The budget of a factory is distributed among raw materials, wages, and maintenance in the ratio 4:3:3. During a year, the cost of raw materials rises by 12%, wages by 8%, and maintenance falls by 10%. What is the overall percentage change in the total budget?

The correct option is 4.2% increase.

Step 1: Understand the initial ratio and assign values to the components of the budget.

The total budget of the factory is distributed among three components:
- Raw materials
- Wages
- Maintenance

The initial ratio of allocation is given as $4:3:3$.

To simplify calculations, let the initial costs be proportional to these ratio values by multiplying them by 100:

$\text{Cost of Raw Materials}=4\times 100=400$

$\text{Cost of Wages}=3\times 100=300$

$\text{Cost of Maintenance}=3\times 100=300$

Step 2: Calculate the total initial budget.

$\text{Total Initial Budget}=400+300+300=1000$

Step 3: Calculate the change in cost for each component.

1. Raw materials rise by 12%:

$\text{Increase in Raw Materials}=12\%\text{of}400=\frac{12}{100}\times 400=48$

2. Wages rise by 8%:

$\text{Increase in Wages}=8\%\text{of}300=\frac{8}{100}\times 300=24$

3. Maintenance falls by 10%:

$\text{Decrease in Maintenance}=10\%\text{of}300=\frac{10}{100}\times 300=30$

Step 4: Determine the net change in the total budget.

$\text{Net Change}=+48+24-30=42$

Since the net result is positive (+42), it represents an overall increase in the budget.

Step 5: Calculate the percentage change in the total budget.

$\text{Overall Percentage Change}=\left(\frac{\text{Net Change}}{\text{Total Initial Budget}}\right)\times 100$

$\text{Overall Percentage Change}=\left(\frac{42}{1000}\right)\times 100=4.2\%$

Therefore, the overall percentage change in the total budget is a 4.2% increase.