Let's analyze and solve the questions step-by-step.

Problem Overview:

1. Initial Charge: £4
2. Cost per Kilometer: £2 per kilometer
3. Distance from Shop to Chloe's House: 4 km
4. Distance from Chloe's House to Nadia's House: 12 km (making the total journey 16 km)

Part (a) - Chloe's Proposal:

Chloe suggests splitting the total cost in proportion to the distances traveled by each person.

Total Cost:

$\text{Total cost} = 4 \, (\text{initial charge}) + 2 \times 16 \, (\text{total kilometers}) = £36.$

Distances Traveled:

• Chloe travels 4 km.
• Nadia travels 12 km.
• Total distance traveled = $4 + 12 = 16 \, \text{km}.$

Proportional Split:

The proportion of cost is based on distance traveled:

• Chloe's share: $\frac{4}{16} = \frac{1}{4}$.
• Nadia's share: $\frac{12}{16} = \frac{3}{4}$.

$\text{Chloe's cost} = \frac{1}{4} \times 36 = £9.$ $\text{Nadia's cost} = \frac{3}{4} \times 36 = £27.$

Part (b) - Nadia's Proposal:

Nadia suggests splitting the initial charge equally, then dividing the rest of the cost based on distance traveled.

Step 1: Split the Initial Charge:

The initial charge (£4) is divided equally: $\text{Chloe's share of initial charge} = £2, \quad \text{Nadia's share of initial charge} = £2.$

Step 2: Split Remaining Cost:

The remaining cost is: $36 - 4 = £32.$ This amount is split proportionally based on distance:

• Chloe's share: $\frac{1}{4} \times 32 = £8$.
• Nadia's share: $\frac{3}{4} \times 32 = £24$.

Total Costs:

• Chloe: $2 + 8 = £10$.
• Nadia: $2 + 24 = £26$.

Part (c) - Chloe's Argument:

Chloe claims she should pay nothing because the taxi was passing her house anyway. Do we agree?

No, this argument is not valid. Chloe still benefits from the journey from the shop to her house, which incurs costs that need to be shared. It would be unfair to exempt her from paying for her portion of the journey.

Final Summary:

1. Part (a): Chloe pays £9, Nadia pays £27.
2. Part (b): Chloe pays £10, Nadia pays £26.
3. Part (c): Chloe's argument is invalid.

Do you want further details on any part of the solution?

5 Related Questions:

1. How would the costs change if the cost per kilometer was increased to £3?
2. What would happen if Chloe and Nadia agreed to split the entire cost equally instead of proportionally?
3. How can the concept of weighted averages apply to proportional cost-sharing?
4. What if Nadia stopped halfway to her house? How would the costs be divided?
5. How do proportional splits relate to real-life fairness in shared expenses?

Tip:

When dividing costs, ensure all participants agree on a method before the journey starts to avoid disagreements later!