MHB Optimizing Loan Interest Rates: Solving for Loan Length

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Lending companies typically provide better rates for larger loans over shorter terms, as illustrated by a borrower facing two options: a $10,080 loan at 10% for a shorter duration and a $7,000 loan at 12% for a longer duration. The problem states that the larger loan is 6 months shorter, yet the total interest paid remains the same. To solve for the length of each loan, it is assumed that these are single payment loans with simple interest. An equation is proposed to find the time variable, but clarification is sought on whether the interest is simple or compound. Understanding the type of interest is crucial for accurately solving the problem.
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The problem: Lending companies often offer better rates when you borrow a larger sum of money for a shorter period of time. A borrower is offered a loan of \$10,080 at 10% and a loan of \$7000 at 12%. If the larger loan is 6 mo shorter but the total interest payed is the same, find the length of each loan. (End of problem.) I am trying to write an equation for t (time) but have not been able to solve it. Thanks in advance!
 
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MisterMeister said:
The problem: Lending companies often offer better rates when you borrow a larger sum of money for a shorter period of time. A borrower is offered a loan of \$10,080 at 10% and a loan of \$7000 at 12%. If the larger loan is 6 mo shorter but the total interest payed is the same, find the length of each loan. (End of problem.) I am trying to write an equation for t (time) but have not been able to solve it. Thanks in advance!
Looks like we need to assume that these are single payment loans,
and that simple interest is involved (no compounding).

n = number of years (7000 loan)
n - .5 = number of years (10080 loan)

(n - .5)*(10080*.10) = n*7000*.12

I'll let you finish that off...
 
Is this simple interest or compound interest? If compound interest, how often compounded?
 

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