# Determine a Simple Interest Rate For a Loan with Known Interest

A friend asks to borrow $500 and offers to

pay you $520 in one month. What annual simple interest rate is this equivalent to? We will

be solving this problem using the simple interest formula, I equals p times r times t. I is

the interest paid. p is the principal, which is the original or starting amount. r is the

interest rate and must be a per year rate written as a decimal. t is time expressed

in years. Looking at the given information, because your friend wants to borrow $500.

The principal is $500. Because he is willing to pay you $520 in one month, the interest

paid would be $520 minus $500, which equals $20. He is paying you $20 of interest to borrow

the $500. The time is one month, but the time must be expressed in years, not months. Because

there are 12 months in 1 year, 1 month is equal to 1/12 of a year. We are asked to determine

the simple interest rate so the unknown is r, the interest rates. Now using the equation

I=p times r times t, we have 20 equals the principal, which is 500, times the unknown

interest rate, r, times the time, which is 1/12 of a year. 500 times r times 1/12 will

be 500/12 r. Let’s write 20 equals 500/12 r. 500/12 simplifies. Let’s work on simplifying

the fraction. The greatest common factor between the top and bottom is 4. So we divide the

top and bottom by 4, which gives us 125/3. So now let’s write the equation as 20 equals

125/3 r. To solve the equation for r, multiply both sides of the equation by the reciprocal

of 125/3, which would be 3/125. On the right side, because these are reciprocals, the product

would be equal to one. One times r is equal to r. So we have r equals, let’s write 20

as 20 over 1. So we have 3 over 125 times 20 over 1. 20 and 125 share a common factor

of 5. There are 4 5’s in 20 and 25 5’s in 125. Notice how we have r equals 12/25. We

do want to express r as a percentage so we could divide and then convert the decimal

to a percentage, but because a percent is a number compared to 100, let’s write this

fraction with a denominator of 100. We will multiply the top by 4 and the bottom by 4,

which would give us 48/100, which means r is equal to 48%. Now we know in this situation,

the annual simple interest rate is equivalent to 48%. I hope you found this helpful.

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