# Math Help - Help with venn diagrams/sets word problem

1. ## Help with venn diagrams/sets word problem

A survey of 225 adults found that during the last year, 45 traveled by plane but not by train, 70 traveled by train but not by plane, 20 traveled by bus but not by plane or by train, 45 traveled by bus and plane, 20 traveled by all three, and 165 traveled by plane or train. How many did not travel by any of these modes of transportation?

I'm a little confused here, I tried mapping it out in a venn diagram but it got pretty messy. The wording is a little confusing. Can someone help me solve this?

2. Originally Posted by zweevu
A survey of 225 adults found that during the last year, 45 traveled by plane but not by train, 70 traveled by train but not by plane, 20 traveled by bus but not by plane or by train, 45 traveled by bus and plane, 20 traveled by all three, and 165 traveled by plane or train. How many did not travel by any of these modes of transportation?

I'm a little confused here, I tried mapping it out in a venn diagram but it got pretty messy. The wording is a little confusing. Can someone help me solve this?
if you draw your Venn diagram with T=train, P=plane, B=bus

$T\cap P\cap B=20$

$B\ less\ (T\cup P)=20$

$(P\cap B)\ less\ T=25$

$P\ less\ (T\cup B)=20$

Label the other 3 subsets X=T only, Y=(T Union B) less P, Z=(P Union T) less B

Then X+Y=70
165=20+20+25+X+Y+Z=65+X+Y+Z

100=X+Y+Z

Then Z=30 and the others follow.

3. First step is to place 20 in the intersection of all three.

Next, 20 travelled by bus, but not by plane or train.
Place 20 in the bus only subset.

Next, 45 travelled by bus and plane is the intersection of "b" and "p".
This consists of two subsets. The first is the intersection of all three,
the second is the intersection of "b" and "p" that excludes "t".

Therefore $(p\cap b)\ without\ t=45-20=25$

Next, 45 travelled by plane but not by train.
This means we can find the number who travelled only by plane,
since we have the numbers who used all 3 modes of travel
and we also have the number who travelled by "bus and plane" only.

Therefore $p\ without\ t=45$

$p\ without\ (t\cup b)=45-25=20$

Four subsets have been filled in.
The others can be labelled X, Y, Z.

In finding Z, use the two clues

$X+Y=70$

$20+20+25+X+Y+Z=165$

Hence, this gives $Z=30$

X and Y do not need to be found, since we have their sum.

Finally,

$225-(X+Y+Z+20+20+20+25)=225-(70+30+20+20+20+25)=225-185=40$