The balance on a car loan after 4 years is $8,996.32. The interest rate is 5.6% compounding annually. What was the initial value of the loan?
An investment made in the stock market decreased at a rate of 4% per year for 5 years. What is the current value of the $1,000,000 investment? Include your calculations in your final answer.
explain how to solve these in steps asap
x (1 - 0.056 )^5 = 8996.32
.944^5 x = 8996.32
x = 8996.32 /.749652
x = 12000.65
same way
10^6 * 0.96^5
To calculate the initial value of the car loan, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final balance ($8,996.32)
P is the initial principal (unknown)
r is the interest rate per period (5.6% or 0.056)
n is the number of compounding periods per year (1, since it is compounded annually)
t is the number of years (4)
Step 1: Rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Step 2: Substitute the given values:
P = $8,996.32 / (1 + 0.056/1)^(1*4)
Step 3: Simplify the expression inside the parentheses:
P = $8,996.32 / (1.056)^4
Step 4: Calculate the result:
P = $8,996.32 / 1.243477
P ≈ $7,237.52
Therefore, the initial value of the car loan was approximately $7,237.52.
Now let's move on to the investment in the stock market:
Step 1: Calculate the annual decrease rate in decimal form:
Decrease rate = 4% = 0.04
Step 2: Apply the decrease rate to the initial investment:
Current value = Initial investment - (Decrease rate * Initial investment * Number of years)
= $1,000,000 - (0.04 * $1,000,000 * 5)
Step 3: Simplify the expression:
Current value = $1,000,000 - $200,000
Current value = $800,000
Therefore, the current value of the $1,000,000 investment after 5 years with a 4% decrease rate is $800,000.
To find the initial value of the car loan and the current value of the investment, you can use the formula for compound interest:
For the car loan:
Step 1: Use the compound interest formula to find the initial value of the loan.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A is the final amount (the loan balance after 4 years)
P is the principal (initial loan amount)
r is the annual interest rate (as a decimal)
n is the number of times interest is compounded per year
t is the number of years
In this case, A = $8,996.32, r = 5.6% = 0.056 (decimal form), n = 1 (compounded annually), and t = 4.
Step 2: Rearrange the formula to solve for P.
P = A / (1 + r/n)^(nt)
Substitute the given values into the formula and solve for P.
For the investment:
Step 1: Use the compound interest formula to find the current value of the investment.
In this case, the investment decreased at a rate of 4% per year, which means the interest rate is -4% = -0.04 (as a decimal).
Step 2: Use the formula A = P(1 + r/n)^(nt), but this time the final amount A is
A = $1,000,000, r = -4% = -0.04, n = 1 (compounded annually), and t = 5.
Substitute the values into the formula and solve for P.
Performing these calculations will give you the initial value of the car loan and the current value of the investment.