SOLUTION: When calculating his income tax an accountant realized that his federal tax was 25% of his income after his provincial tax was paid, and that his provincial tax was 15% of his inco

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: When calculating his income tax an accountant realized that his federal tax was 25% of his income after his provincial tax was paid, and that his provincial tax was 15% of his inco      Log On


   

Question 996057: When calculating his income tax an accountant realized that his federal tax was 25% of his income after his provincial tax was paid, and that his provincial tax was 15% of his income after his federal tax was paid. If the accountant's income before taxes amounted to $120 000, how much federal tax did he have to pay (to the nearest penny)?
Thanks for the help in advance!
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
best i can do is this.

f = federal tax
p = provincial tax.
i = income

f = .25 * (i - p)
p = .15 * (i - f)

those are your formulas.

f = .25 * (i - p) means:
federal tax = .25 * income after you paid the provincial tax.

i = .15 * (i - f) means:
provincial tax = .15 * income after you paid the federal tax.

you want to find what the federal tax was.

that's f = .25 * (i - p)
since p = .15 * (i - f), you can replace p with that to get:
f = .25 * (i - .15 * (i - f))
simplify this by removing the inner parentheses to get:
f = .25 * (i - .15 * i + .15 * f)
now remove the next set of parentheses to get:
f = .25 * i - .25 * .15 * i + .25 * .15 * f
subtract .25 * .15 * f from both sides of the equation to get:
f - .25 * .15 * f = .25 * i - .25 * .15 * i
factor out f from the left side of the equation to get:
f * (1 - .25 * .15) = .25 * i - .25 * .15 * i
factor out .25 * i from the right side of the equation to get:
f * (1 - .25 * .15) = .25 * i * (1 - .15)
simplify to get:
.9625 * f = .2125 * i
divide both sides of the equation by .9625 to get:
f = .2125 * i / .9625
i = 120,000, the formula becomes:
f = .2125 * 120,000 / .9625
solve for f to get:
f = 26493.50649 which is equal to 26493.51 rounded to the nearest penny.

that's your federal tax.

you can calculate your provincial tax in a similar manner.
you don't have to do it, but i did, just to see if it could be done.
it could be done.

your provincial tax winds up to be equal to 14025.97403.

now your federal tax is equal to .25 * your income after provincial taxes have been paid.

that's .25 * (120,000 - 14025.97403) which is equal to 26493.50649.

and your provincial tax is equal to .15 * your income after federal taxes have been paid.

that's .15 * (120,000 - 26493.50649) which is equal to 14025.97403.

that confirms that the solution is more then likely correct.

the solution is:

your federal tax is 26493.51 to the nearest penny.