Emily, a college student, obtained two identical educational loans to support her academic journey for the same period. One loan came from a government-affiliated student loan programme with a 6% annual simple interest rate. Also, she acquired a separate loan from a private financial institution with a 9% annual simple interest rate. Emily successfully paid off the government-based loan 8 months later than its original maturity date. Similarly, she settled the private loan precisely on its predetermined due date. If for each of the loan, she had to pay $8,512 as the amount, what is the principal amount she borrowed from the private financial institution, and what was the duration for which she borrowed the loan from the same institution?

Let P1 and P2 be the identical principals, and let t be the original loan period in years. The government loan was paid 8 months late, so its amount is P1 plus (P1 times 6 times (t plus 2 divided by 3) divided by 100), while the private loan amount is P2 plus (P2 times 9 times t divided by 100). Equating both amounts to 8512 gives P1 times (1 plus 0.06t plus 0.04) equals 8512 and P2 times (1 plus 0.09t) equals 8512; since P1 equals P2, equating the multipliers yields 1.04 plus 0.06t equals 1 plus 0.09t, meaning 0.04 equals 0.03t and t equals 4 divided by 3 years. Substituting t back into the private loan equation gives P2 times (1 plus 0.12) equals 8512, so P2 equals 8512 divided by 1.12, making the principal $7,600.

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