If 6 men and 8 boys can do a piece of work in 10 days, while 26 men and 48 boys can do the same work in 2 days, then the time taken by 10 men and 20 boys for doing the same piece of work will be:
5 days
Work and Time Problem Analysis
This question is about solving a classic work and time problem. It involves determining the time required for a specific group of workers (men and boys) to complete a task, given the time taken by different combinations of these workers. The core idea is to find the individual work rate of a man and a boy and then calculate their combined efficiency for the new group.
Mathematical Formulation of Work Rates
To solve this, we first assign variables to represent the work rate of each individual.
- Let \(M\) denote the amount of work one man can complete in one day.
- Let \(B\) denote the amount of work one boy can complete in one day.
We will assume the total piece of work to be completed is equivalent to 1 unit.
Deriving Equations from Given Information
Condition 1: 6 men and 8 boys can do the work in 10 days.
If 6 men and 8 boys complete the work in 10 days, their combined work rate per day is:
\( \text{Work Rate}_1 = \frac{\text{Total Work}}{\text{Time}} = \frac{1}{10} \)
This can be expressed in terms of \(M\) and \(B\) as:
\( 6M + 8B = \frac{1}{10} \quad (\text{Equation } 1) \)
Condition 2: 26 men and 48 boys can do the same work in 2 days.
Similarly, their combined work rate per day is:
\( \text{Work Rate}_2 = \frac{\text{Total Work}}{\text{Time}} = \frac{1}{2} \)
This gives us the second equation:
\( 26M + 48B = \frac{1}{2} \quad (\text{Equation } 2) \)
Solving the System of Equations for Individual Rates
We now have a system of two linear equations with two variables, \(M\) and \(B\):
- \(6M + 8B = \frac{1}{10}\)
- \(26M + 48B = \frac{1}{2}\)
We can use the method of elimination to solve for \(M\) and \(B\). Let's aim to eliminate \(B\). We can multiply Equation 1 by 6 to make the coefficient of \(B\) equal to 48, matching Equation 2.
Multiplying Equation 1 by 6:
\( 6 \times (6M + 8B) = 6 \times \frac{1}{10} \)
\( 36M + 48B = \frac{6}{10} = \frac{3}{5} \quad (\text{Equation } 3) \)
Now, subtract Equation 3 from Equation 2:
\( (26M + 48B) - (36M + 48B) = \frac{1}{2} - \frac{3}{5} \)
\( 26M - 36M = \frac{5}{10} - \frac{6}{10} \)
\( -10M = -\frac{1}{10} \)
Dividing both sides by -10:
\( M = \frac{-1/10}{-10} = \frac{1}{100} \)
So, one man completes \(\frac{1}{100}\) of the work in one day.
Now substitute the value of \(M\) back into Equation 1 to find the value of \(B\):
\( 6\left(\frac{1}{100}\right) + 8B = \frac{1}{10} \)
\( \frac{6}{100} + 8B = \frac{1}{10} \)
Isolate \(8B\):
\( 8B = \frac{1}{10} - \frac{6}{100} \)
Find a common denominator (100):
\( 8B = \frac{10}{100} - \frac{6}{100} \)
\( 8B = \frac{4}{100} \)
Solve for \(B\):
\( B = \frac{4}{100 \times 8} = \frac{4}{800} = \frac{1}{200} \)
Thus, one boy completes \(\frac{1}{200}\) of the work in one day.
Calculating the Combined Work Rate for the New Group
The question asks for the time taken by 10 men and 20 boys to complete the same work.
First, we calculate the combined work rate of this group in one day:
Work done by 10 men in 1 day = \(10 \times M = 10 \times \frac{1}{100} = \frac{10}{100} = \frac{1}{10}\)
Work done by 20 boys in 1 day = \(20 \times B = 20 \times \frac{1}{200} = \frac{20}{200} = \frac{1}{10}\)
The total work done by 10 men and 20 boys in one day is the sum of their individual contributions:
\( \text{Combined Rate} = (\text{Work by 10 men}) + (\text{Work by 20 boys}) \)
\( \text{Combined Rate} = \frac{1}{10} + \frac{1}{10} \)
\( \text{Combined Rate} = \frac{2}{10} = \frac{1}{5} \)
This signifies that 10 men and 20 boys together can complete \(\frac{1}{5}\) of the total work in a single day.
Final Time Calculation
The relationship between total work, work rate, and time is:
\( \text{Total Work} = \text{Combined Rate} \times \text{Time} \)
Since the total work is 1 unit, and the combined rate is \(\frac{1}{5}\) units per day, we can find the time:
\( 1 = \frac{1}{5} \times \text{Time} \)
To find the time, we rearrange the equation:
\( \text{Time} = \frac{1}{1/5} \)
\( \text{Time} = 5 \text{ days} \)
Therefore, it will take 10 men and 20 boys 5 days to complete the same piece of work.
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