SOLUTION: A takes 10 more days to do a job than B they both do a job in 12 days. in how many days B will take to complete the work if he works alone??

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Question 1013984: A takes 10 more days to do a job than B they both do a job in 12 days. in how many days B will take to complete the work if he works alone??
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
A takes 10 more days to do a job than B. They both do a job in 12 days.
In how many days B will take to complete the work if he works alone??
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Let a = # of days for A to do the job working alone, and
Let b = # of days for B to do the job working alone.

According to the condition, you have this system of two equations in two unknowns a and b:

a = b + 10,                 (1)

. = 1.           (2)

From (1), substitute a = b + 10 into (2). You will get

. = .    (3)

In (3), multiply both sides by b*(b+10) to get off the denominators. You will get

12*b + 12*(b+10) = b*(b+10),   or

24b + 120 = ,    or

 = ,   or

 = .

Factor left side:

(b-20)*(b+6) = 0.

The roots are b = 20 and b = -6.

Only positive b = 20 fits the condition.

Then a = b + 10 = 30.

Answer. It will take 20 days for B to complete the job working alone.

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    - Using fractions to solve word problems on joint work   and
    - Solving more complicated word problems on joint work
in this site.


Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
general formula is R * T = Q
R = rate
T = time
Q = quantity

Q = 1 job.
formula becomes R * T = 1

they can both do the job in 12 days.
formula for both becomes R * 12 = 1
solve for R to get:
R for both = 1/12

A takes 10 more days to do the job than B when both work alone.

T for A = T + 10
T for B = T

formula for A = R * (T + 10) = 1
formula for B = R * T = 1

solve for R to get:

rate for A = 1/(T+10)
rate for B = 1/T

when they both work together, their rates are additive.

R for both = 1/(T+10) + 1/T

since R for both = 1/12. we get:
1/(T+10) + 1/T = 1/12

we can use this formula to solve for T.
the T that we will be solving for is the T for A and the T for B.
multiply both sides of this equation by T * (T + 10) to get:
T + (T + 10) = 1/12 * T * (T + 10)
simplify this equation to get:
T + T + 10 = 1/12 * (T^2 + 10T)
combine like terms and simplify further to get:
2T + 10 = 1/12 * T^2 + 1/12 * 10T
multiply both sides of this equation by 12 to get:
24T + 120 = T^2 + 10T
subtract 24T + 120 from both sides of this equation to get:
0 = T^2 + 10T - 24T - 120
combine like terms and commute the equation to get:
T^2 - 14T - 120 = 0
factor this quadratic to get:
(T + 10) * (T - 20) = 0
solve for T to get:
T = -10 or T = 20
T can't be negative, so T = 20 is the solution.

T = 20 is the time for B.
T = 30 is the time for A because A takes 10 more days to complete the job on his own than B does.

the solution to your problem is that B takes 20 days to complete the job on his own.

you should confirm your answer is correct.

you do this by replacing T for A with 30 and T for B with 20 and solving for R for A and B individually and then solving for both to see if everything checks out.

we now know that T for A is 30 and Q is 1.
we can solve for R for A to get R = 1/30.

similarly we can solve for R for B to get R = 1/20

the RT = Q equation for A becomes 1/30 * 30 = 1 which results in 1 = 1 which confirms the solution is good.

the RT = Q equation for B becomes 1/20 * 20 = 1 which results in 1 = 1 which confirms the solution is good.

the RT = Q equation for A and B together becomes (1/30 + 1/20) * 12 = 1.
this is because, when they work together, their rates are additive and we already know that they take 12 days to complete the job when they work together.
start with:
(1/30 + 1/20) * 12 = 1
multiply both sides of this equation by 20*30 to get:
(20 + 30) * 12 = 20 * 30
combine like terms and simplify to get:
50 * 12 = 600
simplify further to get:
50 * 12 = 600 becomes 600 = 600 which confirms the solution is good.

the solution is:

it will take B 20 days to complete the job if he works alone.






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