Determining the Time for A and B to Complete a Task Together

When it comes to work problems, understanding the principle of combination of work rates can help us to determine the time required for two or more individuals to complete a task together. Let's explore this concept using a practical example. Consider two individuals, A and B, where A can complete a piece of work in 5 days and B in 20 days. The problem is to find out the number of days they will take to finish the work together.

Understanding Individual Work Rates

First, let's compute the individual work rates of A and B. We begin by finding the total amount of work. A can do the work in 5 days, so A's work rate is 1/5 of the work per day. Similarly, B can do the work in 20 days, so B's work rate is 1/20 of the work per day.

Combining the Work Rates

Now, we need to combine their work rates to find out how much work they can do together in one day:

Work rate of A per day 1/5

Work rate of B per day 1/20

Combined work rate per day (frac{1}{5} frac{1}{20})

Adding the Fractions

To add the fractions, we need to find a common denominator. The least common multiple of 5 and 20 is 20. Converting the fractions, we get:

(frac{1}{5} frac{4}{20})

So, the combined work rate is: [frac{4}{20} frac{1}{20} frac{5}{20} frac{1}{4}]

This means that A and B together can complete (frac{1}{4}) of the work in one day. To find out how many days they will take to complete the entire work, we take the reciprocal of their combined work rate:

[ text{Time required} frac{1}{frac{1}{4}} 4 , text{days} ]

Verification of the Combined Work Rate

Another method to verify this is by considering the work as a unit. If A can do the work in 5 days, A does (frac{1}{5}) of the work in one day. Similarly, B does (frac{1}{20}) of the work in one day. Together, they do:

[ frac{1}{5} frac{1}{20} frac{4}{20} frac{1}{20} frac{5}{20} frac{1}{4} ]

So, in one day, A and B together complete (frac{1}{4}) of the work. Therefore, the total time required for them to complete the work together is:

[ text{Time required} frac{1}{frac{1}{4}} 4 , text{days} ]

Conclusion

Thus, A and B will finish the work together in 4 days.

Additional Example

For further practice, consider this example where A alone can do 1/15 of the work in one day, and B alone can do 1/12 of the work in one day. Together, they do:

[ frac{1}{15} frac{1}{12} frac{4}{60} frac{5}{60} frac{9}{60} frac{3}{20} ]

This means together, A and B complete (frac{3}{20}) of the work in one day. Therefore, the time required for them to complete the work together is:

[ text{Time required} frac{1}{frac{3}{20}} frac{20}{3} approx 6.67 , text{days} ]

This confirms the consistency of the method used.

Optimizing Work Rate Problems with SEO Best Practices

When writing about work rate problems for SEO, it's important to structure your content to cater to search engines and readers. Include clear headings, use relevant keywords, and provide step-by-step solutions to ensure maximum readability and user satisfaction.