Gaga borrowed $2500 at 4% p.a compound interest. She repays $600 at the end of each year. Find out during which year the debt will finish.?
To determine in which year Gaga's debt will be completely repaid, we need to calculate the number of years it will take for the total repayments to equal the borrowed amount.
Let's break down the problem step by step:
1. Calculate the compound interest each year:
Compound interest is calculated by multiplying the principal amount (initial borrowed amount) by (1 + interest rate) raised to the power of the number of years. The formula is A = P(1 + r/n)^(nt), where:
A = the future value of the loan (amount repaid),
P = the principal amount (initial borrowed amount),
r = the annual interest rate (in decimal form),
n = the number of compounding periods per year, and
t = the number of years.
In this case, Gaga borrowed $2500 at an annual interest rate of 4% (or 0.04 in decimal form). Assuming the interest is compounded annually (n = 1), the equation becomes A = $2500(1 + 0.04)^t.
2. Calculate the total repayments each year:
Gaga repays $600 at the end of each year. So, for each year t, the total repaid amount is simply $600 multiplied by the number of years.
3. Find the year when the total repayments equal the borrowed amount:
Set the total repaid amount for a given year (600t) equal to the future value of the loan (2500(1 + 0.04)^t). Solve the equation to find the value of t.
Let's perform the calculations:
600t = 2500(1 + 0.04)^t
To find the year when the debt will be finished, we need to solve this equation. Unfortunately, this equation cannot be solved directly using algebraic methods. We need to use an iterative method, such as the trial and error method or a graphing calculator.
Using a trial and error method, we can try different values of t until we find the year that satisfies the equation. Starting with t = 1, we can substitute this value into the equation:
600(1) = 2500(1 + 0.04)^1
600 = 2500(1.04)
600 ≈ 2600
The total repayments of $600 in the first year have not yet equaled the borrowed amount of $2500. We need to try a larger value of t.
Next, let's try t = 2:
600(2) = 2500(1 + 0.04)^2
1200 = 2500(1.08)
1200 ≈ 2700
Again, the total repayments have not yet equaled the borrowed amount. Let's try t = 3 and continue this process until we find the year where the total repayments meet or exceed the borrowed amount.
600(3) = 2500(1 + 0.04)^3
1800 = 2500(1.12)
1800 < 2800
After testing t = 1, 2, and 3, we can conclude that Gaga's debt will be finished during the third year since the total repayments in that year (1800) will exceed the borrowed amount (2500). Thus, the debt will be finished in the third year.