Calculate the profit available for sharing among partners for the first year.

The total profit for the first year is Ksh 4,800,000.
First, 25% of the profit is put back into the business:

$$\text{Amount for business} = 0.25 \times 4,800,000 = \text{Ksh } 1,200,000$$

The remaining profit is:

$$\text{Remaining profit} = 4,800,000 - 1,200,000 = \text{Ksh } 3,600,000$$

Next, 20% of the remaining profit is set aside for taxes and insurance:

$$\text{Amount for taxes/insurance} = 0.20 \times 3,600,000 = \text{Ksh } 720,000$$

The profit available for sharing among partners is:

$$\text{Profit for sharing} = 3,600,000 - 720,000 = \text{Ksh } 2,880,000$$

The partners' contributions are Eliza: Ksh 90,000, Nekesa: Ksh 130,000, and Tanui: Ksh $x$.
The total contribution is $90,000 + 130,000 + x = 220,000 + x$.
Tanui's share of the profit is Ksh 960,000. The profit is shared in the ratio of their contributions.

$$\frac{\text{Tanui's share}}{\text{Profit for sharing}} = \frac{\text{Tanui's contribution}}{\text{Total contribution}}$$ $$\frac{960,000}{2,880,000} = \frac{x}{220,000 + x}$$

Simplify the fraction on the left:

$$\frac{1}{3} = \frac{x}{220,000 + x}$$

Cross-multiply to solve for $x$:

$$220,000 + x = 3x$$ $$220,000 = 2x$$ $$x = \frac{220,000}{2}$$ $$x = \text{Ksh } 110,000$$

a) (i) Tanui invested $\boxed{\text{Ksh } 110,000}$.

The contributions are Eliza: Ksh 90,000, Nekesa: Ksh 130,000, and Tanui: Ksh 110,000.
The ratio is:

$$\text{Eliza} : \text{Nekesa} : \text{Tanui} = 90,000 : 130,000 : 110,000$$

Divide each number by 10,000 to simplify the ratio:

$$9 : 13 : 11$$

a) (ii) The ratio of contributions by the three partners is $\boxed{9:13:11}$.

Nekesa invested an additional Ksh 30,000.
New contributions:

• Eliza: Ksh 90,000
• Nekesa: $130,000 + 30,000 = \text{Ksh } 160,000$
• Tanui: Ksh 110,000 New total contribution:

$$90,000 + 160,000 + 110,000 = \text{Ksh } 360,000$$

The gross profit increased in the ratio 6:5.

$$\text{New gross profit} = \frac{6}{5} \times 4,800,000 = 6 \times 960,000 = \text{Ksh } 5,760,000$$

First, 25% of the new gross profit is put back into the business:

$$\text{Amount for business} = 0.25 \times 5,760,000 = \text{Ksh } 1,440,000$$

The remaining profit is:

$$\text{Remaining profit} = 5,760,000 - 1,440,000 = \text{Ksh } 4,320,000$$

Next, 20% of the remaining profit is set aside for taxes and insurance:

$$\text{Amount for taxes/insurance} = 0.20 \times 4,320,000 = \text{Ksh } 864,000$$

The profit available for sharing among partners is:

$$\text{Profit for sharing} = 4,320,000 - 864,000 = \text{Ksh } 3,456,000$$

Nekesa's contribution is Ksh 160,000. The total contribution is Ksh 360,000.

$$\text{Nekesa's share} = \frac{\text{Nekesa's contribution}}{\text{Total contribution}} \times \text{Profit for sharing}$$ $$\text{Nekesa's share} = \frac{160,000}{360,000} \times 3,456,000$$

Simplify the fraction $\frac{160,000}{360,000} = \frac{16}{36} = \frac{4}{9}$:

$$\text{Nekesa's share} = \frac{4}{9} \times 3,456,000$$ $$\text{Nekesa's share} = 4 \times 384,000$$ $$\text{Nekesa's share} = \text{Ksh } 1,536,000$$

b) Nekesa's share of the profit at the end of the second year is $\boxed{\text{Ksh } 1,536,000}$.