Working together two people can cut a lawn of grass in 5 hours. One person can do the job alone in 1 hour less than the other. How long would it take the faster person to do the job?

If the faster person takes x hours, then the slower person takes (x+1) hours. So, how much of the job gets done in an hour?

1/x + 1/(x+1) = 1/5

Now just find x.

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To solve this problem, we can use the concept of work rates. Let's assume that the slower person can complete the job in x hours. Therefore, the faster person can complete the job in (x-1) hours since they can do it in 1 hour less.

Now, we know that the work rate can be calculated by dividing the amount of work done by the time taken. So, if the slower person's work rate is 1/x (meaning they can do 1 job in x hours), then the faster person's work rate is 1/(x-1).

Given that both people working together can complete the job in 5 hours, we can set up an equation using their work rates. The equation is:

1/x + 1/(x-1) = 1/5

To solve this equation and find the value of x, we need to simplify it. We can multiply each term by 5x(x-1) to remove the fractions:

5(x-1) + 5x = x(x-1)

Now, we can expand and simplify the equation:

5x - 5 + 5x = x^2 - x

Combine like terms:

10x - 5 = x^2 - x

Rearrange the equation into a quadratic form:

x^2 - 11x + 5 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula. Assuming it can't be easily factored, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 1, b = -11, and c = 5:

x = (-(-11) ± √((-11)^2 - 4(1)(5))) / (2(1))

x = (11 ± √(121 - 20)) / 2

x = (11 ± √101) / 2

Now we have two potential values for x. However, since one person takes x hours and the other takes (x-1) hours, x must be greater than 1. Therefore, x = (11 + √101) / 2 is not a valid solution.

The valid solution is x = (11 - √101) / 2. This represents the slower person's time, which is approximately 0.1935 hours, or about 11 minutes and 37 seconds.

Since the faster person takes 1 hour less, their time is x-1 = (11 - √101) / 2 - 1. This is approximately 0.1935 - 1 = -0.8065 hours, or about -48 minutes and 23 seconds.

However, a negative time doesn't make sense in this context. Therefore, the answer is that the faster person would take approximately 11 minutes and 37 seconds to complete the job.