Question

to borrow money, you pawn your guitar. based on the value of the guitar, the pawnbroker loans you $960. one month later, you get the guitar back by paying the pawnbroker $1270. what annual interest rate did you pay?
you will pay a simple interest rate of \\(\square\\)%
(round to the nearest whole number as needed )

Explanation:

Step1: Find the interest paid

The interest \( I \) is the amount paid minus the principal. So \( I = 1270 - 960 = 310 \) dollars.

Step2: Recall the simple interest formula

The simple interest formula is \( I = Prt \), where \( P \) is principal, \( r \) is annual rate, \( t \) is time in years. Here, \( t=\frac{1}{12} \) (1 month is \( \frac{1}{12} \) of a year), \( P = 960 \), \( I = 310 \).

Step3: Solve for \( r \)

From \( I = Prt \), we get \( r=\frac{I}{Pt} \). Substitute values: \( r=\frac{310}{960\times\frac{1}{12}} \). First, calculate \( 960\times\frac{1}{12}=80 \). Then \( r = \frac{310}{80}=3.875 \). Multiply by 100 to get percentage: \( 3.875\times100 = 387.5\% \)? Wait, no, wait. Wait, I made a mistake. Wait, the amount paid is $1270, principal is $960, so interest is \( 1270 - 960 = 310 \)? Wait, no, that can't be. Wait, no, when you pawn, you borrow \( P = 960 \), and pay back \( A=1270 \) after 1 month. So simple interest formula is \( A=P(1 + rt) \), where \( t \) is in years. So \( 1270=960(1 + r\times\frac{1}{12}) \). Let's solve for \( r \). First, divide both sides by 960: \( \frac{1270}{960}=1+\frac{r}{12} \). Calculate \( \frac{1270}{960}\approx1.3229 \). Then subtract 1: \( 0.3229=\frac{r}{12} \). Multiply both sides by 12: \( r = 0.3229\times12\approx3.875 \), so \( r\approx387.5\% \)? That seems very high, but maybe pawn shops have high interest. Wait, but let's check again. Wait, maybe I messed up the interest. Wait, the interest is \( 1270 - 960 = 310 \) over 1 month. So monthly interest rate is \( \frac{310}{960}\approx0.3229 \), so annual rate is \( 0.3229\times12\approx3.875 \), so 387.5%, which rounds to 388%? Wait, but that seems extremely high. Wait, maybe the problem is that I misread the numbers. Wait, the loan is $960, pay back $1270 in one month? That's a huge interest. Alternatively, maybe the numbers are $960 loan, pay back $1270? Wait, maybe it's a typo, but assuming the numbers are correct. Wait, let's recalculate. \( I = 1270 - 960 = 310 \), \( t = \frac{1}{12} \) year, \( P = 960 \). So \( r=\frac{I}{Pt}=\frac{310}{960\times\frac{1}{12}}=\frac{310\times12}{960}=\frac{3720}{960}=3.875 \), so 387.5%, which rounds to 388%. But that seems very high, but maybe that's correct for pawn shops. Wait, but let's check the steps again.

Wait, let's start over. Simple interest formula: \( I = Prt \), where \( I \) is interest, \( P \) principal, \( r \) annual rate, \( t \) time in years.

Given:

• \( P = 960 \) (amount borrowed)
• \( A = 1270 \) (amount repaid)
• \( t = \frac{1}{12} \) years (1 month)
• Interest \( I = A - P = 1270 - 960 = 310 \)

We need to find \( r \).

From \( I = Prt \), solve for \( r \):

\( r = \frac{I}{Pt} \)

Substitute values:

\( r = \frac{310}{960 \times \frac{1}{12}} \)

Calculate denominator: \( 960 \times \frac{1}{12} = 80 \)

So \( r = \frac{310}{80} = 3.875 \) (this is the annual rate, since \( t \) was in years). To express as a percentage, multiply by 100: \( 3.875 \times 100 = 387.5\% \), which rounds to 388% when rounded to the nearest whole number.

Wait, but that seems extremely high, but maybe that's the case with pawnbrokers. Let's verify with the simple interest formula. If \( r = 387.5\% = 3.875 \) per year, then in 1 month (\( t = \frac{1}{12} \)), the interest is \( I = 960 \times 3.875 \times \frac{1}{12} = 960 \times \frac{3.875}{12} = 80 \times 3.875 = 310 \), which matches \( 1270 - 960 = 310 \). So that's correct. So the annual interest rate is approximately 388%.

Answer:

\boxed{388}