It takes john 12 hours to plow a field. After he had been working for 3 hours, his brother Mike came to help. Together they finished the job in 5 hours. How long would it take mike to plow the field working alone?

How would I set this problem up in fraction form? And would I need to multiply it out soon afterword?

job done = time x rate

let job done = 1, since it is a constant

John's rate --- 1/12
Mike's rate ---- 1/x
combined rate = 1/12 + 1/x = (x+12)/(12x)

after John worked for 3 hours
job done = 3(1/12) = 1/4
so job left to be done = 3/4

(3/4) / ((x+12)/(12x)) = 5
(3/4)(12x)/(x+12) = 5
36x/(x+12) = 20
36x = 20x + 240
x = 15

Alone, Mike could do it in 15 hours

check my arithmetic

To set up the problem in fraction form, we need to determine the rate of work for each person. Let's denote the amount of work John can do per hour as "J" and the amount of work Mike can do per hour as "M". Since John takes 12 hours to plow the field working alone, his work rate can be represented as 1/12 of the field per hour. So we have:

John's work rate per hour: J = 1/12

When John worked alone for 3 hours, he completed 3 * (1/12) = 3/12 of the field.

Now, let's consider the time when both John and Mike worked together. In this case, we can add their individual work rates to get their combined work rate. So:

Combined work rate per hour: J + M = 1/5

Together, John and Mike finished the remaining 9/12 of the field in 5 hours. Therefore, their combined work rate can be expressed as (9/12) รท 5 = (9/12) * (1/5) = 9/60.

Since we know that J = 1/12, we can substitute it in the equation for the combined work rate:

(1/12) + M = 9/60

To solve for M, we can isolate it by subtracting 1/12 from both sides:

M = 9/60 - 1/12

Simplifying this expression, we get:

M = (9 - 5)/60 = 4/60 = 1/15

Therefore, Mike can plow the field alone in 15 hours.

You do not need to multiply the fraction out, but you may prefer to simplify it if necessary. In this case, we simplified 4/60 to 1/15.