Understanding the Work and Time Problem
This problem involves calculating the time required to complete a piece of work given different combinations of men and boys working on it. We need to determine the relative work efficiency of men and boys to solve this.
Defining Variables and Work Rate
Let's denote the amount of work done by one man in one day as $M$ units, and the amount of work done by one boy in one day as $B$ units.
The total amount of work ($W$) can be calculated by multiplying the number of workers by their individual work rate and the number of days they work.
Setting Up Equations from Given Information
We are given two scenarios:
- Scenario 1: 6 men and 8 boys complete the work in 10 days.
The total work done is the sum of work done by men and boys over the given time.
Work done per day = $6M + 8B$
Total Work ($W$) = $(6M + 8B) \times 10$
Equation 1: $W = 60M + 80B$
- Scenario 2: 26 men and 48 boys complete the same work in 2 days.
Work done per day = $26M + 48B$
Total Work ($W$) = $(26M + 48B) \times 2$
Equation 2: $W = 52M + 96B$
Calculating Relative Efficiency of Men and Boys
Since the total work ($W$) is the same in both scenarios, we can equate the expressions from Equation 1 and Equation 2:
$60M + 80B = 52M + 96B$
Now, let's rearrange the terms to find the relationship between $M$ and $B$:
$60M - 52M = 96B - 80B$
$8M = 16B$
Divide both sides by 8:
$M = 2B$
This result indicates that one man does the same amount of work in one day as two boys do.
Calculating Total Work in Terms of Boys' Work
We can now substitute the relationship $M = 2B$ into either Equation 1 or Equation 2 to find the total work ($W$) in terms of the work done by boys.
Using Equation 1:
$W = 60M + 80B$
$W = 60(2B) + 80B$
$W = 120B + 80B$
$W = 200B$
So, the total work required is equivalent to 200 boys working for one day.
Calculating Work Done by the Target Group
The question asks for the time taken by 15 men and 20 boys to complete the same work.
First, let's find the combined work rate of this group in terms of boys' work:
Work rate of 15 men and 20 boys = $15M + 20B$
Substitute $M = 2B$:
Work rate = $15(2B) + 20B$
Work rate = $30B + 20B$
Work rate = $50B$
This means the group of 15 men and 20 boys can complete work equivalent to 50 boys working for one day.
Calculating the Time Taken
To find the time taken, we divide the total work by the work rate of the group:
Time = $\frac{\text{Total Work}}{\text{Work rate of the group}}$
Time = $\frac{200B}{50B}$
Time = $4$ days
Therefore, 15 men and 20 boys can complete the same piece of work in 4 days.
Important Questions from Time & Work (Notes)
- A fresh water tap fills a fish tank in 40 minutes. The same tank is filled by a salt water tap in 120 minutes. If both the taps are open, how many minutes will it take to fill the tank?
Three pipes A, B and C can fill a tank in $10$, $15$ and $20$ hours respectively. Pipe A was opened at $6$ AM, pipe B at $7$ AM and pipe C at $8$ AM. At what time was the tank completely filled, if pipe C needs a break of $1$ hour after remaining open for $3$ hours?
A tank has four pipes $P_1$, $P_2$, $P_3$ and $P_4$. The tank can be filled in $15$ minutes by pipes $P_1$, $P_2$, $P_3$ together. It can be filled in $20$ minutes by pipes $P_2$, $P_3$, $P_4$ together and it can be filled by pipes $P_1$, $P_4$ together in $30$ minutes. If all the pipes are opened together, then in how much time will the tank be filled?
$5$ men and $4$ women can earn ₹ $20000$ in $8$ days. $10$ men and $7$ women can earn ₹ $23,750$ in $5$ days. In how many days will $5$ men and $6$ women earn ₹ $12,000$?
- $6$ men or $5$ women earn ₹ $16,800$ in $4$ days. How much will $4$ women and $6$ men earn in one day?
