Fraction Operations

The painter uses $\frac{2}{5}$ of the $40$ liters for the exterior.

$\text{Paint used for exterior}=\frac{2}{5}\times 40=2\times \frac{40}{5}=2\times 8=16$ liters.

$\text{Remaining paint}=\text{Total paint}-\text{Paint used for exterior}$

$\text{Remaining paint}=40-16=24$ liters.

(Alternatively, if $\frac{2}{5}$ was used, $1-\frac{2}{5}=\frac{3}{5}$ remains. So, $\frac{3}{5}\times 40=3\times 8=24$ liters.)

Of the remaining $24$ liters, he uses $\frac{3}{4}$ for the interior.

$\text{Paint used for interior}=\frac{3}{4}\times 24=3\times \frac{24}{4}=3\times 6=18$ liters.

$\text{Paint left over}=\text{Remaining paint (after exterior)}-\text{Paint used for interior}$

$\text{Paint left over}=24-18=6$ liters.

So, $6$ liters of paint are left over.

$3\frac{1}{2}-1\frac{3}{4}$

First, find a common denominator for the fractions. The common denominator for 2 and 4 is 4.

$3\frac{1}{2}=3\frac{2}{4}$

So, $3\frac{2}{4}-1\frac{3}{4}$.

Since $\frac{2}{4}$ is less than $\frac{3}{4}$, we need to borrow from the whole number part of $3\frac{2}{4}$.

$3\frac{2}{4}=2+1+\frac{2}{4}=2+\frac{4}{4}+\frac{2}{4}=2\frac{6}{4}$

Now, $2\frac{6}{4}-1\frac{3}{4}=(2-1)+(\frac{6}{4}-\frac{3}{4})=1+\frac{3}{4}=1\frac{3}{4}$.

Convert $1\frac{3}{4}$ to an improper fraction: $\frac{(1\times 4)+3}{4}=\frac{7}{4}$.

The expression now becomes: $\frac{7}{4}÷\frac{7}{8}+\frac{1}{3}\times \frac{9}{10}$.

First, the division: $\frac{7}{4}÷\frac{7}{8}$

To divide fractions, multiply by the reciprocal of the second fraction:

$\frac{7}{4}\times \frac{8}{7}$

Cancel out the 7s and simplify $\frac{8}{4}$ to 2:

$1\times 2=2$.

Next, the multiplication: $\frac{1}{3}\times \frac{9}{10}$

Multiply the numerators and the denominators:

$\frac{1\times 9}{3\times 10}=\frac{9}{30}$

Simplify the fraction by dividing both numerator and denominator by 3:

$\frac{9÷3}{30÷3}=\frac{3}{10}$.

The expression now becomes: $2+\frac{3}{10}$.

$2+\frac{3}{10}=2\frac{3}{10}$.

So, the value of the expression is $2\frac{3}{10}$.

Total paint available: $12\frac{1}{2}=\frac{(12\times 2)+1}{2}=\frac{25}{2}$ gallons.

Paint required per room: $1\frac{3}{4}=\frac{(1\times 4)+3}{4}=\frac{7}{4}$ gallons.

Divide the total paint by the paint required per room:

$\text{Number of rooms}=\frac{25}{2}÷\frac{7}{4}$

To divide by a fraction, multiply by its reciprocal:

$\frac{25}{2}\times \frac{4}{7}=\frac{25\times 4}{2\times 7}=\frac{100}{14}$

Simplify the fraction: $\frac{100}{14}=\frac{50}{7}$.

Convert to a mixed number: $\frac{50}{7}=7\frac{1}{7}$.

Since the painter can only paint full rooms, the maximum number of full rooms is $7$.

First, find out how much paint is used for $7$ full rooms:

$\text{Paint used}=7\times 1\frac{3}{4}=7\times \frac{7}{4}=\frac{49}{4}$ gallons.

Convert $\frac{49}{4}$ to a mixed number: $\frac{49}{4}=12\frac{1}{4}$ gallons.

Now, subtract the paint used from the total paint available:

$\text{Paint left over}=\text{Total paint}-\text{Paint used}$

$\text{Paint left over}=12\frac{1}{2}-12\frac{1}{4}$

Find a common denominator for the fractions ($\frac{1}{2}$ and $\frac{1}{4}$), which is 4:

$12\frac{2}{4}-12\frac{1}{4}$

$=(12-12)+(\frac{2}{4}-\frac{1}{4})=0+\frac{1}{4}=\frac{1}{4}$ gallon.

So, the painter can paint $7$ full rooms, and $\frac{1}{4}$ gallon of paint will be left over.