Question : 4 men and 8 women complete a job in 10 days and 5 men and 24 women complete the same work in 4 days. In how many days will 1 man and 1 woman complete the same work?
Option 1: $63 \frac{1}{3}$ days
Option 2: $69 \frac{7}{9}$ days
Option 3: $67 \frac{1}{3}$ days
Option 4: $62 \frac{2}{9}$ days
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Correct Answer: $62 \frac{2}{9}$ days
Solution :
Let 1 man's and 1 woman's 1 day's work be $x$ and $y$ respectively.
4 men and 8 women complete a job in 10 days.
Then $4x + 8y = \frac{1}{10}$...........(i)
5 men and 24 women complete the same work in 4 days.
Then $5x + 24y = \frac{1}{4}$...............(ii)
Multiplying equation (i) by 5 and equation (ii) by 4, we get,
$20x + 40y = \frac{1}{2}$..............(iii)
$20x + 96y = 1$.............(iv)
Subtracting (iii) from (iv)
⇒ $56y = \frac{1}{2}$
⇒ $y = frac{1}{112}$
Putting the value of y in equation (iii) we get,
$20x + \frac{5}{14} =\frac{1}{2}$
⇒ $20x = \frac{1}{2} -\frac{5}{14}$
⇒ $20x = \frac{2}{14}$
⇒ $x = \frac{1}{140}$
1 man and 1 woman in 1 day can complete $= \frac{1}{112} + \frac{1}{140} = \frac{5+4}{560} = \frac{9}{560}$ th of the work
$\therefore$ 1 man and 1 woman can complete the work in $\frac{560}{9} = 62\frac{2}{9}$ days
Hence, the correct answer is $62 \frac{2}{9}$ days.
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