Introduction to Work Rate Calculation

Imagine a project that needs to be completed by a group of individuals working together. In this article, we will explore how to determine the project completion time for a team consisting of 4 girls and 3 boys. This is a classic problem in work rate calculations. Let's break it down step-by-step to understand the solution.

Understanding Work Rate

The concept of work rate is fundamental when dealing with such problems. The work rate is the amount of work done per unit of time. For example, if a person can complete a work in 'n' days, their work rate is 1/n per day.

Step 1: Determine the Work Rate of Girls and Boys

From the problem statement, we know:

4 girls can complete the work in 18 days. 6 boys can also complete the work in 18 days.

We can express the total work done in terms of 'girl-days' and 'boy-days'.

Work Done by Girls

If 4 girls can complete the work in 18 days, the work done by one girl in one day (her work rate) is:

Rate of 1 girl 1/72 work/day frac{1}{4 times 18} frac{1}{72}

This is because 4 girls in 18 days complete 1 work, so each girl does (frac{1}{72}) of the work per day.

Work Done by Boys

Similarly, if 6 boys can complete the work in 18 days, the work done by one boy in one day (his work rate) is:

Rate of 1 boy 1/108 work/day frac{1}{6 times 18} frac{1}{108}

This is because 6 boys in 18 days complete 1 work, so each boy does (frac{1}{108}) of the work per day.

Step 2: Calculate the Combined Work Rate of 4 Girls and 3 Boys

Next, we need to find the combined work rate of 4 girls and 3 boys.

Work Rate of 4 Girls

The combined work rate of 4 girls is:

Rate of 4 girls 4 (times) 1/72 frac{4}{72} frac{1}{18}) work/day

Work Rate of 3 Boys

The combined work rate of 3 boys is:

Rate of 3 boys 3 (times) 1/108 frac{3}{108} frac{1}{36}) work/day

Step 3: Combine the Rates

Now we combine the work rates to find the total work rate of the team:

Total combined rate (frac{1}{18} frac{1}{36})

To add these fractions, we need a common denominator. The least common multiple of 18 and 36 is 36:

(frac{1}{18} frac{2}{36})

So, the combined rate becomes:

Total combined rate (frac{2}{36} frac{1}{36} frac{3}{36} frac{1}{12}) work/day

Step 4: Calculate the Total Time to Complete the Work

With the combined work rate of (frac{1}{12}) work per day, the total time to complete the work is the reciprocal of the work rate:

Time to complete the work (frac{1 text{ work}}{frac{1}{12} text{ work/day}} 12 text{ days})

Conclusion

Therefore, 4 girls and 3 boys working together can complete the work in 12 days.

This approach can be generalized to solve similar work rate problems. Understanding and applying the work rate formula will help you tackle more complex scenarios in project management and resource allocation.