Solving Work Rate Problems: A Comprehensive Guide on Men and Children Finishing a Work Task

Understanding and solving work rate problems, specifically those involving men and children, can significantly enhance your problem-solving skills, especially in scenarios requiring quick and accurate estimations. In this guide, we will walk through a detailed step-by-step solution to a classic work rate problem: determining how many days 9 men will take to finish the entire work based on the combined productivity of smaller teams.

Problem Statement

2 men and 5 children can finish portion of work in 5 days, while 3 men and 4 children can finish portion of the work in 3 days. We need to determine how many days 9 men will take to finish the entire work.

Step-by-Step Solution

Step 1: Set Up Equations Based on Given Information

Let nM be the work done by one man in one day and nC be the work done by one child in one day.

From the first piece of information: 2 men and 5 children can finish of the work in 5 days. From the second piece of information: 3 men and 4 children can finish of the work in 3 days.

The total work done in the given days can be expressed as:

5 days with 2 men and 5 children: 5(2M 5C) frac{1}{2} 3 days with 3 men and 4 children: 3(3M 4C) frac{1}{3}

Step 2: Solve the Equations

We now have two equations to solve:

1. 2M 5C

2. 3M 4C

We can solve these equations simultaneously by first eliminating one of the variables.

Step 3: Eliminate One Variable

Multiply Equation 1 by 3:

3(2M 5C) 3(frac{1}{10}) 6M 15C frac{3}{10}

Multiply Equation 2 by 2:

2(3M 4C) 2(frac{1}{9}) 6M 8C frac{2}{9}

Subtract the second equation from the first:

(6M 15C) - (6M 8C) (frac{3}{10}) - (frac{2}{9})

This simplifies to:

7C (frac{3}{10}) - (frac{2}{9})

To subtract the fractions, find a common denominator, which is 90:

(frac{3}{10}) (frac{27}{90}) and (frac{2}{9}) (frac{20}{90})

So, (frac{3}{10}) - (frac{2}{9}) (frac{27}{90}) - (frac{20}{90}) (frac{7}{90})

Thus, 7C (frac{7}{90}) implies C (frac{1}{90})

Step 4: Substitute the Value of C Back to Find M

Substituting C (frac{1}{90}) back into Equation 1:

2M 5left(frac{1}{90}right) (frac{1}{10})

This simplifies to:

2M (frac{5}{90}) (frac{1}{10})

Convert (frac{1}{10}) to a fraction with a denominator of 90:

(frac{1}{10}) (frac{9}{90})

Thus, 2M (frac{5}{90}) (frac{9}{90})

Subtract (frac{5}{90}) from both sides:

2M (frac{9}{90}) - (frac{5}{90}) (frac{4}{90})

Implies M (frac{2}{90}) (frac{1}{45})

Step 5: Calculate the Total Work and the Time Taken by 9 Men

Now we know:

- Work done by one man in one day, M (frac{1}{45})

- Work done by one child in one day, C (frac{1}{90})

Let's calculate how much work 9 men can do in one day:

Work done by 9 men in one day 9M 9 times (frac{1}{45}) (frac{9}{45}) (frac{1}{5})

Step 6: Find the Total Work

The total work W can be computed from either equation. From 2M 5C :

W 10 times (2M 5C) 10 times (frac{1}{10}) 1

Step 7: Calculate the Time Taken by 9 Men to Finish the Entire Work

If 9 men can complete (frac{1}{5}) of the work in one day, the total number of days (D) required to finish the entire work is:

D (frac{W}{text{Work done by 9 men in one day}}) (frac{1}{frac{1}{5}}) 5 text{ days}

Thus, 9 men will take 5 days to finish the entire work.

Conclusion

This comprehensive guide demonstrates a systematic approach to solving work rate problems involving multiple workers, providing a clear understanding of the underlying principles and calculations. By following these steps, you can apply the same method to other similar problems, enhancing your problem-solving skills in mathematics and real-world scenarios.

Related Keywords

Solving work rate problems, men and children, simultaneous equations