Question 1044238
sit = state income tax
fit = federal income tax
tsi = taxable state income
tfi = taxable federal income


you are given that:


sit = .10 * (100,000 - fit)
fit = 10/19 * (100,000 - sit)


we'll work with sit first.


sit = .10 * (100,000 - fit)
fit = 10/19 * (100,000 - sit)


replace fit with 10/19 * (100,000 - sit) and you get:


sit = .10 * (100,000 - 10/19 * (100,000 - sit))


simplify this to get:


sit = .10 * (100,000 - 10/19 * 100,000 + 10/19 * sit)


simplify further to get:


sit = .10 * 100,000 - .10 * 10/19 * 100,000 + .10 * 10/19 * sit


simplify further to get:


sit = 10,000 - 1/19 * 100,000 + 1/19 * sit


simplify further to get:


sit = 10,000 - 100,000/19 + 1/19 * sit


subtract 1/19 * sit from both sides of the equation to get:


sit - 1/19 * sit = 10,000 - 100,000/19


simplify further to get 18/19 * sit = 190,000/19 - 100,000/19


simplify further to get 18/19 * sit = 90,000/19


multiply both sides of this equation by 19/18 to get:


sit = 19/18 * 90,000/19


simplify further to get sit = 90,000/18


simplify further to get sit = 5,000


you do a similar thing with federal income tax, except now you know what the value of sit is.


start with:


fit = 10/19 * (100,000 - sit)


replace sit with 5,000 to get:


fit = 10/19 * (100,000 - 5,000)


simplify to get:


fit = 10/19 * 95,000


simplify further to get:


fit = 50,000


your solution is:


federal income tax = 50,000
state income tax = 5,000