Question

Naya and Scott have a cleaning business. Naya can clean an office suite in 20 minutes less time than it takes Scott to do the same job. Working together, they can complete the job in 90 minutes. Approximately how long does it take Naya to complete the job if she works alone? Assume both people work at steady rates and their rates are not affected by either working alone or working together.

Answer 1

Naya takes approximately 80 minutes to clean the office suite on her own.

Step-by-step explanation:

To solve this problem, we can set up a system of equations. Let's assume that Scott takes x minutes to clean the office suite.

According to the problem, Naya can clean the office suite in 20 minutes less time than Scott, so Naya takes (x-20) minutes to clean the suite.

When they work together, they can complete the job in 90 minutes. This means that their combined rate of work is 1 job per 90 minutes, or 1/90 of the job per minute. Since Naya's rate of work is 1 job per (x-20) minutes and Scott's rate of work is 1 job per x minutes, we can write the equation:

1/(x-20) + 1/x = 1/90

To solve this equation, we can multiply through by the common denominator of x(x-20) to get:

x + (x-20) = (x-20)x/90

Simplifying gives:

2x - 20 = (x^2 - 20x)/90

Multiplying through by 90 to eliminate the fractions gives:

180x - 1800 = x^2 - 20x

Rearranging terms gives the quadratic equation:

x^2 - 200x + 1800 = 0

This equation can be factored as (x-100)(x-18) = 0, or solved using the quadratic formula. The solutions are x = 100 and x = 18, but since Naya takes 20 minutes less than Scott, the solution x = 18 is not valid. Therefore, Scott takes 100 minutes to clean the office suite, and Naya takes (100-20) = 80 minutes to clean it.