Question 1044238: (2) Owen and Sarah's pre-tax income is $100,000. Suppose that state income tax is 10% of taxable state income and that federal income tax is 10/19 of taxable federal income. Taxable federal income is $100,000 less state income tax. Taxable state income is $100,000 less federal income tax. How much will Owen and Sarah pay in state and federal income tax? Hint: Let TF = taxable federal income and TS = taxable state income.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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
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