SOLUTION: P can complete a task in x days while Q takes (x + 12) days to do the same task. a) What fraction of the task can each of them complete in a day? b) If both of them work together

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Question 1204175: P can complete a task in x days while Q takes (x + 12) days to do the same task.
a) What fraction of the task can each of them complete in a day?
b) If both of them work together, they can
complete the task in 8 days. Find x.
Found 2 solutions by Edwin McCravy, Theo:
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
P can complete a task in x days while Q takes (x + 12) days to do the same task.
a) What fraction of the task can each of them complete in a day?
P can complete 1 task in x days, so P can complete 1/x of a task in 1 day.
Q can complete 1 task in (x + 12) days, so Q can complete 1/(x + 12) of a task
in 1 day.

b) If both of them work together, they can
complete the task in 8 days. Find x.
P can complete 1/x of a task in 1 day, so P can complete 8/x of a task in 8
days.
Q can complete 1/(x + 12) of a task in 1 day, so Q can complete 8/(x + 12) of a
task in 8 days.
So together, they can complete 8/x + 8/(x+12) of a task in 8 days

That's 1 task, so 



LCD = x(x + 12)













x+8=0; x-12=0
  x=-8;   x=12

x cannot be negative.

x = 12.

Now that we have x, we can answer (a) as a numerical fraction:

a) What fraction of the task can each of them complete in a day?
P can complete 1 task in 12 days, so P can complete 1/12 of a task in 1 day.
Q can complete 1 task in (12 + 12) = 24 days, so Q can complete 1/(12 + 12) = 1/24 of a task in 1 day.
Edwin
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
let p = the rate that P works per day.
let q = the rate that Q works per day.

rate * time = quantity.

for P, you get p * x = 1, resulting in p = 1/x
for Q, you get q * (x + 12) = 1, resulting in q = 1/(x + 12).

you are given that they can both complete the task in 8 days when working together.
the equation for that is 8 * (p + q) = 1
simplify to get 8p + 8q = 1

since p = 1/x and q = 1/(x + 12), you get:
8 * 1/x + 8 * 1/(x + 12) = 1
simplify to get 8/x + 8/(x + 12) = 1
multiply both sides of this equation by x * (x + 12) to get:
8 * (x + 12) + 8 * x = x * (x + 12)
simplify to get:
8x + 96 + 8x = x^2 + 12x
combine like terms to get:
16x + 96 = x^2 + 12x
subtract the left side of the equation from both sides of the equation to get:
0 = x^2 + 12x - 16x - 96
combine like terms to get:
0 = x^2 - 4x - 96
solve this quadratic equation to get:
x = 12 or x = -8.

since x has to be positive, then x = 12.
since p = 1/x, then p = 1/12
since q = 1/(x + 12), then q = 1/24

since 8p + 8q = 1, then replace p and q with their respective values to get:
8 * 1/12 + 8 * 1/24 = 1
simplify to get 8/12 + 8/24 = 1
simplify further to get 1 = 1, confirming the value of p and q are good.

answers to your questions are:

a) What fraction of the task can each of them complete in a day?

P can complete 1/12 of the task in one day.
Q can complete 1/24 of the taxk in one day.


b) If both of them work together, they can complete the task in 8 days. Find x.

x is equal to 12.


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