SOLUTION: At first two different pipes running together will fill a tank in 20/3 minutes. The rate that water runs through each of the pipes is then adjusted. If one pipe, running alone, tak
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Question 320359: At first two different pipes running together will fill a tank in 20/3 minutes. The rate that water runs through each of the pipes is then adjusted. If one pipe, running alone, takes 1 minute less to fill the tank at its new rate, and the other pipe, running alone, takes 2 minutes more to fill the tank at its new rate, then the two running together will fill the tank in 7 minutes. Find in what time the tank will be filled by each pipe running alone at the new rates.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
At first two different pipes running together will fill a tank in 20/3 minutes.
The rate that water runs through each of the pipes is then adjusted.
If one pipe, running alone, takes 1 minute less to fill the tank at its new
rate,
and the other pipe, running alone, takes 2 minutes more to fill the tank
at its new rate,
then the two running together will fill the tank in 7 minutes.
:
Find in what time the tank will be filled by each pipe running alone at the new rates.
:
let a = first pipe original time alone (in minutes)
let b = second pipe original time alone
then
(a-1) = first pipe adjusted rate time alone
(b+2) = 2nd pipe adjusted rate time alone
:
Let the completed job = 1
:
Original, working together equation
+ = 1
We convert this to
+ = 1
multiply by 3ab, results
20b + 20a = 3ab
20b - 3ab = -20a
b(20-3a) = -20a
b = ; use this form for substitution
:
Adjusted, working together equation
+ = 1
multiply by (a-1)(b+2), results
7(b+2) + 7(a-1) = (a-1)(b+2)
7b + 14 + 7a - 7 = ab + 2a - b - 2
Combine
7b + b + 7a - 2a + 14 - 7 + 2 = ab
8b + 5a + 9 = ab
Multiply by 3,
24b + 15a + 27 = 3ab
The two equations = 3ab, therefore
24b + 15a + 27 = 20b + 20a
24b - 20b + 15a - 20a = -27
4b - 5a = -27
replace b with
4() - 5a = -27
- 5a = -27
Get rid of all those negatives, multiply by -1
+ 5a = 27
Multiply by (20-3a), results
80a + 5a(20-3a) = 27(20-3a)
80a + 100a - 15a^2 = 540 - 81a
180a - 15a^2 = 540 - 81a
Combine on the right
0 = 15a^2 - 81a - 180a + 540
A quadratic equation
15a^2 - 261a + 540 = 0
Simplify, divide by 3
5a^2 - 87a + 180 = 0
Factor this
(5a-12)(a-15) = 0
Reasonable solution here
a = 15 hrs pipe a originally
then
15 - 1 = 14 hrs pipe a alone at the new rate
:
Find b original value
b =
b =
b =
b =
b = +12 hrs pipe b originally
then
12 + 2 = 14 hrs pipe b alone at the new rate
:
:
Check solution in the adjusted equation
+ = 1
:
:
Seems like there should be an easier way to do this, but I see no one else has
worked this problem in 12 hrs, so here it is.
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