Astudent who was ineligible for financial aid needed to borrow 775,000 for tuition, books, and living expenses. some money was borrowed at 8%, some at 9%, and some at 10%. how much was borrowed at each interest rate if the annual interest owed was 67,500 and the amount borrowed at 8% was four times the amount borrowed at 10%. 8%: 9%: 10%:

$75,000 was borrowed at 10%; $300,000 was borrowed at 8%; and $400,000 was borrowed at 9%.

Since we are only looking at one year, and we assume the interest is compounded no more than annually, we can use the formula for simple interest:

I = prt

Let p be the amount borrowed at 10%.  Then 4p, or 4 times the amount borrowed at 10%, is borrowed at 8% interest.  The remaining amount, (775000-4p-p) or (775000-5p) is borrowed at 9%.  The amount of interest for each portion of the loan is given by:

I=p(0.1)(1)
I=4p(0.08)(1)
I=(775000-5p)(0.09)(1)

Adding these together for the total amount of interest, $67,500, we have:

67500=p(0.1)(1)+4p(0.08)(1)+(775000-5p)(0.09)(1)

Simplifying the right hand side we have:
67500=0.1p+0.32p+69750-0.45p

Combining like terms gives us:
67500= -0.03p+69750

Subtract 69750 from both sides:
67500-69750= -0.03p+69750-69750
-2250= -0.03p

Divide both sides by -0.03:
-2250/-0.03 = -0.03p/-0.03

75000 = p

So $75000 is the amount borrowed at 10%; 4(75000)=$300,000 borrowed at 8%; and 775000-75000-30000= $400,000 borrowed at 9%.