Here are the solutions to the questions: a) How far is it from home to school if you travel through the direct route? b) i) Since you qualify for the bursary, how much will you pay?

The travel route forms a right-angled triangle with legs of 8 km (west) and 15 km (south). The direct route is the hypotenuse.

Let $d$ be the direct distance.

b) i) Since you qualify for the bursary, how much will you pay?

Original school fees: shs 900,000
Bursary discount: 60% of school fees
Discount amount = $0.60 \times 900,000 \text{ shs} = 540,000 \text{ shs}$
School fees after discount = $900,000 \text{ shs} - 540,000 \text{ shs} = 360,000 \text{ shs}$

Original admission fees: shs 100,000
Bursary benefit: Free admission
Admission fees after bursary = $0 \text{ shs}$

Original uniform fees: shs 350,000
Bursary discount: shs 87,500
Uniform fees after discount = $350,000 \text{ shs} - 87,500 \text{ shs} = 262,500 \text{ shs}$

Total amount = School fees (after discount) + Admission fees (after discount) + Uniform fees (after discount)
Total amount = $360,000 \text{ shs} + 0 \text{ shs} + 262,500 \text{ shs}$
Total amount = $\boxed{\text{shs 622,500}}$

b) ii) Will your guardian afford the school expenses according to her budget? Justify your answer.

Guardian's budget = shs 700,000
Total expenses with bursary = shs 622,500

Since $622,500 \text{ shs} < 700,000 \text{ shs}$, the guardian will afford the school expenses.
Justification: The total cost of shs 622,500 is less than the guardian's budget of shs 700,000.

c) How much will those who are to pay school fees of shs 900,000 pay per installment, according to your plans?

Plan 1: Two installments

First installment = $\frac{11}{15} \times 900,000 \text{ shs}$
First installment = $11 \times 60,000 \text{ shs}$
First installment = $\boxed{\text{shs 660,000}}$

Second installment = $900,000 \text{ shs} - 660,000 \text{ shs}$
Second installment = $\boxed{\text{shs 240,000}}$

Plan 2: Three equal installments

Each installment = $\frac{900,000 \text{ shs}}{3}$
Each installment = $\boxed{\text{shs 300,000}}$

d) Which payment plan would you recommend for them and why?

I would recommend Plan 2.
This plan requires three equal installments of shs 300,000 each, spread across the term (beginning, visitation, and end of term). This approach breaks down the total cost into smaller, more manageable payments, which is generally easier for budgeting and cash flow compared to Plan 1's large initial payment of shs 660,000.