Work and Time Problem: Calculation Steps
This problem involves calculating the time required for a specific group to complete a work based on the combined efforts of men and women.
Setting Up Work Rate Equations
Let $W$ represent the work rate of one woman (amount of work done per day) and $M$ represent the work rate of one man.
The total work is assumed to be 1 unit. The formula used is Work = Rate × Time.
- From the first condition: 6 women and 4 men complete the work in 10 days. $ (6W + 4M) \times 10 = 1 $ $ 60W + 40M = 1 \quad \quad (1) $
- From the second condition: 4 women and 5 men complete the work in 10 days. $ (4W + 5M) \times 10 = 1 $ $ 40W + 50M = 1 \quad \quad (2) $
Solving for Individual Work Rates
We need to solve the system of linear equations (1) and (2) to find the values of $W$ and $M$.
- Multiply equation (1) by 5 and equation (2) by 4 to make the coefficient of $M$ the same: $ 5 \times (60W + 40M = 1) \implies 300W + 200M = 5 \quad \quad (1') $ $ 4 \times (40W + 50M = 1) \implies 160W + 200M = 4 \quad \quad (2') $
- Subtract equation (2') from equation (1'): $ (300W - 160W) + (200M - 200M) = 5 - 4 $ $ 140W = 1 $ $ W = \frac{1}{140} $ So, one woman completes $ \frac{1}{140} $ of the work per day.
- Substitute the value of $W$ back into equation (1): $ 60 \left( \frac{1}{140} \right) + 40M = 1 $ $ \frac{60}{140} + 40M = 1 $ $ \frac{3}{7} + 40M = 1 $ $ 40M = 1 - \frac{3}{7} $ $ 40M = \frac{4}{7} $ $ M = \frac{4}{7 \times 40} $ $ M = \frac{1}{70} $ So, one man completes $ \frac{1}{70} $ of the work per day.
Calculating Time for 8 Women
Now, we need to find the time it takes for 8 women to complete the work.
- Calculate the combined work rate of 8 women: $ \text{Rate of 8 women} = 8 \times W = 8 \times \frac{1}{140} = \frac{8}{140} = \frac{2}{35} $ The rate of 8 women is $ \frac{2}{35} $ of the work per day.
- Calculate the time required using the formula Time = Work / Rate: $ \text{Time} = \frac{1}{\text{Rate of 8 women}} = \frac{1}{\frac{2}{35}} = \frac{35}{2} $ $ \text{Time} = 17.5 \text{ days} $
Therefore, 8 women will complete the work in 17.5 days.
Similar Questions
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Important Questions from Time and Work
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Had been one menless, then the number of days required to do a piece of work would have been one more. If the number of Man. Days required to complete the work is 56, how many workers were there?
A can do a piece of work in 16 hours, B and C can do it in 8 hours while A and C can do it 12 hours. How long will B alone take to do it?
- lf 12 men can do a work in 20 days, in how many days will the work be done by 15 men-
