Calculate Emerick's monthly taxable income and PAYE, given his salary and allowances, using the provided tax brackets.

Taxable income includes the basic salary and all taxable allowances. The transport allowance is stated as non-taxable.

$$\text{Taxable Income} = \text{Basic Salary} + \text{House Allowance} + \text{Medical Allowance}$$ $$\text{Taxable Income} = \text{Kshs } 38,780 + \text{Kshs } 10,900 + \text{Kshs } 3,850$$ $$\text{Taxable Income} = \text{Kshs } 53,530$$

The monthly taxable income is $\boxed{\text{Kshs } 53,530}$.

We use the provided tax table, interpreting "Tax rate (Kshs per Kshs)" as a percentage.

• First band: Kshs $1 - 12,000$ at $2\%$ $$\text{Tax}_1 = 12,000 \times 0.02 = \text{Kshs } 240$$
• Second band: Kshs $12,001 - 22,500$ (range of 10,500) at $3\%$ $$\text{Tax}_2 = (22,500 - 12,000) \times 0.03 = 10,500 \times 0.03 = \text{Kshs } 315$$
• Third band: Kshs $22,501 - 33,000$ (range of 10,500) at $4\%$ $$\text{Tax}_3 = (33,000 - 22,500) \times 0.04 = 10,500 \times 0.04 = \text{Kshs } 420$$
• Fourth band: Kshs $33,001 - 43,500$ (range of 10,500) at $5\%$ $$\text{Tax}_4 = (43,500 - 33,000) \times 0.05 = 10,500 \times 0.05 = \text{Kshs } 525$$
• Fifth band: Remaining income above Kshs 43,500 at $6\%$ The taxable income is Kshs 53,530. $$\text{Taxable income in this band} = 53,530 - 43,500 = \text{Kshs } 10,030$$ $$\text{Tax}_5 = 10,030 \times 0.06 = \text{Kshs } 601.80$$

Total gross tax (PAYE before relief):

$$\text{Gross PAYE} = \text{Tax}_1 + \text{Tax}_2 + \text{Tax}_3 + \text{Tax}_4 + \text{Tax}_5$$ $$\text{Gross PAYE} = 240 + 315 + 420 + 525 + 601.80 = \text{Kshs } 2,101.80$$

Subtract the personal tax relief of Kshs 1,054:

$$\text{Net PAYE} = \text{Gross PAYE} - \text{Personal Tax Relief}$$ $$\text{Net PAYE} = 2,101.80 - 1,054 = \text{Kshs } 1,047.80$$

Emerick's monthly Pay As You Earn (PAYE) is $\boxed{\text{Kshs } 1,047.80}$.

First, calculate the total gross monthly earnings (basic salary + all allowances, including non-taxable).

$$\text{Gross Earnings} = \text{Basic Salary} + \text{House Allowance} + \text{Medical Allowance} + \text{Transport Allowance}$$ $$\text{Gross Earnings} = 38,780 + 10,900 + 3,850 + 5,200 = \text{Kshs } 58,730$$

Next, calculate the total deductions.

$$\text{Total Deductions} = \text{NHIF} + \text{Bank Loan} + \text{WCPS} + \text{Net PAYE}$$ $$\text{Total Deductions} = 3,000 + 7,568 + 780 + 1,047.80 = \text{Kshs } 12,395.80$$

Finally, calculate the net monthly salary.

$$\text{Net Monthly Salary} = \text{Gross Earnings} - \text{Total Deductions}$$ $$\text{Net Monthly Salary} = 58,730 - 12,395.80 = \text{Kshs } 46,334.20$$

Emerick's net monthly salary is $\boxed{\text{Kshs } 46,334.20}$.