Question 1111270
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Let S be the rate of work of the small hose (= volume of water per 1 hour).

Let L be the rate of work of the large hose (= volume of water per 1 hour).


Then you have these two equation from the condition:

 3L +  8S = 0.5,    (1)
10L + 10S = 1       (2)

where 0.5 denotes half of pool volume and 1 denotes whole volume.


To solve the system, multiply eq(1) by 10 (both sides)  and  multiply eq(2) by 3 (both sides). You will get

30L + 80S = 5,      (3)
30L + 30S = 3.      (4)


Now subtract eq(4) from eq(3)  (both sides).


You will get

     50S = 2  ====>  S = {{{2/50}}} = {{{1/25}}}.


Thus you found that the rate of work of the small hose is {{{1/25}}} of the pool volume per hour.

It means that the small hose can fill the pool in 25 hour, working alone.



Now from equation (2)

10L + 10*(1/25) = 1,   or   10L = 1 - 10/25 = 15/25 = 3/5  ====>  L = {{{3/50}}}.


Thus you found that the rate of work of the large hose is {{{3/50}}} of the pool volume per hour.

It means that the small hose can fill the pool in {{{50/3}}} hours = {{{16}}}{{{2/3}}} hours = 16 hours and 40 minutes, working alone.
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Solved.


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It is a typical joint work problem.


There is a wide variety of similar solved joint-work problems with detailed explanations in this site. &nbsp;See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Word-problems-WORKING-TOGETHER-by-Fractions.lesson>Using Fractions to solve word problems on joint work</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Solving-more-complicated-word-problems-on-joint-work.lesson>Solving more complicated word problems on joint work</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Selected-problems-from-the-archive-on-joint-work-word-problems.lesson>Selected joint-work word problems from the archive</A> 



Read them and get be trained in solving joint-work problems.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic 
"<U>Rate of work and joint work problems</U>" &nbsp;of the section &nbsp;"<U>Word problems</U>".



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.