Out of 250 students who failed in an examination, it was revealed that 128 failed in mathematics,87 failed in physics and 134 failed in chemistry.31 failed in

Question

maths and physics,54 failed in chemistry and mathematics,30 failed in chemistry and physics.find how many candidates failed in: [a]in all the three subjects.[b]in mathematics but not in physics.[c] in chemistry but not in mathematics.[d]in physics but not in chemistry or in mathematics.[e]in chemistry or in mathematics but not in physics.hint;user venn diagram

Answer 1

The hint to draw a Venn diagram is a good one. Since I can't do that here, I will describe my diagram. It is 3 circles that overlap to create 7 regions: Failed math only, failed physics only, failed chemistry only, failed math and chemistry but not physics, failed math and physics but not chemistry, failed physics and chemistry but not math, and failed all three. Let the number that failed all three equal x. We then have the following:

Failed math only = 128 - (31-x) - (54-x) - x = 43 + x

Failed physics only = 87 - (31-x) - (30-x) - x = 26 + x

Failed chemistry only = 134 - (54-x) - (30-x) - x = 50 + x

Failed math and chemistry but not physics = 54 - x

Failed math and physics but not chemistry= 31 - x

Failed physics and chemistry but not math = 30 - x

Failed all three = x

 

The sum of these expressions is 43 + x + 26 + x + 50 + x + 54 - x + 31 - x + 30 - x + x = 234 + x

This expression must equal 250. So 234 + x = 250. Solving for x gives x = 16

 

a) 16

b) 128 - 31 = 97

c) 134 - 54 = 80

d) physics only = 26 + 16 = 42

e) math only + chemistry only + math and chemistry but not physics = 43 + x + 50 + x + 54 - x = 163

Answer 2

I have a Venn diagram with three circles intersecting, two side by side and one underneath in the middle.

I'll try to explain each step as I do it on my paper.

top left is math, top right is physics, bottom is chemistry

there are 7 sections, 3 for only one subject, 3 for two subjects, and the center for failing all three

128 failed math so 128 goes in the math only section

87 failed physics so 87 goes in the physics only section

134 failed chemistry so 134 goes in the chemistry only section

31 failed math and physics, so 31 goes in the math & physics overlap

therefore math only becomes 128-31=97 and physics only becomes 87-31=56

54 failed chemistry and math, so 54 goes in the chemistry & math overlap

therefore math only becomes 97-54=43 and chemistry only becomes 134-54=80

30 failed chemistry and physics, so 30 goes in the chemistry & physics overlap

therefore chemistry only becomes 80-30=50 and physics only becomes 56-30=26

the center is empty because we have not determined how many failed all three subjects yet

let y=# of students failing all three subjects

now add all of our numbers together:

43+31+54+26+30+50+y which must equal 250 students

43+31+54+26+30+50+y=250

234+y=250

y=250-234

y=16 students who failed all three subjects and this number goes in the center of the Venn diagram

if you look at the math and chemistry only section it becomes 54-16=38

if you look at the math and physics section only it becomes 31-16=15

if you look at the physics and chemistry section only it becomes

30-16=14

failing math but not physics is 43+54=97

failing chemistry but not math is 50+30=80

failing physics only is 26

failing in chemistry or math but not physics is 50+54+43=147

 

Answer 3

Just a quick note to add to Ryan's excellent solution:

 

I created Venn diagram for this problem, which can be viewed here:

 

http://i.imgur.com/pNHlzDE.png

 

By the time I finished that, Ryan had already posted his solution, which is briefer than mine was since I took more of a step-by-step approach.