SOLUTION: A father and his son can clean the house together in 8 hours. When the son works alone, it takes him an hour longer to clean than it takes his dad alone. Find how long it takes the

Algebra ->  Rate-of-work-word-problems -> SOLUTION: A father and his son can clean the house together in 8 hours. When the son works alone, it takes him an hour longer to clean than it takes his dad alone. Find how long it takes the      Log On


   

Question 1089560: A father and his son can clean the house together in 8 hours. When the son works alone, it takes him an hour longer to clean than it takes his dad alone. Find how long it takes the son to clean alone.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
FATHER                 1%2Ft

SON                    1%2F%28t%2B1%29

FATHER AND SON          1%2F8

-
1%2Ft%2B1%2F%28t%2B1%29=1%2F8
-


ALTERNATIVE:
FATHER                 1%2F%28x-1%29
SON                    1%2Fx
FATHER AND SON          1%2F8

1%2F%28x-1%29%2B1%2Fx=1%2F8
-
8x%2B8%28x-1%29=x%28x-1%29
8x%2B8x-8=x%5E2-x
16x-8=x%5E2-x
x%5E2-17x%2B8=0
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x=%2817%2Bsqrt%2817%5E2-4%2A8%29%29%2F2
x=%2817%2Bsqrt%28257%29%29%2F2
-
highlight%28x=16.516%29


-
1%2F%2816.5-1%29%2B1%2F16.5=0.0645%2B0.60606=0.125, 
close enough to 1%2F8

Answer by ikleyn(52773) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let x be the time in hours for son to complete the job working alone.
Then the time for the father is x-1 hours.

The son's    rate of work is  1%2Fx      of the job per hour.

The father's rate of work is  1%2F%28x-1%29  of the job per hour.

Their combined rate of work is  1%2Fx+%2B+1%2F%28x-1%29.

From the other side, the condition says that their combined rate of work is 1%2F8   of the job per hour.


It gives you an equation 

1%2Fx+%2B+1%2F%28x-1%29 = 1%2F8.


To solve it, multiply both sides by 8x*(x-1). You will get

8(x-1) + 8x = x*(x-1).

Simplify and solve for x:

x^2 - 17x + 8 = 0,

x%5B1%2C2%5D = %2817+%2B-+sqrt%2817%5E2+-+4%2A8%29%29%2F2 = %2817+%2B-+sqrt%28257%29%29%2F2 = %2817+%2B-+16.03%29%2F2.

Only positive root works: x = %2817%2B16.03%29%2F2 = 16.5 hours = 16 hours and 30 minutes (approximately).

Check.  1%2F16.5+%2B+1%2F15.5 = 0.1251;   1%2F8 = 0.125.

               Check is good !.

Answer.  It will take approximately 16.5 hours for the son to make this job working alone.

Solved.


It is a typical joint work problem.

There is a wide variety of similar solved joint-work problems with detailed explanations in this site.  See the lessons
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Selected joint-work word problems from the archive


Read them and get be trained in solving joint-work problems.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic
"Rate of work and joint work problems"  of the section  "Word problems".