Question ID: #802
A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:
- (1) 24 days
- (2) 42 days
- (3) 30 days
- (4) 36 days
Solution:
Let time taken by Mason A alone $= x$ days.
Then time taken by Mason B alone $= x + 24$ days (Since A takes 24 days less).
Work done by A in 1 day $= \frac{1}{x}$.
Work done by B in 1 day $= \frac{1}{x+24}$.
Together they finish in $22.5 = \frac{45}{2}$ days.
Combined work in 1 day $= \frac{1}{22.5} = \frac{2}{45}$.
Equation:
$$ \frac{1}{x} + \frac{1}{x+24} = \frac{2}{45} $$
$$ \frac{x + 24 + x}{x(x+24)} = \frac{2}{45} $$
$$ \frac{2x + 24}{x^2 + 24x} = \frac{2}{45} $$
Divide numerator by 2:
$$ \frac{x + 12}{x^2 + 24x} = \frac{1}{45} $$
Cross multiply:
$$ 45(x + 12) = x^2 + 24x $$
$$ 45x + 540 = x^2 + 24x $$
$$ x^2 – 21x – 540 = 0 $$
Factorize the quadratic equation:
Find factors of $-540$ that sum to $-21$.
$36 \times 15 = 540$.
$(x – 36)(x + 15) = 0$.
$$ x = 36 \quad \text{or} \quad x = -15 $$
Since time cannot be negative, $x = 36$.
Ans. (4)
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