Calculate Sarah's monthly income tax based on her salary and given tax bands.

Sarah's monthly taxable income is the sum of her basic salary, house allowance, and commuter allowance.

$$\text{Taxable Income} = \text{Basic Salary} + \text{House Allowance} + \text{Commuter Allowance}$$ $$\text{Taxable Income} = 72940 + 16160 + 8000$$ $$\text{Taxable Income} = 97100 \text{ Ksh.}$$

a) (i) Sarah's monthly taxable income:
The monthly taxable income is $\boxed{97100 \text{ Ksh.}}$.

The tax bands are:

• Up to 13100: 10%
• 13101 - 22600: 15%
• 22601 - 32100: 20%
• 32101 - 41600: 25%
• 41601 and over: 30%

Calculate the tax for each band:

• Band 1 (0 - 13100): Tax = $13100 \times 0.10 = 1310 \text{ Ksh.}$
• Band 2 (13101 - 22600): Width = $22600 - 13100 = 9500 \text{ Ksh.}$ Tax = $9500 \times 0.15 = 1425 \text{ Ksh.}$
• Band 3 (22601 - 32100): Width = $32100 - 22600 = 9500 \text{ Ksh.}$ Tax = $9500 \times 0.20 = 1900 \text{ Ksh.}$
• Band 4 (32101 - 41600): Width = $41600 - 32100 = 9500 \text{ Ksh.}$ Tax = $9500 \times 0.25 = 2375 \text{ Ksh.}$
• Band 5 (41601 and over): Income remaining for this band = $\text{Total Taxable Income} - 41600$ Income remaining = $97100 - 41600 = 55500 \text{ Ksh.}$ Tax = $55500 \times 0.30 = 16650 \text{ Ksh.}$

Total income tax = Sum of tax from all bands

$$\text{Total Income Tax} = 1310 + 1425 + 1900 + 2375 + 16650$$ $$\text{Total Income Tax} = 23660 \text{ Ksh.}$$

a) (ii) Total income tax charged on Sarah's monthly income:
The total income tax charged is $\boxed{23660 \text{ Ksh.}}$.

Given that Sarah's net monthly tax is Ksh. 21260.

$$\text{Net Monthly Tax} = \text{Total Income Tax} - \text{Tax Relief}$$ $$\text{Tax Relief} = \text{Total Income Tax} - \text{Net Monthly Tax}$$ $$\text{Tax Relief} = 23660 - 21260$$ $$\text{Tax Relief} = 2400 \text{ Ksh.}$$

b) Determine the monthly tax relief allowed:
The monthly tax relief allowed is $\boxed{2400 \text{ Ksh.}}$.

The first income tax band is expanded by 20%.
Original first band width = 13100 Ksh.
New first band width = $13100 \times 1.20 = 15720 \text{ Ksh.}$
New Band 1: Up to 15720 (10%)

The next three tax bands retain their size (9500 Ksh. each).
New Band 2: $15721 - (15720 + 9500) = 15721 - 25220$ (15%)
New Band 3: $25221 - (25220 + 9500) = 25221 - 34720$ (20%)
New Band 4: $34721 - (34720 + 9500)