A = larger, faster pipe
B = smaller, slower pipe
let's say B can fill the tank in hmmm 8hrs... so in 1hr, it has only done 1/8 of the job, in 2hrs it has done 2/8 of the job and in 3hrs has done 3/8 of the job and so on, to finish the job, it needs to do 8/8 of the job or 8/8=1 whole
we dunno how long it took the B pipe to do it though, let's say it took "t" hours, so in 1hr it had done 1/t of the job
now, if B took "t" hours to do the whole job, pipe A is faster and thus it did it 5hrs less than that, so, A can do it in "t - 5" hours
so, in 1hr, A had done 1/(t-5) of the job
now, we know the rates in 1hr of each pipe, we know together, they can do the job in 6hrs
so
![\bf \textit{in 1hr, both pipes have done} \\\\\\ \begin{array}{llllll} \cfrac{1}{t}&+&\cfrac{1}{t-5}&=&\cfrac{1}{6}\\ smaller&&larger&&job\ done\\ B&&A \end{array} \\\\\\ \textit{let's add the left-side, our LCD is t(t-5)} \\\\\\ \cfrac{t-5+t}{t(t-5)}=\cfrac{1}{6}\implies \cfrac{2t-5}{t(t-5)}=\cfrac{1}{6}\implies 6(2t-5)=t(t-5) \\\\\\ 12t-30=t^2-5t\implies 0=t^2-17t+30 \\\\\\ 0=(t-15)(t-2)\implies \begin{cases} 0=t-15\implies &\boxed{15=t}\\ 0=t-2\implies &2=t \end{cases}](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bin%201hr%2C%20both%20pipes%20have%20done%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Cbegin%7Barray%7D%7Bllllll%7D%0A%5Ccfrac%7B1%7D%7Bt%7D%26%2B%26%5Ccfrac%7B1%7D%7Bt-5%7D%26%3D%26%5Ccfrac%7B1%7D%7B6%7D%5C%5C%0Asmaller%26%26larger%26%26job%5C%20done%5C%5C%0AB%26%26A%0A%5Cend%7Barray%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ctextit%7Blet%27s%20add%20the%20left-side%2C%20our%20LCD%20is%20t%28t-5%29%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7Bt-5%2Bt%7D%7Bt%28t-5%29%7D%3D%5Ccfrac%7B1%7D%7B6%7D%5Cimplies%20%5Ccfrac%7B2t-5%7D%7Bt%28t-5%29%7D%3D%5Ccfrac%7B1%7D%7B6%7D%5Cimplies%206%282t-5%29%3Dt%28t-5%29%0A%5C%5C%5C%5C%5C%5C%0A12t-30%3Dt%5E2-5t%5Cimplies%200%3Dt%5E2-17t%2B30%0A%5C%5C%5C%5C%5C%5C%0A0%3D%28t-15%29%28t-2%29%5Cimplies%20%0A%5Cbegin%7Bcases%7D%0A0%3Dt-15%5Cimplies%20%26%5Cboxed%7B15%3Dt%7D%5C%5C%0A0%3Dt-2%5Cimplies%20%262%3Dt%0A%5Cend%7Bcases%7D)
well, clearly, if both pipes take 6hrs, the smaller B can't do it in 2hrs by itsef, thus 15 = t