 (quarterly compounding)
t = 10 years
Plugging in the values, we get:
P = 20000 / (1 + 0.03/4)^(4*10)
Simplifying further:
P = 20000 / (1 + 0.0075)^(40)
P = 20000 / (1.0075)^40
Using a calculator:
P ≈ 20000 / 1.357595
P ≈ $14,738.01
Therefore, you would need to deposit approximately $14,738.01 now to reach your $20,000 goal in 10 years.
To solve this problem, we can use the formula for future value of an investment with compound interest:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future value (the goal amount you want to reach, which is $20,000 in this case)
PV = Present value (the amount you need to deposit today)
r = Annual interest rate (3% in this case)
n = Number of compounding periods per year (quarterly compounding, so 4 quarters per year)
t = Number of years (10 years in this case)
Let's substitute the given values into the formula:
$20,000 = PV * (1 + 0.03/4)^(4*10)
Now, let's solve for PV:
$20,000 = PV * (1 + 0.0075)^(40)
To isolate PV, divide both sides of the equation by (1.0075)^40:
PV = $20,000 / (1.0075)^40
Using a calculator, you can find that (1.0075)^40 is approximately 1.31719.
Now, let's calculate the present value (PV):
PV ≈ $20,000 / 1.31719
PV ≈ $15,186.17
Therefore, you would need to deposit approximately $15,186.17 today to reach your $20,000 goal in 10 years.
To solve this problem, 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 deposit
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
In this case, we know:
A = $20,000
r = 3% = 0.03 (since APR is given, we divide by 100)
n = 4 (quarterly compounding)
t = 10 years
We need to find P (the initial deposit). Rearranging the formula, we get:
P = A / (1 + r/n)^(nt)
Substituting the given values, we have:
P = 20,000 / (1 + 0.03/4)^(4*10)
Now let's calculate it step by step:
Step 1: Calculate (1 + r/n)
(1 + 0.03/4) = 1.0075
Step 2: Calculate (nt)
(4 * 10) = 40
Step 3: Calculate (1 + r/n)^(nt)
1.0075^40 = 1.349858807
Step 4: Calculate the final result
20,000 / 1.349858807 = $14,813.18 (rounded to the nearest cent)
So, you would need to deposit approximately $14,813.18 now to reach your $20,000 goal in 10 years with quarterly compounding and an APR of 3%.