The total initial cost of labour, materials, and contractor's charges is $64,000.00. The ratio of labour, materials, and contractor's charges is 12:15:5. The cost of labour increased by x%. The cost o

The total initial cost is $64,000.00. The ratio of labour, materials, and contractor's charges is$12 : 15 : 5$. The total number of ratio parts is$12 + 15 + 5 = 32$.

Initial cost of labour ($L_0$):

$$L_0 = \frac{12}{32} \times 64000 = 12 \times 2000 = 24000$$

Initial cost of materials ($M_0$):

$$M_0 = \frac{15}{32} \times 64000 = 15 \times 2000 = 30000$$

Initial contractor's charges ($C_0$):

$$C_0 = \frac{5}{32} \times 64000 = 5 \times 2000 = 10000$$

The cost of labour increased by $x \%$. The new cost of labour ($L_1$) is:

$$L_1 = L_0 \left(1 + \frac{x}{100}\right) = 24000 \left(1 + \frac{x}{100}\right)$$

The cost of materials increased by $2x \%$. The new cost of materials ($M_1$) is:

$$M_1 = M_0 \left(1 + \frac{2x}{100}\right) = 30000 \left(1 + \frac{2x}{100}\right)$$

The contractor's charges ($C_1$) remained the same:

$$C_1 = C_0 = 10000$$

After the increase, the cost of labour was two-thirds the cost of materials:

$$L_1 = \frac{2}{3} M_1$$

Substitute the expressions for $L_1$ and $M_1$:

$$24000 \left(1 + \frac{x}{100}\right) = \frac{2}{3} \times 30000 \left(1 + \frac{2x}{100}\right)$$ $$24000 \left(1 + \frac{x}{100}\right) = 20000 \left(1 + \frac{2x}{100}\right)$$

Divide both sides by 1000:

$$24 \left(1 + \frac{x}{100}\right) = 20 \left(1 + \frac{2x}{100}\right)$$

Divide both sides by 4:

$$6 \left(1 + \frac{x}{100}\right) = 5 \left(1 + \frac{2x}{100}\right)$$

Expand the equation:

$$6 + \frac{6x}{100} = 5 + \frac{10x}{100}$$

Multiply by 100 to clear denominators:

$$600 + 6x = 500 + 10x$$

Rearrange terms to solve for $x$:

$$600 - 500 = 10x - 6x$$ $$100 = 4x$$ $$x = \frac{100}{4}$$ $$x = 25$$

a) The value of $x$:

$$\boxed{\text{25}}$$

Substitute $x=25$ into the expressions for $L_1$ and $M_1$:

$$L_1 = 24000 \left(1 + \frac{25}{100}\right) = 24000 (1 + 0.25) = 24000 \times 1.25 = 30000$$ $$M_1 = 30000 \left(1 + \frac{2 \times 25}{100}\right) = 30000 \left(1 + \frac{50}{100}\right) = 30000 (1 + 0.5) = 30000 \times 1.5 = 45000$$

The contractor's charges remain $C_1 = 10000$.
The new total cost of building the restaurant is the sum of the new costs:

$$\text{New total cost} = L_1 + M_1 + C_1 = 30000 + 45000 + 10000 = 85000$$

b) The new cost of building the restaurant: $\boxed{\text{\$85,000.00}}$