Can Anyone Help Me With This Compound Interest Question?

[Elena_W]

I needed help solving this, A financial obligation requires the payment of $500 in 9 months, $700 in fifteen months, and $600 in 27 months. When can the obligations be discharged by a single payment of $1600 if interest is 10% compounded quarterly? (Let the focal date be now.)

[Jonathan-Joe]

Original amount X
After 9 months (3 interest periods) amount = X(1.1)^3 = 1.331X
With repayment amount =1.331X – 500
After 15 months (2 further int pers) amount = (1.331X – 500)(1.1)^2 = 1.61051X – 605
With repayment amount = 1.61051X – 605 – 700 = 1.61051X – 1305
After 27 months (4 further int pers) amount = (1.61051X – 1305)(1.1)^4 = 2.357947691X - 1910.6505
With repayment amount = 2.357947691X - 1910.6505 – 600 = 2.357947691X - 2510.6505
With debt repaid 2.357947691X - 2510.6505 = 0
2.357947691X = 2510.6505, X = 1064.7608976156884559149450614339

Check
1064.76 1.1^3 1417.20 -500 917.20
917.20 1.1^2 1109.81 -700 409.81
409.81 1.1^4 600.00 -600 0

Original amount owed 1064.76, add interest for y periods
1064.76(1.1)^y = 1600
(1.1)^y = 1.5027
y(log(1.1)) = log(1.5027)
y = log (1.5027) / log(1.1) = .1769 / .04139 = 4.2729
Answer $ 1600 will discharge the debt after 4 quarters and (about) 25 days.