Question

Arden, an employee of Sweet Shoppe Industries, receives a bonus of $4,000 for stellar work this period. Arden's boss wants the net check to be $4,000. Arden contributes 3 percent to a pre-tax 401(k) deduction.

Required:
Calculate the gross pay amount that would result in $4,000 paid to Arden. (The tax rate on bonuses is 22 percent. Social Security (6.2%) and Medicare taxes (1.45%) must be added to this rate.)

Answer 1

To determine the gross pay amount that would result in $4,000 net pay, we need to account for the pre-tax 401(k) deduction and the various taxes that will be deducted from the gross pay.

### Given Information:
- Desired net pay (take-home pay) = $4,000
- Pre-tax 401(k) deduction = 3%
- Bonus tax rate = 22%
- Social Security tax rate = 6.2%
- Medicare tax rate = 1.45%

### Combined Tax Rates:
First, we need to calculate the total effective tax rate (including Social Security and Medicare taxes) on the gross pay after the 401(k) deduction:

1. **Tax rate on the bonus:**
- Bonus tax = 22%

2. **Social Security tax:**
- 6.2%

3. **Medicare tax:**
- 1.45%

Total effective tax rate (excluding the 401(k) deduction) = 22% + 6.2% + 1.45% = 29.65%.

However, since Arden is also contributing to the 401(k), we need to account for that first. Here's how to tackle it:

Let **G** be the gross pay before any deductions.

### Calculation Steps:

1. **Calculate the 401(k) deduction:**
\[
\text{401(k) Deduction} = 0.03 \times G
\]

2. **Calculate the taxable income after the 401(k) deduction:**
\[
\text{Taxable Income} = G - \text{401(k) Deduction} = G - (0.03G) = 0.97G
\]

3. **Calculate the total taxes on the taxable income:**
\[
\text{Total Tax} = 0.2965 \times 0.97G \quad \text{(using the combined tax rate)}
\]

4. **Calculate net pay:**
The net pay, which is the gross pay minus the taxes and the 401(k) deduction, should equal $4,000. Therefore:
\[
\text{Net Pay} = G - \text{Total Tax} - \text{401(k) Deduction}
\]
Substitute the first two parts:
\[
4000 = G - (0.2965 \times 0.97G) - (0.03G)
\]

5. **Combine terms:**
\[
4000 = G - 0.2879G - 0.03G
\]
Simplifying this gives:
\[
4000 = G(1 - 0.2879 - 0.03)
\]
\[
4000 = G(1 - 0.3179)
\]
\[
4000 = G(0.6821)
\]

6. **Solve for G (Gross pay):**
\[
G = \frac{4000}{0.6821} \approx 5854.01
\]

### Final Calculation:
So, the gross pay amount that would result in a net pay of $4,000 is approximately **$5,854.01**.