working together 2 people can mow a large lawn in 4 hours. One person can do the job alone one hour faster than the other person. How long does it take each person working alone to mow the lawn?

rate of faster person = 1/x

rate of slower person = 1/(x+1)

combined rate = 1/x + 1/(x+1) = (2x+1)/(x(x+1))

so time with combined rate = 1/[(2x+1)/(x(x+1))
= x(x+1)/(2x+1)

so x(x+1)/(2x+1) = 4
x^2 + x = 8x + 4
x^2 - 7x -4 = 0
x = (7 ± √ 65)/2
= 7.53 or a negatiave

So one takes 7.53 hours, the other 8.53 hours

To find out how long each person takes to mow the lawn individually, we need to set up equations based on the given information.

Let's assume the faster person takes x hours to mow the lawn alone. Then, the slower person would take x + 1 hour to mow the same lawn.

Now, let's calculate their individual rates of work. The rate at which someone works is determined by dividing the amount of work they do by the time it takes them to complete it.

For the faster person:
Rate = 1 lawn / x hours

For the slower person:
Rate = 1 lawn / (x + 1) hours

Now, let's calculate their combined rate when working together. We know that it takes them 4 hours to mow the lawn together, so their combined rate can be calculated as:
Combined Rate = 1 lawn / 4 hours

Since we know that the rates add up when people work together, we can set up the equation:
1 lawn / x hours + 1 lawn / (x + 1) hours = 1 lawn / 4 hours

To solve this equation, we need to find a common denominator for the fractions:

[(x + 1) + x] / [x(x + 1)] = 1 / 4

Simplifying the above equation, we get:
2x + 1 / [x(x + 1)] = 1 / 4

Cross-multiplying to eliminate the denominators, we get:
4(2x + 1) = x(x + 1)

Expanding and simplifying, we have:
8x + 4 = x^2 + x

Rearranging the equation to the standard quadratic form, we get:
x^2 - 7x - 4 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, the equation factors as follows:
(x - 8)(x + 1) = 0

So, x can be either 8 or -1. However, time cannot be negative, so we discard -1.

Therefore, the faster person takes 8 hours to mow the lawn alone, and the slower person (x + 1) takes 8 + 1 = 9 hours to mow the lawn alone.

In summary, it takes the faster person 8 hours to mow the lawn alone and the slower person 9 hours to mow the lawn alone.