KCSE 2025 Maths P2 Q19 — Income Tax (PAYE), Tax Bands & Relief

KCSE 2025 Form 4 Commercial Arithmetic

Published

The Question

“The monthly income tax rates for a certain year were: 0 to 24,000 taxed at 10%, 24,001 to 32,333 at 25%, 32,334 to 500,000 at 30%, 500,001 to 800,000 at 32.5%, and over 800,000 at 35%. In March that year Judy's monthly earnings were a basic salary of Ksh 104,644, a house allowance of Ksh 25,000 and a commuter allowance of Ksh 12,000, and she was entitled to a monthly tax relief of Ksh 2,400. (a)(i) Calculate Judy's taxable income that month. (a)(ii) Calculate the tax payable by Judy that month. (b) In July that year Judy's basic salary changed while her allowances and tax relief stayed the same, and her net tax that month was Ksh 39,483.35. Calculate her new basic salary.”

1

Find the taxable income (part a i)

Taxable income is the basic salary plus every allowance, because allowances are treated as income by the taxman. Add the basic salary, the house allowance and the commuter allowance together to get the figure that the tax bands are applied to.

Taxable income=104,644+25,000+12,000\text{Taxable income} = 104{,}644 + 25{,}000 + 12{,}000
=Ksh 141,644= \text{Ksh } 141{,}644
2

Tax the first two bands

Income is taxed slab by slab, not all at one rate. The whole of the first band, the first 24,000, is charged at 10%. The second band runs from 24,001 to 32,333, so its width is 32,333 minus 24,000, which is 8,333, and that width is charged at 25%. Work out the tax in each of these two bands.

Band 1: 24,000×10%=2,400\text{Band 1: } 24{,}000 \times 10\% = 2{,}400
Band 2: (32,333−24,000)×25%=8,333×25%=2,083.25\text{Band 2: } (32{,}333 - 24{,}000) \times 25\% = 8{,}333 \times 25\% = 2{,}083.25
3

Tax the top band and total the gross tax

Everything above 32,333 falls in the 30% band, since her income of 141,644 is well below the 500,000 ceiling of that band. Subtract 32,333 from the taxable income to get the amount taxed at 30%, work out that tax, then add all three band amounts to get the gross tax before relief.

Band 3: (141,644−32,333)×30%=109,311×30%=32,793.30\text{Band 3: } (141{,}644 - 32{,}333) \times 30\% = 109{,}311 \times 30\% = 32{,}793.30
Gross tax=2,400+2,083.25+32,793.30=37,276.55\text{Gross tax} = 2{,}400 + 2{,}083.25 + 32{,}793.30 = 37{,}276.55
4

Subtract the relief for tax payable (part a ii)

Tax relief is a fixed amount removed from the gross tax to give the actual tax handed over. Subtract Judy's monthly relief of 2,400 from the gross tax to get the tax payable that month.

Tax payable=37,276.55−2,400=Ksh 34,876.55\text{Tax payable} = 37{,}276.55 - 2{,}400 = \text{Ksh } 34{,}876.55
5

Work backwards from July's net tax (part b)

In July we are given the net tax, which is the tax after relief, so first add the 2,400 relief back to recover the gross tax. The first two bands are always fully used and give a fixed amount of 2,400 plus 2,083.25, which is 4,483.25. Whatever gross tax is left over that fixed part was charged at 30%, so subtract the fixed part and divide by 0.3 to find how much income sat in the 30% band.

Gross tax=39,483.35+2,400=41,883.35\text{Gross tax} = 39{,}483.35 + 2{,}400 = 41{,}883.35
Fixed (bands 1 and 2)=2,400+2,083.25=4,483.25\text{Fixed (bands 1 and 2)} = 2{,}400 + 2{,}083.25 = 4{,}483.25
Income in 30% band=41,883.35−4,483.250.3=37,400.100.3=124,667\text{Income in 30\% band} = \frac{41{,}883.35 - 4{,}483.25}{0.3} = \frac{37{,}400.10}{0.3} = 124{,}667
6

Rebuild the taxable income and the basic salary

Add back the 32,333 that was already taxed in the lower two bands to get the new taxable income. Then reverse the very first step: the allowances of 25,000 and 12,000 total 37,000, so subtract them from the taxable income to leave the new basic salary.

New taxable income=124,667+32,333=157,000\text{New taxable income} = 124{,}667 + 32{,}333 = 157{,}000
New basic salary=157,000−(25,000+12,000)=Ksh 120,000\text{New basic salary} = 157{,}000 - (25{,}000 + 12{,}000) = \text{Ksh } 120{,}000

Final Result

Judy's taxable income in March was Ksh 141,644, and the tax payable that month was Ksh 34,876.55. In July her new basic salary was Ksh 120,000.

Why this method works

PAYE is a slab (marginal) system, so each portion of income is taxed only at the rate for the band it falls in rather than one flat rate on the whole amount. That is why you build the tax up band by band, and it is also why the first two bands always contribute the same fixed amount once income is large enough to fill them. Relief is just a flat rebate on the gross tax, so it can be added back to reverse the process. Because the only variable part is the 30% band, dividing the leftover tax by 0.3 recovers exactly the income sitting in that band, and adding back the lower bands and stripping out the allowances walks the calculation all the way back to the basic salary.

✓

Check July forward: taxable income 157,000 gives band 3 tax of (157,000 - 32,333) x 30% = 124,667 x 30% = 37,400.10; gross tax = 4,483.25 + 37,400.10 = 41,883.35; net tax = 41,883.35 - 2,400 = 39,483.35, which matches.