Expert: Surajit Basu - 9/29/2009

Question
there were 840 pumpkins in a pumpkin patch,but it was difficult for farmer Joe to find the perfect pumpkin.Every third pumpkin was too small,every fourth pumpkin was too green,every fifth pumpkin had a broken stem and every sixth pumpkin had the wrong shape.How many perfect pumpkins did farmer Joe find in the pumpkin patch?.

Answer
I shall solve some similar problems, and leave you to solve the given one.

The problem I shall solve is:
"There were 180 pumpkins in a pumpkin patch,but it was difficult for farmer Joe to find the perfect pumpkin.Every third pumpkin was too small,every fourth pumpkin was too green.How many perfect pumpkins did farmer Joe find in the pumpkin patch?."

There were 180 pumpkins in a pumpkin patch.
Every third pumpkin was too small, every fourth pumpkin was too green.
Imagine that they were in a line.
1 2 3 4 5 6 7 8 9 10 11 12 etc...... 180
Let's mark the small ones with a little S, and the green ones with a little G.
   s g   s   g s       sg
1 2 3 4 5 6 7 8 9 10 11 12 etc...... 180
Notice in the first 12, there are 4 s pumpkins ( 1/3rd of 12)
There are 4 g pumpkins ( 1/4th of 12)
And how many good ones are there: 6 ( 1,2,5,7,10,11)
not 12 - 3 - 4 because the 12th pumpkin is both s and g!
so, 12 - 3 - 4 + 1 (since we counted it twice)

Thus, for the 180 set, we get
No. of s pumpkins is 1/3rd of 180 = 60
No. of g pumpkins is 1/4th of 180 = 45
no. of both s and g pumpkins is 1/3*1/4 (i.e 1/12th) of 180 = 15
So, no of good ones is
Total number of pumpkins - No. of s pumpkins - No. of g pumpkins + No. of both s and g pumpkins (since we counted them twice)
i.e. 180 - 60 - 45 + 15 = 90.


Now try this:
"There were 180 pumpkins in a pumpkin patch,but it was difficult for farmer Joe to find the perfect pumpkin.Every third pumpkin was too small,every fourth pumpkin was too green,every fifth pumpkin had a broken stem.How many perfect pumpkins did farmer Joe find in the pumpkin patch?."

Think. Try.


Imagine that they were in a line.
1 2 3 4 5 6 7 8 9 10 11 12 etc...... 180
Let's mark the small ones with a little S, and the green ones with a little G, the broken stem with a B.
   s g B s   g s  B     sg
1 2 3 4 5 6 7 8 9 10 11 12 etc...... 180

Thus, for the 180 set, we get the ones to exclude:
No. of s pumpkins is 1/3rd of 180 = 60
No. of g pumpkins is 1/4th of 180 = 45
No. of g pumpkins is 1/5th of 180 = 36

Double counted in these, so need to include
No. of both s and g pumpkins is 1/3*1/4 (i.e 1/12th) of 180 = 15
No. of both s and b pumpkins is 1/3*1/5 (i.e 1/15th) of 180 = 12
No. of both b and g pumpkins is 1/4*1/5 (i.e 1/20th) of 180 = 9

But now we have to watch out for all the ones which have all three (sbg!).
We counted them thrice out in the first set, and counted them thrice in the next set (double counted)

So, need to add in all the triple counted ones
No. of s,b, g pumpkins is 1/3*1/4*1/5 (i.e 1/60th) of 180 = 3


So, no of good ones is
Total number of pumpkins - ( No. of s pumpkins - No. of g pumpkins ...)
+ No. of double counted pumpkins (since we counted them twice)
- No. of triple counted pumpkins
i.e. 180 - (60 + 45 + 36) + (15+12+9) - 3
= 180 - 142 + 36 - 3
= 71

got it?
now try the problem?
stuck?
email me at surajit@gmail.com