if you want the interest part of the payment to be equal to 1/3 the payment, you have to first find the payment.
i used the calculator at https://arachnoid.com/finance/ to find the payment.
input to the calculator was:
present value = -10000
future value = 0
number of time periods = 10 years
interest rate per time period = 13% per year
paments are made at the end of each time period.
the calculator tells me that the payment required at the end of each year is equal to 1842.90.
if you want the interest part of the payment to be equal to 1/3 the payment, then the interest part of the payment needs to be equal to 1/3 * 1842.90 = 614.30
that will occur when the remaining balance on the loan is equal to 614.3 / .13 = 4725.39, or the remaining balance closest to that.
i used excel to get the remaining balance at the end of each year, along with the interest part of the payment and the equity part of the payment.
excel gives an end of year account for each of the 10 years.
the formula used is remaining balance for the year is the remaining balance for the previous year * 1.13 minus the payment for the year.
for example:
end of year 6 remaining balance is 5481.64.
end of year 7 remaining balance is 5481.64 * 1.13 minus 1842.90 = 4351.36.
you only see the numbers rounded to the nearest penny.
the calculations are done with the actual numbers stored that are not rounded to the nearest penny.
the end of year remaining balance closest to 4725 is the remaining balance for end of year 7, which is 4351.36.
the interest on that remaining balance is taken out as part of the payment for end of year 8.
that interest was calculated as 4351.36 * .13 = 565.68.
since they asked for the year in which the interest portion of the payment was most equal to 1/3 the payment, i would have to say that would be year 8.