.
Francis builds a fish pond for his fishes, goldfishes and a variety of carp. He decided to use two different hoses
to easily fill his fish pond with water. Using together the two hoses, it takes him 12 minutes to fill the fishpond.
If he used one hose, he would be able to fill the pond 10 minutes faster than the other hose.
How long does it take for each hose to fill the fish pond by itself?
~~~~~~~~~~~~~~~~
Let x = the unknown time for "one" hose, in minutes.
then the time for the "other" hose is ((x+10) minutes.
The individual rates of filling are of the tank volume per minute (for "one" hose)
and of the tank volume per minute (for the "other" hose).
The combined rate of filling is of the tank volume.
So, your balance equation is
+ = . (*)
To solve it, multiply both sides by 12*x*(x+10). You will get
12(x+10) + 12x = x*(x+10).
Simplify it step by step
12x + 120 + 12x = x^2 + 10x
x^2 - 14x - 120 = 0.
(x-20)*(x+6) = 0.
The roots are x= 20 and x= -6.
Of the two roots, only positive x= 20 is meaningful and makes sense.
ANSWER. "One" hose needs 20 minutes. The "other" hose needs 20+10 = 30 minutes.
CHECK. I will check if the equation (*) is valid.
+ = + = = . ! Correct !
Solved.
-----------------
It is a standard and typical joint work problem.
There is a wide variety of similar solved joint-work problems with detailed explanations in this site. See the lesson
- Using quadratic equations to solve word problems on joint work
Read it and get be trained in solving joint-work problems.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this textbook under the topic
"Rate of work and joint work problems" of the section "Word problems".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.