A can do a work in 18 days, B in 27 days. They start together, but after 6 days A leaves. In how many more days will B finish the remaining work?
12 days
Take the total work as the LCM of the two completion times, \(\text{LCM}(18,27)=54\) units, so that each worker’s daily rate comes out a whole number.
A finishes in 18 days, so A’s rate is \(\dfrac{54}{18}=3\) units/day; B finishes in 27 days, so B’s rate is \(\dfrac{54}{27}=2\) units/day.
Working together, their combined rate is \(3+2=5\) units/day, so in the first 6 days they complete \(5\times6=30\) units.
The work still remaining is \(54-30=24\) units.
After A leaves, B tackles this alone at 2 units/day.
Time for B to finish the rest is \(\dfrac{24}{2}=12\) days.
Key concept: setting total work to the LCM of the times makes rates integers, so combined work, remaining work, and the finishing time follow by simple arithmetic.
Hence, B needs 12 days more.
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