Step-by-Step Solution

Step-by-Step Solution

Step 1: Define variables and formulate rate equation

Let $x$ be the number of days Abdul Hadi takes alone ($x > 0$).

• Abdul Hadi's rate per day $= \dfrac{1}{x}$
• Abdullah's rate per day $= \dfrac{1}{x + 10}$
• Combined rate per day $= \dfrac{1}{12}$

$\frac{1}{x} + \frac{1}{x + 10} = \frac{1}{12}$

Step 2: Simplify and form the quadratic equation

Combine fractions on the left: $\frac{(x + 10) + x}{x(x + 10)} = \frac{1}{12}$

$\frac{2x + 10}{x^2 + 10x} = \frac{1}{12}$

Cross-multiply: $x^2 + 10x = 12(2x + 10)$

$x^2 + 10x = 24x + 120$

$x^2 - 14x - 120 = 0$

Step 3: Factorize for $x$

$(x - 20)(x + 6) = 0$

$x = 20 \quad \text{or} \quad x = -6$

Since time cannot be negative ($x > 0$), we reject $x = -6$.

Step 4: Check solution

• Abdul Hadi's daily rate $= \dfrac{1}{20}$
• Abdullah's daily rate $= \dfrac{1}{30}$
• Combined daily rate $= \dfrac{1}{20} + \dfrac{1}{30} = \dfrac{3 + 2}{60} = \dfrac{5}{60} = \dfrac{1}{12}$ $\checkmark$

$\boxed{\text{Abdul Hadi takes } 20 \text{ days to finish the work alone}}$