Chapter 7.1, Problem 7CYU

To determine

To Explain:

Dora has 8% of her earnings deducted from her paycheck for a college savings plan.She can choose to take the deduction either before taxes are withheld , which reduces her taxable income, or after taxes are withheld.Dora’s tax rate is 17.5%. If her pay before taxes and deductions is $950 , find whether she save more money if the deductions are taken before or after taxes are withheld .

Answer to Problem 7CYU

Dora will save same amount of money in both of the cases, either deductions are taken before or after taxes are withheld.

Explanation of Solution

Given:

Amount deducted from Dora’s paycheck for a college savings plan = 8% of her earnings

Tax rate = 17.5%

Dora’s pay before taxes and deductions = $950

Calculation:

Let Dora’s total income before all deductions and taxes = x

Now we create functions for the amount left with Dora after and before taxes :

  $\begin{array}{l}f(x)=100\%x-8\%x=0.92x\mathrm{...........}[money\text{ left after 8\% deduction]}\\ g(x)=100\%x-17.5\%x=0.825x\mathrm{.....}[\text{money left after 17}\text{.5\% tax paid]}\end{array}$

Now , $f\circ g\text{ and }g\circ f$ determine the amount left from Dora’s income , if tax is paid first and if deduction is done first respectively.

So , we calculate $f\circ g\text{ and }g\circ f$ to find how much amount is left with Dora , in both of the cases:

  $\begin{array}{l}\left(f\circ g\right)(x)=f(0.825x)=0.92(0.825x)=0.759x\\ \left(g\circ f\right)(x)=g\left(0.92x\right)=0.825(0.92x)=0.759x\end{array}$

So, in both of the cases , Dora will save same about of money from her total income . It does not matter whether deductions are taken before or after taxes are withheld.

The other way to solve this can be directly by the percentage :

Dora’s total income = $950

8% of Income = $950\cdot 8\%=950(0.08)=\$76$

Amount left after 8% deductions = 950-76 = $874

Tax rate = 17.5%

Tax to be paid on $874 = $874\cdot 17.5\%=874(0.175)=\$152.95$

Amount left after 17.5% tax payment from the left amount after deductions = $\$874-\$152.95=\$721.05$

Dora’s total income = $950

Tax rate = 17.5%

Tax to be paid on $950 = $\$950\cdot 17.5\%=950(0.175)=\$166.25$

Amount left after 17.5% tax payment from the total income = $\$950-\$166.25=\$783.75$

8% of deductions from the amount left after the tax has been paid = $783.75\cdot 8\%=783.75(0.08)=\$62.7$

Amount left after the 8% deductions = $\$783.75-\$62.7=\$721.05$

So, in both of the cases :

If deductions are taken before or after the taxes are withheld are $721.05 , which is equal in both of the cases.