SOLUTION: (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. Taxabl

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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