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.