Problem solving using venn diagrams?

Question

in a troop of 64 scouts, 37 had earned First Aid Merit Badge, 26 earned Lifesaving Merit Badge, and 12 had earned both of these badges. How many scouts had earned:

First aid but not lifesaving?

lifesaving but not first aid?

at least one of the badges?

neither of the badges?

Answer 1

To solve this problem using venn diagram :

Draw :

1) A rectangle : it denotes all the 64 irrespective of whether they have  a badge or not .

2) Two circles overlapping each other inside the rectangle:

one of them (say A) represents the scouts who have first aid badge . the second one (say B) represents the ones who have the lifesaving badge

The intersection of the two circles A and B  represents the scouts who have both (so inside the intersection write 12)

______________________________________...

Now 37 scouts who have earned first aid badge includes the 12 who got both . Subtract 12 from 37 and write it in the portion of the circle where it does not intersect with the other circle :

37-12= 25

This gives the number of scouts who have first aid but not lifesaving

this is the answer for the first question.

For this question shade the whole of circle A except the intersection

The same way, in circle B the portion which is not intersected with A

represents the ones with only life saving badges which gives

26 -12 = 14   scouts

This is the answer to the second question

For this shade the whole of circle B except the portion of intersection of A.

Now the third question asks for atleast one of the badges

This means that the scout can have two or one

So it includes the circles A and B

There are 25 students who got only first aid , 14 students who got only life saving and 12 who got both . Add all

25+14+12

=51 scouts have got atleast one badge

For this question shade the two circles completely .

The scouts who have got no badges is 64-51=13

For this shade the rectangle expect the portion of the two circles. This represents the number of the scouts who have not got both the badges

Answer 2

For first aid but not lifesaving, we just need to know all of the people who got first aid badges (37) and subtract all of the people who got lifesaving badges and first aid badges (12), giving us 25.

For lifesaving but not first aid, we just need to know all of the people who got lifesaving badges (26) and subtract all of the people who got lifesaving badges and first aid badges (12), giving us 14.

For at least one of the badges, we can add up all of the people who got either one of the badges (37 + 26 = 63) and subtract all of the people who got both (because we counted them twice).  That's 12, so 63 - 12 = 51, and so 51 people got at least one badge.

For neither of the badges, we take the total number of campers and subtract the ones who got at least one badge, which is 51.  So 64 - 51 = 13, and 13 people got no badges.

Answer 3

First of all, count the number of categories (or "badges" here) when solving a problem with a Venn diagram.  Here, there are two.

By hand, you can draw a large rectangle and two circles inside of it which overlap each other.  

  

Inside of the left part of the left circle, write "F" for first aid.

Inside of the right part of the right circle, write "L" for lifesaving.

In between F and L, where the circles overlap, write "B" for both.

Completely outside of the circles, write "N" for neither.

So, the circle with F and B contains all of the First Aid badges.

The circles with B and L contains all of the Lifesaving badges.

At this point, think "from the inside out".

12 earned both badges.  So, write "12" inside region B.  (I will simply say "B = 12" here in this typing.)

Because 37 earned the First Aid badge, then F + B = 37.  Since B = 12, then F = 25.

Because 26 earned the Lifesaving badge, the B + L = 26.  Since B = 12, then L = 14.

So, F + B + L = 25 + 12 + 14 = 51.

As there are a total of 64 scouts, N = 64 - 51 = 13.

Now, find which regions are in the question and count the total number of scouts in the specific regions.

(For example, "Not both of the badges" would be everything

outside B.  And F + L + N = 25 + 14 + 13 = 52.)