Question 1146364
rate * time = quantity of work completed.
h = rate of hot water.
c = rate of cold water.
quantity of work completed is 1 full tub.


when they work together, the rate becomes (c + h) and formula becomes 6 * (c + h) = 1
when only the cold water is running, the formula becomes 14 * c = 1


you have two equations that need to be solved simultaneously.
they are:
6 * (c +  h) = 1
14 * c = 1


simplify the first equation and leave the second equation as is to get:
6 * c + 6 * h = 1
14 * c = 1
multiply both sides of the first equation by 14 and multiply both sides of the second equation by 6 to get:
84 * c + 84 * h = 14
84 * c = 6
subtract the second equation from the first to get:
84 * h = 8
solve for h to get:
h = 8/84 = 2/21


from 14c = 1, solve for c to get c = 1/14


6 * (h + c) = 1 becomes 6 * h + 6 * c = 1 which becomes 6 * 1/14 + 6 * 2/21 = 1.
simplify to get 6/14 + 12/21 = 1
place both fractions under the common denominator of 42 to get:
18/42 + 24/42 = 1
combine like terms to get 42/42 = 1
simplify to get 1 = 1


this confirms the solution is correct.


the rate of the cold water is 1/14 of the tub in 1 minute.
the rate of the hot water is 2/21 of the tub in 1 minute.


the formula when they work together is (1/14 + 2/21) * T = 1
solve for T to get T = 1 / (1/14 + 2/21) = 1 / (3/42 + 4/21) = 1 / (7/42) = 1 * 42 / 7 = 1 * 6 = 6 minutes.


this agrees with the problem statement that says it takes 6 minutes to fill the tub when they are working together.


when the hot water alone is working, the formula becomes 2/21 * T = 1
solve for T to get T = 1 / (2/21) = 1 * 21 / 2 = 10.5 minutes.


that's your solution.