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Question 1027961: A takes 10 days less than the time taken by B to finish an embroidery work. If both A and B together can finish the work in 12 days. Find the time taken by B alone to finish the work.
Found 2 solutions by Theo, mananth:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = quantity
if the time it takes for B to finish a quantity of work is x, then the time it takes for A to finish the same quantity of work is x-10.
the formula for A becomes rate * (x-10) = quantity
the formula for B becomes rate * x = quantity.
the quantity of work produced is equal to 1 embroidery work.
for A, the formula becomes rate * (x-10) = 1
solve for rate to get rate = 1/(x-10)
for B, the formula becomes rate * x = 1
solve for rate to get rate = 1/x
the rate for A becomes 1/(x-10)
the rate for B becomes 1/x
when they work together, their rates are additive.
since the time taken for both to finish the work is 12 days, the formula of rate * time = quantity for when they are working together becomes:
(1/(x-10) + 1/x) * 12 = 1
(1/(x-10) + 1/x) is their combined rate.
12 is the time it takes.
1 is the quantity of work produced.
simplify this equation to get:
12/(x-10) + 12/x = 1
multiply both sides of this equation by (x-10)*x to get:
12*x + 12*(x-10) = 1*x*(x-10)
simplify further to get 12x + 12x - 120 = x^2 - 10x
combine like terms to get 24x - 120 = x^2 - 10x
subtract 24x from both sides of the equation and add 120 to both sides of the equation to get 0 = x^2 - 24x - 10x + 120
combine like terms to get 0 = x^2 - 34x + 120
factor the quadratic equation to get:
0 = (x-4) * (x-30)
solve for x to get x = 4 or x = 30.
when x = 4, the rate for A becomes 1/(4 - 10) = 1/(-6) which is negative.
since rate can't be negative, then x = 4 is not a viable solution and is discarded.
when x = 30, you get:
rate for A = 1/(x-10) = 1/20
rate for B = 1/x = 1/30
the rate * time formula for both of them working together becomes:
start with (1/(x-10) + 1/x) * 12 = 1
replace 1/(x-10) with 1/20 and replace 1/x with 1/30 to get:
(1/20 + 1/30) * 12 = 1
put 1/20 and 1/30 under the common denominator of 60 to get:
(3/60 + 2/60) * 12 = 1
combine the like terms to get 5/60 * 12 = 1
simplify to get 60/60 = 1
simplify further to get 1 = 1
this confirms the rates are good.
the rate for A is 1/20
the rate for B is 1/30
when A works alone, the formula for rate * time = quantity becomes 1/20 * time = 1
solve for time to get time = 1 * 20 = 20 days
when B works alone, the formula for rate * time = quantity becomes 1/30 * time = 1
solve for time to get time = 1 * 30 = 30 days.
you are asked how long it takes for B to finish the work alone.
it would take B 30 days to finish the work alone.
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website! A x days
B x+ 10 days
A 1/x of the job in hour
B 1/(x+ 10 )of the job in 1 day
Together they take 1/x+ 1/(x+ 10 )of the job in 1 day
Together they do the job in 12 days 12 days
Together they do 1/ 12 of the job in 1 day
1/x + 1/(x+ 10 )= 1/ 12
LCD =x(x+ 10 )
(x+ 10 )+x= 1/ 12 (x + 10 )x
( 2 x+ 10 ) * 12 = x^2+ 10 x
24 x+ 120 = x^2+ 10 x
x^2 -14 x -120 = 0
Find the roots of the equation
x^2-20x+6x-120=0
x(x-20)+6(x-20)=0
(x-20)(x+6)=0
x= 20 OR -6
Ignore negative
x= 20
A takes 10 days
B takes 20 days to do the job alone
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