# How long does it take these boys to paint a fence?

I came across the following question:

If 3 boys can paint a fence in 2 days what part of the job can be completed by two boys in 1 day?

The answer to this problem is $\frac13$.

I can't manage to tackle this problem. Could anyone tell me what would be the quickest way to solve this problem (This question is a standardized test based question). Any help would be appreciated.

Three boys paint it in two days. That means that it takes six times the amount of work a single boy can do in a day to finish painting (assuming every boy paints the same amount, of course); so one boy will take three times as long as three boys: six days. So in one day, a single boy can paint $\frac{1}{6}$th of the fence. Two boys will paint twice as much, that is $\frac{2}{6} = \frac{1}{3}$ of the fence.

• @ArturiMagidin Thanks for the explanation. So far I realize that 1 boy takes 6 days to paint. Could you explain how you got 1/6th of the fence ? – Rajeshwar Jun 1 '12 at 3:31
• @Rajeshwar: If it takes you six days to paint the whole fence, then how much fence are you painting each day? – Arturo Magidin Jun 1 '12 at 3:33

Ask yourself how much of the fence one boy can paint in one day. Then two boys will be able to paint twice as much.

• Well 2 boys can complete the fence in 3 days – Rajeshwar Jun 1 '12 at 3:23
• @Rajeshwar: Why? – Arturo Magidin Jun 1 '12 at 3:23
• if 3 boys = 2 days then 2 boys complete the fence is 3 day.. But i cant figure out the next part.. – Rajeshwar Jun 1 '12 at 3:24
• @ArturoMagidin is that wrong ? – Rajeshwar Jun 1 '12 at 3:24
• It isn't wrong, but you didn't say why. – MJD Jun 1 '12 at 3:25

Imagine that the fence comprises 12 portions. In one day 6 portions will be completed by 3 boys. Meaning that 1 boy paints 2 portions. 2 boys will paint 4 portions. We need to calculate the part of job completed by 2 boys in one day I.e 4/12 =1/3