<PRE><B>Question 1027961</B>
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&#39;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.








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