# Venn Diagrams

• Apr 23rd 2010, 05:46 AM
wahhdoe
Venn Diagrams
I have a question with 3 parts, I have done the first 2 parts but i cannot make sense of the last part and was wondering if someone could please help me?

Q: 75 pupils, 48 pupils like ready salted, 35 like cheese & onion

i) if 7 pupils do not like either flavor, how many like both?
A: 15

ii) *what is the greatest possible number of pupils who could like both?
A: 35

**what is the smallest possible number of pupils who could like both?
A: 8

Now this is the part that i cannot figure out:

The number of pupils who like both is twice the number who do not like either;

iii) what fraction of the pupils who like ready salted do not like cheese & onion?
• Apr 23rd 2010, 07:35 AM
Soroban
Hello, wahhdoe!

Quote:

75 pupils: 48 pupils like ready salted, 35 like cheese & onion.

i) If 7 pupils do not like either flavor, how many like both?

iia) What is the greatest possible number of pupils who could like both?

iib) What is the smallest possible number of pupils who could like both?

All correct . . . Good work!

Now this is the part that i cannot figure out:

The number of pupils who like both is twice the number who do not like either;

iii) What fraction of the pupils who like ready salted do not like cheese & onion?

I often use a chart instead of a Venn diagram . . .

. . $\begin{array}{c||c|c||c|}
& \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline
\text{Salted} & & & 48 \\ \hline \sim\text{Salted} & & & 27 \\ \hline\hline
\text{Total} & 35 & 40 & 75\\ \hline \end{array}$

We are told: . $\text{(both)} \;=\;2 \times \text{(neither)}$

. . $\begin{array}{c||c|c||c|}
& \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline
\text{Salted} & {\color{red}2x} & & 48 \\ \hline \sim\text{Salted} & & {\color{red}x} & 27 \\ \hline\hline
\text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Reading across, we can fill in the empty cells:

. . $\begin{array}{c||c|c||c|}
& \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline
\text{Salted} & 2x & {\color{red}48-2x} & 48 \\ \hline \sim\text{Salted} & {\color{red}27-x} & x & 27 \\ \hline\hline
\text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Reading down, we have two equations:

. . $\begin{Bmatrix}2x + (27-x) &=& 35 \\ (48-2x) + x &=& 40 \end{Bmatrix} \quad\Rightarrow\quad x \:=\:8$

The chart becomes:

. . $\begin{array}{c||c|c||c|}
& \text{Cheese} & \sim\text{Cheese} & \text{Total} \\ \hline\hline
\text{Salted} & 16 & 32 & 48 \\ \hline \sim\text{Salted} & 19 & 8 & 27 \\ \hline\hline
\text{Total} & 35 & 40 & 75\\ \hline \end{array}$

Now you can answer the question.