A project's initial cost of $64,000 is split between labour, materials, and contractor's charges in the ratio 12:15:5. Labour costs increase by x%, materials by 2x%, and contractor's charges remain th

The total estimated cost is \64,000.00$. The ratio of labour : materials : contractor's charges is$12 : 15 : 5$. The sum of the ratio parts is$12 + 15 + 5 = 32$.

Initial cost of labour (L): $L = \frac{12}{32} \times \$64,000 = 12 \times \$2,000 = \$24,000$

Initial cost of materials (M): $M = \frac{15}{32} \times \$64,000 = 15 \times \$2,000 = \$30,000$

Initial contractor's charges (C): $C = \frac{5}{32} \times \$64,000 = 5 \times \$2,000 = \$10,000$

The cost of labour increased by $x \%$.
New labour cost (L'): $L' = L \left(1 + \frac{x}{100}\right) = \$24,000 \left(1 + \frac{x}{100}\right)$

The cost of materials increased by $2x \%$.
New materials cost (M'): $M' = M \left(1 + \frac{2x}{100}\right) = \$30,000 \left(1 + \frac{2x}{100}\right)$

The contractor's charges remained the same.
New contractor's charges (C'): $C' = C = \$10,000$

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

$$L' = \frac{2}{3} M'$$

Substitute the expressions for L' and M':

$$24,000 \left(1 + \frac{x}{100}\right) = \frac{2}{3} \times 30,000 \left(1 + \frac{2x}{100}\right)$$ $$24,000 \left(1 + \frac{x}{100}\right) = 20,000 \left(1 + \frac{2x}{100}\right)$$

Divide both sides by 1,000:

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

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 the denominators:

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

Rearrange and solve for $x$:

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

a) The value of $x$:

$$\boxed{x = 25}$$

First, calculate the new individual costs using $x=25$.
New labour cost (L'): $L' = \$24,000 \left(1 + \frac{25}{100}\right) = \$24,000 \times (1 + 0.25) = \$24,000 \times 1.25 = \$30,000$ New materials cost (M'): $M' = \$30,000 \left(1 + \frac{2 \times 25}{100}\right) = \$30,000 \times (1 + 0.50) = \$30,000 \times 1.50 = \$45,000$ New contractor's charges (C'): $C' = \$10,000$ The new total cost of building the restaurant is the sum of the new labour, materials, and contractor's charges:

$$\text{New Total Cost} = L' + M' + C'$$

$\text{New Total Cost} = \$30,000 + \$45,000 + \$10,000$ $\text{New Total Cost} = \$85,000$

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