Your brother lends you $125 at an annual interest rate of 25%. If you want to pay no more than $62.50 in simple interest, in how many years will you need to pay off the loan?
The Correct Answer and Explanation is :
To determine how many years you need to pay off a loan of $125 with a simple interest rate of 25% while keeping the total interest paid to no more than $62.50, we can use the formula for simple interest:
[
I = P \times r \times t
]
Where:
- (I) = Interest paid
- (P) = Principal amount (the initial amount of the loan)
- (r) = Annual interest rate (in decimal)
- (t) = Time in years
In this case:
- (P = 125)
- (r = 0.25) (since 25% as a decimal is 0.25)
- (I \leq 62.50)
Substituting the known values into the formula gives us:
[
I = 125 \times 0.25 \times t
]
Now, since we want the interest (I) to be no more than $62.50, we set up the inequality:
[
125 \times 0.25 \times t \leq 62.50
]
Calculating the left side:
[
125 \times 0.25 = 31.25
]
So the inequality becomes:
[
31.25 \times t \leq 62.50
]
To find (t), divide both sides by 31.25:
[
t \leq \frac{62.50}{31.25}
]
Calculating the right side:
[
t \leq 2
]
This means you can take up to 2 years to pay off the loan without exceeding $62.50 in interest.
In summary, if you want to ensure that the total interest paid on your $125 loan at a 25% annual interest rate does not exceed $62.50, you should plan to pay off the loan within 2 years. After 2 years, the interest accrued would exactly reach $62.50, which is the limit you’ve set. If you repay the loan sooner than that, the interest paid will be less.