perplexus.info :: Just Math : Three problems

Three problems (Posted on 2004-01-16)

In a contest of intelligence, three problems A, B and C were posed.

Among the contestants there were 25 who solved at least one problem each.

Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C.

The number of participants who solved only problem A was one more than the number who solved problem A and at least one other problem.

Of all students who solved just one problem, half did not solve problem A.

How many students solved only problem B?

Submitted by Ravi Raja

Rating: 3.6000 (5 votes)

Solution: (Hide)

6 students solved only problem B
X => Students who solved only problem A
Y => Students who solved only problem B
Z => Students who solved only problem C
P => Students who solved both problem B and problem C
From 4 :
Students who solved only problem A = Students who solved only problem B + Students who solved only problem C
X = Y + Z
From 3 :
Students who solved problem A and at least one other = X - 1
From 2 :
(Y + P) = 2 * (Z + P)
Y + P = 2 * Z + 2 * P
Z = (Y - P) / 2
From 1:
X + X - 1 + Y + Z + P = 25
2*X + Y + Z + P = 26
2*(Y + Z) + Y + Z + P = 26 (from 4)
3*Y + 3*Z + P = 26
3*Y + 3* (Y - P) / 2 + P = 26 (from 2)
6*Y + 3*Y - 3*P + 2*P = 52
9*Y - P = 52
Y = (52 + P) / 9
Now, it is obvious that all values are integer. Hence, P must be 2 and Y must be 6.
So 6 students solved only problem B.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
answer	K Sengupta	2007-12-23 12:07:24
solution	SilverKnight	2004-01-16 10:51:05
solution	Charlie	2004-01-16 10:34:12

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