Let t = time taken by the apprentice to paint the room alone.

It takes the man an hour less than the time the apprentice takes to paint the same room.

Man's time = t - 1

Fraction of the time taken by the apprentice in an hour = $\frac{1}{\mathrm{t}}$

Fraction of the time taken by the man in an hour = $\frac{1}{\mathrm{t-1}}$

Time by both = 72 minutes = $\frac{72}{60}$ hours = 1.2 hours

Fraction of the time in an hour by both = $\frac{1}{1.2}$

$\frac{1}{\mathrm{t}}$ + $\frac{1}{\mathrm{t-1}}$ = $\frac{1}{1.2}$

Solve for the time t taken by the apprentice to paint the room alone.

Multiply both sides by the L.C.M of the denominators t, t - 1 and 1.2. The L.C.M will be t x (t - 1) x 1.2

t x (t - 1) x 1.2 x $\frac{1}{\mathrm{t}}$ + t x (t - 1) x 1.2 x $\frac{1}{\mathrm{t-1}}$ = t x (t - 1) x 1.2 x $\frac{1}{1.2}$

(t - 1) x 1.2 x 1 + t x 1.2 x 1 = t x (t - 1) x 1

1.2t - 1.2 + 1.2t = t² - t

2.4t - 1.2 = t² - t

t² - t - 2.4t + 1.2 = 0

t² - 3.4t + 1.2 = 0

Multiply each sides by 5 to make the decimals whole numbers.

5 x t² - 3.4t x 5 + 1.2 x 5 = 0 x 5

5t² - 17t + 6 = 0

Split the b (- 17) such that two numbers when you multiply you will get (a x c = 5 x 6 = 30) and when you add you will get b (-17)

The numbers are -15 and -2

5t² - 15t - 2t + 6 = 0

5t(t - 3) - 2(t - 3) = 0

(t - 3)(5t - 2) = 0

When t - 3 = 0, t = 3

When 5t - 2 = 0, t = $\frac{2}{5}$ = 0.4

Since the man's time is an hour less than the apprentice, the time is greater than 1 because if less than 1, the man's time (t - 1) will be negative.

Hence t = 3 hours

∴ It takes 3 hours for the apprentice to paint the room alone.

Total expenses = $\frac{1}{5}$ + $\frac{1}{3}$ + $\frac{1}{8}$

Total expenses = $\frac{\mathrm{24\; x\; 1\; +\; 40\; x\; 1\; +\; 15\; x\; 1}}{120}$

Total expenses = $\frac{\mathrm{24\; +\; 40\; +\; 15}}{120}$

Total expenses = $\frac{79}{120}$

The whole (annual salary) is 1

Fraction left = 1 - $\frac{79}{120}$

Every number is being divided by 1

1 = $\frac{1}{1}$

Fraction left = $\frac{1}{1}$ - $\frac{79}{120}$

Fraction left = $\frac{\mathrm{120\; x\; 1\; -\; 1\; x\; 79}}{120}$

Fraction left = $\frac{\mathrm{120\; -\; 79}}{120}$

Fraction left = $\frac{41}{120}$

Amount left = GH₵ 4100

The whole (fraction) for the annual salary is 1.

If $\frac{41}{120}$ = GH₵ 4100

1 = (GH₵ 4100 x 1) ÷ $\frac{41}{120}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

1 = (GH₵ 4100 x 1) x $\frac{120}{41}$ = GH₵ 100 x 120 = GH₵ 12,000

∴ He earns GH₵ 12,000 per annum.

Method I

Using proportionality

Total expenses = $\frac{1}{6}$ + $\frac{1}{3}$ + $\frac{1}{4}$

Total expenses = $\frac{\mathrm{2\; x\; 1\; +\; 4\; x\; 1\; +\; 3\; x\; 1}}{12}$

Total expenses = $\frac{\mathrm{2\; +\; 4\; +\; 3}}{12}$

Total expenses = $\frac{9}{12}$ = $\frac{3}{4}$

The fraction of the whole salary is 1

Remaining fraction = 1 - $\frac{3}{4}$

Every number is being divided by 1

1 = $\frac{1}{1}$

Remaining fraction = $\frac{1}{1}$ - $\frac{3}{4}$

Remaining fraction = $\frac{\mathrm{4\; x\; 1\; -\; 1\; x\; 3}}{4}$

Remaining fraction = $\frac{\mathrm{4\; -\; 3}}{4}$

Remaining fraction = $\frac{1}{4}$

Amount left = GH₵ 285

Using proportionality

1 represents the whole salary

$\frac{1}{4}$ represents the amount left

If $\frac{1}{4}$ = 285

1 = 1 x 285 ÷ $\frac{1}{4}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

1 = 1 x 285 x $\frac{4}{1}$ = 1,140

∴ the woman's monthly salary is GH₵ 1,140

Method II

Using a variable to represent the salary

Total expenses = $\frac{1}{6}$ + $\frac{1}{3}$ + $\frac{1}{4}$

Total expenses = $\frac{\mathrm{2\; x\; 1\; +\; 4\; x\; 1\; +\; 3\; x\; 1}}{12}$

Total expenses = $\frac{\mathrm{2\; +\; 4\; +\; 3}}{12}$

Total expenses = $\frac{9}{12}$ = $\frac{3}{4}$

She spent $\frac{3}{4}$ of the salary

Let y = salary

of means multiplication.

Amount spent = $\frac{3}{4}$ x y = $\frac{\mathrm{3y}}{4}$

Amount left = Salary - Amount spent

Amount left = y - $\frac{\mathrm{3y}}{4}$

But amount left = GH₵ 285

y - $\frac{\mathrm{3y}}{4}$ = 285

Get rid of the fraction by multiplying both sides by the denominator, 4

4 x y - $\frac{\mathrm{3y}}{4}$ x 4 = 285 x 4

4y - 3y = 285 x 4

y = 1,140

∴ the woman's monthly salary is GH₵ 1,140