SOLUTION: A cistern can be filled by two pipes. The small pipe alone will take 24 minutes longer than the larger pipe to fill the cistern alone. The small pipe alone will take 32 minutes lon

Algebra ->  Rate-of-work-word-problems -> SOLUTION: A cistern can be filled by two pipes. The small pipe alone will take 24 minutes longer than the larger pipe to fill the cistern alone. The small pipe alone will take 32 minutes lon      Log On


   

Question 1006498: A cistern can be filled by two pipes. The small pipe alone will take 24 minutes longer than the larger pipe to fill the cistern alone. The small pipe alone will take 32 minutes longer to fill the cistern alone when the two pipes are operating together. How long will it take for the larger pipe to fill the cistern alone?
Found 2 solutions by josmiceli, MathTherapy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +t%5B1%5D+ = time in minutes for the larger pipe
to fill cistern when filling alone
Rate of filling = +1%2Ft%5B1%5D+
Let +t%5B2%5D+ = time in minutes for both pipes to
fill cistern when filling together
---------------------------
+t%5B1%5D+%2B+24+ = time in minutes for smaller pipe to
fill cistern when filling alone
Rate of filling = +1%2F%28+t%5B1%5D+%2B+24+%29+
+t%5B2%5D+%2B+32+ = this is also time in minutes for smaller pipe to
fill cistern when filling alone
----------------------------
+t%5B1%5D+%2B+24+=+t%5B2%5D+%2B+32+
+t%5B2%5D+=+t%5B1%5D+-+8+
-----------------------------
Add their rates of filling alone to get rate filling together
+1%2Ft%5B1%5D+%2B+1%2F%28+t%5B1%5D+%2B+24+%29+=+1%2F%28+t%5B1%5D+-+8+%29+
Multiply both sides by +t%5B1%5D%2A%28+t%5B1%5D+%2B+24+%29%2A%28+t%5B1%5D+-+8+%29+
---------------------------------------------


+t%5B1%5D%5E2+-+16t%5B1%5D++=+192+
Complete the square
++t%5B1%5D%5E2+-+16t%5B1%5D+%2B+%2816%2F2%29%5E2+=+192+%2B+%2816%2F2%29%5E2+
++t%5B1%5D%5E2+-+16t%5B1%5D+%2B+64+=+192+%2B+64+
+%28+t%5B1%5D+-+8+%29%5E2+=+256+
+%28+t%5B1%5D+-+8+%29%5E2+=+16%5E2+
+t%5B1%5D+-+8+=+16+
+t%5B1%5D+=+24+
The larger pipe takes 24 min filling alone
check:
+1%2Ft%5B1%5D+%2B+1%2F%28+t%5B1%5D+%2B+24+%29+=+1%2F%28+t%5B1%5D+-+8+%29+
+1%2F24+%2B+1%2F%28+24+%2B+24+%29+=+1%2F%28+24+-+8+%29+
+1%2F24+%2B+1%2F48+=+1%2F16+
+2%2F48+%2B+1%2F48+=+3%2F48+
+3%2F48+=+3%2F48+
OK

Answer by MathTherapy(10858) About Me  (Show Source):
You can put this solution on YOUR website!
A cistern can be filled by two pipes. The small pipe alone will take 24 minutes longer than the larger pipe to fill the cistern alone. The small pipe alone will take 32 minutes longer to fill the cistern alone when the two pipes are operating together. How long will it take for the larger pipe to fill the cistern alone?
Let time larger pipe takes be L

Then larger pipe can fill 1%2FL of cistern in 1 minute

Smaller pipe can fill cistern in (L + 24) mins, or 1%2F%28L+%2B+24%29 of cistern in 1 minute

Since smaller pipe takes 32 minutes longer to fill cistern than when both pipes are on, then the time it

takes to fill cistern when both are on = L + 24 – 32, or L – 8 minutes. Both can fill 1%2F%28L+-+8%29 of cistern in 1 min

The following 1-minute RATES equation is thus formed: 1%2FL+%2B+1%2F%28L+%2B+24%29+=+1%2F%28L+-+8%29

(L + 24)(L – 8) + L(L – 8) = L(L + 24) -------- Multiplying by LCD, L(L + 24)(L – 8)

L%5E2+%2B+16L+-+192+%2B+L%5E2+-+8L+=+L%5E2+%2B+24L

L%5E2+%2B+L%5E2+%2B+16L+-+8L+-+192+=+L%5E2+%2B+24L

2L%5E2+%2B+8L+-+192+=+L%5E2+%2B+24L

2L%5E2+-+L%5E2+%2B+8L+-+24L+-+192+=+0

L%5E2+-+16L+-+192+=+0

(L - 24)(L + 8) = 0

L, or time larger pipe takes = highlight_green%2824%29 minutes	  OR       L = - 8 (ignore)